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In mathematics, a '''uniformly bounded representation''' of a [[locally compact group]] on a [[Hilbert space]] is a homomorphism into the bounded invertible operators which is continuous for the [[strong operator topology]]. In 1947 [[Béla Szőkefalvi-Nagy]] established that any uniformly bounded representation of the integers or the real numbers is '''unitarizable''', i.e. conjugate by an invertible operator to a [[unitary representation]]. For the integers this gives a criterion for an invertible operator to be similar to a unitary operator: the [[operator norm]]s of all the positive and negative powers must be uniformly bounded. The result on unitarizability of uniformly bounded representations was extended in 1950 by [[Jacques Dixmier|Dixmier]], Day and Nakamura-Takeda to all locally compact [[amenable group]]s, following essentially the method of proof of Sz-Nagy. The result is known to fail for non-amenable groups such as  SL(2,'''R''') and the free group on two generators. {{harvtxt|Dixmier|1950}} conjectured that a locally compact group is amenable if and only if every uniformly bounded representation is unitarizable. 
==Statement==
Let ''G'' be a locally compact [[amenable group]] and let ''T''<sub>''g''</sub> be a homomorphism of ''G'' into ''GL''(''H''), the group of a invertible operators on a Hilbert space such that
 
*for every ''x'' in ''H'' the vector-valued ''gx'' on ''G'' is continuous;
*the operator norms of the operators ''T''<sub>''g''</sub> are uniformly bounded.
 
Then there is a positive invertible operator ''S'' on ''H'' such that ''S'' ''T''<sub>''g''</sub> ''S''<sup>–1</sup> is unitary for every ''g'' in ''G''.
 
As a consequence, if ''T'' is an invertible operator with all its positive and negative powers unformly bounded in operator norm, then ''T'' is conjugate by a positive invertible operator to a unitary.
==Proof==
By assumption the continuous functions
 
:<math>\displaystyle{f_{x,y}(g)=(T_g^{-1}x,T_g^{-1}y),}</math>
 
generate a separable unital C* subalgebra ''A'' of the uniformly bounded continuous functions on ''G''. By construction the algebra is invariant under left translation. By amenability there is an invariant state φ on ''A''. It follows that
 
:<math>\displaystyle{(x,y)_0 =\varphi(f_{x,y})}</math>
 
is a new inner product on ''H'' satisfying
 
:<math>\displaystyle{M^{-1}\|x\| \le \|x\|_0 \le M\|x\|}</math>
 
where
 
:<math>\displaystyle{M=\sup_g \|T_g\| < \infty.}</math>
 
So there is a positive invertible operator ''P'' such that
 
:<math>\displaystyle{(x,y)_0 =(Px,y).}</math>
 
By construction
 
:<math>\displaystyle{(T_gx,T_gy)_0=(x,y)_0.}</math>
 
Let ''S'' be the unique positive square root of ''P''. Then
 
:<math>\displaystyle{(ST_g x,ST_gy)=(PT_gx,T_gy)=(Px,y) =(Sx,Sy).}</math>
 
Applying ''S''<sup>–1</sup> to ''x'' and ''y'', it follows that
 
:<math>\displaystyle{(ST_gS^{-1} x, ST_gS^{-1} y)=(x,y).}</math>
 
Since the operators
 
:<math>\displaystyle{U_g=ST_gS^{-1}}</math>
are invertible, it follows that they are unitary.
 
==Examples of non-unitarizable representations==
===SL(2,R)===
The [[complementary series representation|complentary series]] of irreducible unitary representations of SL(2,R) was introduced by {{harvtxt|Bargmann|1947}}. These representations can be realized on functions on the circle or on the real line: the Cayley transform provides the unitary equivalence between the two realizations.<ref>{{harvnb|Sugiura|1980|391-393}}</ref>
 
In fact for 0 < σ < 1/2 and ''f'', ''g'' continuous functions on the circle define
 
:<math>\displaystyle{(f,g)_\sigma={1\over 4\pi^2} \int_{-\pi}^\pi\int_{-\pi}^\pi f(s)\overline{g(t)} k_\sigma(s-t) \,ds\, dt,}</math>
 
where
 
:<math>\displaystyle{k_\sigma(s)=(1-\cos s)^{\sigma-1/2}.}</math>
 
Since the function ''k''<sub>σ</sub> is integrable, this integral converges. In fact
 
:<math>\displaystyle{(f,g)_\sigma \le \|f\|\cdot \|g\|,}</math>
 
where the norms are the usual L<sup>2</sup> norms.
 
The functions
 
:<math>\displaystyle{f_m(t)=e^{imt}}</math>
 
are orthogonal with
 
:<math>\displaystyle{(f_m,f_m)_\sigma=\prod_{i=1}^{|m|} {i-1/2-\sigma\over i-1/2+\sigma}= {\Gamma(1/2 +\sigma)\Gamma(|m|+1/2-\sigma)\over \Gamma(1/2-\sigma)\Gamma(m+1/2+\sigma)}.}</math>
 
Since these quantities are positive, (''f'',''g'')<sub>σ</sub> defines an inner product. The Hilbert space completion is denoted by ''H''<sub>σ</sub>.
 
For ''F'', ''G'' continuous functions of compact support on '''R''', define
 
:<math>\displaystyle{(F,G)_\sigma^\prime=\int_{-\infty}^\infty\int_{-\infty}^\infty F(x)\overline{G(y)} |x-y|^{2\sigma-1}\,dx\, dy.}</math>
 
Since, regarded as distributions, the Fourier transform of |''x''|<sup>2σ – 1</sup> is C<sub>σ</sub>|''t''|<sup>–2σ</sup> for some positive constant C<sub>σ</sub>, the above expression can be rewritten:
 
:<math>\displaystyle{(F,G)_\sigma^\prime=C_\sigma\int_{-\infty}^\infty \widehat{F}(t)\overline{\widehat{G}(t)} |t|^{-2\sigma}\, dt.}</math>
 
Hence it is an inner product. Let ''H'''<sub>σ</sub> denote its Hilbert space completion.
The Cayley transform gives rise to an operator ''U'':
 
:<math>\displaystyle{Uf(x)=2^{\sigma/2 - 3/4} \pi^{-1} |x+i|^{1-2\sigma} f\left({x-i\over x+i}\right).}</math>
 
''U'' extends to a isometry of ''H''<sub>σ</sub> onto ''H'' '<sub>σ</sub>. Its adjoint is given by
 
:<math>\displaystyle{U^*F(e^{it})=2^{3/4-\sigma/2} \pi |1- e^{it}|^{1-2\sigma} F\left({1+e^{it}\over 1-e^{it}}\right).}</math>  
 
The Cayley transform exchanges the actions by [[Möbius transformation]]s of SU(1,1) on '''S'''<sup>1</sup> and of SL(2, '''R''') on '''R'''.  
 
The operator ''U'' interwtines corresponding actions of SU(1,1) on ''H''<sub>σ</sub> and SL(2,'''R''') on ''H'' '<sub>σ</sub>.
 
For ''g'' in SU(1,1) given by
 
:<math>\displaystyle{g=\begin{pmatrix} \alpha & \beta \\ \overline{\beta} & \overline{\alpha}\end{pmatrix},}</math>
 
with
 
:<math>\displaystyle{|\alpha|^2 - |\beta|^2=1,}</math>
 
and ''f'' continuous, set
 
:<math>\displaystyle{\pi_\sigma(g^{-1}) f(z)=|\overline{\beta}z+\overline{\alpha}|^{1-2\sigma}f\left({\alpha z +\beta\over\overline{\beta}z +\overline{\alpha}}\right).}</math>
 
For ''g''' in SL(2,'''R''') given by
 
:<math>\displaystyle{g^\prime=\begin{pmatrix} a & b \\ c & d \end{pmatrix},}</math>
 
with ''ad'' – ''bc'' = 1, set
 
:<math>\displaystyle{\pi^\prime_\sigma((g^\prime)^{-1}) F(x) =|cx+d|^{1-2\sigma} F\left({ax+b\over cx +d}\right).}</math>
 
If ''g'' ' corresponds to ''g'' under the Cayley transform then
 
:<math>\displaystyle{U\pi_\sigma(g)U^* =\pi^\prime_\sigma(g^\prime).}</math>
 
Polar decomposition shows that SL(2,R) = ''KAK'' with ''K'' = SO(2) and ''A'' the subgroup of positive diagonal matrices. ''K'' corresponds to the diagonal matrices in SU(1,1). Since evidently ''K'' acts unitarily on ''H''<sub>σ</sub> and ''A'' acts unitarily on ''H'' '<sub>σ</sub>, both representations are unitary. The representations are irreducible because the action of the Lie algebra on the basis vectors ''f''<sub>m</sub> is irreducible. This family of irreducible unitary representations is called the '''complementary series'''.
 
{{harvtxt|Ehrenpreis|Mautner|1955}} constructed an analytic continuation of this family of representations as follows.<ref>{{harvnb|Lohoué|1980}}</ref> If ''s'' = σ + iτ, g lies in SU(1,1) and ''f'' in ''H''<sub>σ</sub>, define
 
:<math>\displaystyle{\pi_s(g^{-1}) f(z)=|\overline{\beta}z+\overline{\alpha}|^{1-2s}f\left({\alpha z +\beta\over\overline{\beta}z +\overline{\alpha}}\right).}</math>
 
Similarly if ''g'' ' lies in SL(2,'''R''') and ''F'' in  ''H'' '<sub>σ</sub>, define
 
:<math>\displaystyle{\pi^\prime_s((g^\prime)^{-1}) F(x) =|cx+d|^{1-2s} F\left({ax+b\over cx +d}\right).}</math>
 
As before the unitary ''U'' intertwines these two actions. ''K'' acts unitarily on ''H''<sub>σ</sub> and ''A'' by a uniformly bounded representation on ''H'' '<sub>σ</sub>. The action of the standard basis of the complexification Lie algebra on this basis can be computed:<ref>{{harvnb|Bargmann|1947|p=613}}</ref>
 
:<math>\displaystyle{\pi_s(L_0)f_m=mf_m, \,\, \pi_s(L_{-1})f_m=-(m+1/2+s)f_{m+1},\,\, \pi_s(L_1)f_m=-(m-1/2-s)f_{m-1}.}</math>
 
If the representation were unitarizable for τ ≠ 0, then the similarity operator ''T'' on ''H''<sub>σ</sub> would have to commute with ''K'', since ''K'' preserves the original inner product. The vectors ''Tf''<sub>''m''</sub> would therefore still be orthogonal for the new inner product and 
the operators
 
:<math>\displaystyle{L_i^\prime=TL_iT^{-1}}</math>
 
would satisfy the same relations for
 
:<math>\displaystyle{f_m^\prime = Tf_m=\lambda_m f_m.}</math>
 
In this case
 
:<math> \displaystyle{[L_m^\prime,L_n^\prime]=(m-n)L_{m+n}^\prime,\,\,(L_i^\prime)^*=L_{-i}^\prime.}</math>
 
It is elementary to verify that infinitesimally such a representation cannot exist if τ ≠ 0.<ref>See:
*{{harvnb|Bargmann|1947}}
*{{harvnb|Howe|Tan|1992}}
*{{harvnb|Lang|1985|p=122-123}}</ref>
 
Indeed let ''v''<sub>0</sub> = ''f'' '<sub>0</sub> and set
 
:<math>\displaystyle{v_1=L^\prime_{-1}v_0.}</math>
 
Then
 
:<math>\displaystyle{L^\prime_1 v_1=c v_{0}}</math>
 
for some constant ''c''. On the other hand
 
:<math>\displaystyle{\|v_{1}\|^2=(L_{-1}^\prime v_0,v_{1})=(v_0,L_1^\prime v_{1})=\overline{c}\|v_0\|^2.}</math>
 
Thus ''c'' must be real and positive. The formulas above show that  
 
:<math>\displaystyle{c={1\over 4}-s^2={1\over 4}-\sigma^2 +\tau^2 -2i\sigma\tau,}</math>
 
so the representation π<sub>''s''</sub> is unitarizable only if τ = 0.
 
===Free group on two generators===
The group ''G'' = SL(2,'''R''') contains the discrete group Γ = SL(2,'''Z''') as a closed subgroup of finite covolume, since this subgroup acts on the upper half plane with a fundamental domain of finite hyperbolic area.<ref>See:
*{{harvnb|Serre|1977}}
*{{harvnb|Gelfand|Graev|Pyatetskii-Shapiro|1969}}</ref> The group SL(2,'''Z''') contains a subgroup of index 12 isomorphic to '''F'''<sub>2</sub> the free group on two generators.<ref>See:
*{{harvnb|Magnus|Karrass|Solitar|1976}}
*{{harvnb|Serre|1983}}</ref> Hence ''G'' has a subgroup Γ<sub>1</sub> of finite covolume, isomorphic to '''F'''<sub>2</sub>. If ''L'' is a closed subgroup of finite covolume in a locally compact group ''G'', and π is non-unitarizable uniformly bounded representation of ''G'' on a Hilbert space ''L'', then its restriction to ''L'' is uniformly bounded and non-unitarizable. For if not, applying a bounded invertible operator, the inner product can be made invariant under ''L''; and then in turn invariant under ''G'' by redefining
 
:<math>\displaystyle{(x,y)_1=\int_{H\backslash G} (gx,gy) \, dg.}</math>
 
As in the previous proof, uniform boundedess guarantees that the norm defined by this inner product is
equivalent to the original inner product. But then the original representation would be unitarizable on ''G'', a contradiction. The same argument works for any discrete subgroup of ''G'' of finite covolume. In particular the [[surface group]]s, which are cocompact subgroups, have uniformly bounded representations that are not unitarizable.
 
There are more direct constructions of uniformly bounded representations of free groups that are non-unitarizable: these are surveyed in {{harvtxt|Pisier|2001}}. The first such examples are described in  
{{harvtxt|Figà-Talamanca|Picardello|1983}}, where an analogue of the complementary series is constructed.  
 
Later {{harvtxt|Szwarc|1988}} gave a related but simpler construction, on the Hilbert space ''H'' = <math>\ell</math><sup>2</sup>('''F'''<sub>2</sub>), of a holomorphic family of uniformly bounded representations π<sub>''z''</sub> of '''F'''<sub>2</sub> for  |z| < 1; these are non-unitarizable when  1/√3 < |''z''| < 1 and ''z'' is not real. Let ''L''(''g'') denote the reduced word length on '''F'''<sub>2</sub> for a given set of generators ''a'', ''b''. Let ''T'' be the bounded operator defined on basis elements by
 
:<math>\displaystyle{Te_1=0,\,\, Te_g=e_{g^\prime},}</math>
 
where ''g'' ' is obtained by erasing the last letter in the expression of ''g'' as a reduced word; identifying ''F''<sub>2</sub> with the vertices of its [[Cayley graph]], a rooted tree,<ref>{{harvnb|Serre|1983}}</ref> this corresponds to passing from a vertex to the next closest vertex to the origin or root. For |z| < 1
 
:<math>\displaystyle{\pi_z(g) = (I-zT)^{-1}\lambda(g)(I-zT)}</math>
 
is well-defined on finitely supported functions. {{harvtxt|Pytlik|Szwarc|1986}} had earlier proved that it extends to a uniformly bounded representation on ''H'' satisfying
 
:<math>\displaystyle{\|\pi_z(g)\|\le {1+|z|\over 1-|z|}.}</math>
 
In fact it is easy to check that the operator
λ(''g'')''T''λ(''g'')<sup>–1</sup> – ''T'' has finite rank, with range''V''<sub>''g''</sub>, the finite-dimensional space of functions supported on the set of vertices joining ''g'' to the origin. For on any function vanishing on this finite set, ''T'' and λ(''g'')''T''λ(''g'')<sup>–1</sup> are equal; and they both leave invariant ''V''<sub>''g''</sub>, on which they acts as contractions and adjoints of each other. Hence if ''f'' has finite support and norm 1,
 
:<math>\displaystyle{\|\pi_z(g)f\|=\|\lambda(g)f+\sum_{n=0}^\infty z^{n+1} T^n[T,\lambda(g)]f\|\le 1 + 2 \sum_{n=0}^n |z|^{n+1} ={1+|z|\over 1-|z|}.}</math>
 
For |z| < 1/√3, these representations are all similar to the regular representation λ. If on the other hand 1/√3 < |z| <1, then the operator
 
:<math>\displaystyle{D=\pi_z(a)+\pi_z(a^{-1}) + \pi_z(b) +\pi_z(b^{-1})}</math>
 
satisfies
 
:<math> \displaystyle{Df=(3z +z^{-1})f}</math>
 
where ''f'' in ''H'' is defined by
 
:<math>\displaystyle{f(1)=1,\,\, f(g)={3\over 4} (3z)^{-L(g)} \,\,(g\ne 1).}</math>
 
Thus, if ''z'' is not real, ''D'' has an eigenvalue which is not real. But then π<sub>''z''</sub> cannot be unitarizable, since otherwise ''D'' would be similar to a self-adjoint operator.
 
==The Dixmier Problem==
[[Jacques Dixmier]] asked in 1950 whether amenable groups are characterized by '''unitarizability''', i.e. the property that all their uniformly bounded representations are unitarizable.  This problem remains open to this day.
 
An elementary [[induced representation|induction]] argument shows that a subgroup of a unitarizable group remains unitarizable.  Therefore, the [[von Neumann conjecture]] would have implied a positive answer to Dixmier's problem, had it been true.  In any case, it follows that a counter-example to Dixmier's conjecture could only be a non-amenable group without free subgroups. In particular, Dixmier's conjecture is true for all [[linear group]]s by the [[Tits alternative]].
 
Considerable progress has been made by [[Gilles Pisier|Pisier]] who linked unitarizability to a notion of factorization length. This allowed him to solve a modified form of the Dixmier problem.
 
The potential gap between unitarizability and amenability can be further illustrated by the following open problems, all of which become elementary if "unitarizable" were replaced by "amenable":
 
*Is the [[direct product of groups|direct product]] of two unitarizable groups unitarizable?
 
*Is a directed union of unitarizable groups unitarizable?
 
*If <math>G</math> contains a normal amenable subgroup <math>N</math> such <math>G/N</math> is unitarizable, does it follow that <math>G</math> is unitarizable? (It is elementary that <math>G</math> is unitarizable if <math>N</math> is so and <math>G/N</math> is amenable.)
 
==Notes==
{{reflist|2}}
 
==References==
*{{citation|last=Sz-Nagy|first=Béla|title=On uniformly bounded linear transformations in Hilbert space|journal=Acta Univ. Szeged. Sect. Sci. Math.|volume= 11|year=1947|pages= 152–157
|url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=5883&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=744baf4652e1c219&style=}}
*{{citation|last=Dixmier|first= Jacques|title= Les moyennes invariantes dans les semi-groupes et leurs applications|journal= Acta Sci. Math. Szeged|volume= 12|year=1950|pages= 213–227
|url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=4759&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=7c2f7e4e2a4b554f&style=}}
*{{citation|last=Day|first=Mahlon M.|title=Means for the bounded functions and ergodicity of the bounded representations of semi-groups|journal=
Trans. Amer. Math. Soc.|volume= 69|year=1950|pages= 276–291|url=http://www.jstor.org/stable/1990358?origin=crossref}}
*{{citation|last=Nakamura|first=Masahiro|last2=Takeda|first2=Ziro|title=Group representation and Banach limit|journal=[[Tôhoku Mathematical Journal]]|volume= 3|year=1951|pages= 132–135|url=http://dx.doi.org/10.2748/tmj/1178245513}}
*{{citation|last=Pisier|first=Gilles|authorlink=Gilles Pisier|title=Similarity Problems and Completely Bounded Maps|volume= 1618
|series=Lecture Notes in Mathematics|edition=2nd|publisher=Springer|year= 2001|id=ISBN 3540415246}}
*{{citation|last=Pisier|first=Gilles|authorlink=Gilles Pisier|series=Progress in Mathematics|year= 2005|volume= 248|pages= 323-362|title=Are Unitarizable Groups Amenable?|url=http://arxiv.org/abs/math/0405282}}
*{{citation|last=Ehrenpreis|first= L.|last2= Mautner|first2= F. I.|
title=Uniformly bounded representations of groups |journal=Proc. Nat. Acad. Sci. U. S. A.|volume= 41|year=1955|pages= 231–233}}
*{{citation|last=Lohoué|first= N.|title= Estimations L<sup>p</sup> des coefficients de représentation et opérateurs de convolution|journal= Adv. in Math. |volume=38 |year=1980|pages= 178–221}}
*{{citation|last=Bargmann|first= V.|title= Irreducible unitary representations of the Lorentz group|journal= Ann. of Math.|volume= 48|year=1947|pages=568–640}}
*{{citation|title=Unitary Representations and Harmonic Analysis: An Introduction|volume=44|series=North-Holland Mathematical Library|first=Mitsuo |last=Sugiura|edition=2nd|publisher=Elsevier|year= 1990|id=ISBN 0444885935}}
*{{citation|title=Non-abelian Harmonic Analysis: Applications of SL(2,'''R''')|first=Roger|last= Howe|first2= Eng-chye|last2= Tan|
year=1992|publisher=Springer-Verlag|series=Universitext|ISBN= 0-387-97768-6}}
*{{citation|first=Serge|last=Lang|authorlink=Serge Lang|title=SL(2,'''R''')|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|volume=105|
year=1985|ISBN= 0-387-96198-4}}
*{{citation|last=Serre|first= Jean-Pierre|title=Cours d'arithmétique|edition=2nd|series= Le Mathématicien|volume= 2|publisher= Presses Universitaires de France|year= 1977}}
*{{citation|last=Gelfand|first= I. M.|last2=Graev|first2= M. I.|last3= Pyatetskii-Shapiro|first3= I. I.|title=Representation theory and automorphic functions|publisher= Academic Press|year=1969|id=ISBN 0-12-279506-7}}
*{{citation|last=Serre|first= Jean-Pierre|title=Arbres, amalgames, SL<sub>2</sub>|series= Astérisque|volume=46|publisher=Société Mathématique de France|year= 1977}}
*{{citation|last=Magnus|first= Wilhelm|last2= Karrass|first2= Abraham|last3= Solitar|first3= Donald |title=Combinatorial group theory. Presentations of groups in terms of generators and relations|edition=2nd|publisher= Dover Publications|year= 1976|id=ISBN 0-486-43830-9}}
*{{citation|last=Figà-Talamanca|first=Alessandro|last2= Picardello|first2= Massimo A.|title=Harmonic analysis on free groups|series= Lecture Notes in Pure and Applied Mathematics|volume= 87|publisher= Marcel Dekker|year= 1983}}
*{{citation|last=Pytlik|first= T.|last2= Szwarc|first2= R.|title= An analytic family of uniformly bounded representations of free groups|journal= Acta Math.|volume= 157 |year=1986|pages=287–309|
url=http://www.springerlink.com/content/kn3277r1t2548646/?MUD=MP}}
*{{citation|last=Szwarc|first=Ryszard|title=
An analytic series of irreducible representations of the free group|journal=
Ann. Inst. Fourier |volume= 38 |year=1988|pages= 87–110}}
[[Category:Operator theory]]
[[Category:Functional analysis]]
[[Category:Representation theory]]

Latest revision as of 23:44, 19 August 2013

In mathematics, a uniformly bounded representation of a locally compact group on a Hilbert space is a homomorphism into the bounded invertible operators which is continuous for the strong operator topology. In 1947 Béla Szőkefalvi-Nagy established that any uniformly bounded representation of the integers or the real numbers is unitarizable, i.e. conjugate by an invertible operator to a unitary representation. For the integers this gives a criterion for an invertible operator to be similar to a unitary operator: the operator norms of all the positive and negative powers must be uniformly bounded. The result on unitarizability of uniformly bounded representations was extended in 1950 by Dixmier, Day and Nakamura-Takeda to all locally compact amenable groups, following essentially the method of proof of Sz-Nagy. The result is known to fail for non-amenable groups such as SL(2,R) and the free group on two generators. Template:Harvtxt conjectured that a locally compact group is amenable if and only if every uniformly bounded representation is unitarizable.

Statement

Let G be a locally compact amenable group and let Tg be a homomorphism of G into GL(H), the group of a invertible operators on a Hilbert space such that

  • for every x in H the vector-valued gx on G is continuous;
  • the operator norms of the operators Tg are uniformly bounded.

Then there is a positive invertible operator S on H such that S Tg S–1 is unitary for every g in G.

As a consequence, if T is an invertible operator with all its positive and negative powers unformly bounded in operator norm, then T is conjugate by a positive invertible operator to a unitary.

Proof

By assumption the continuous functions

fx,y(g)=(Tg1x,Tg1y),

generate a separable unital C* subalgebra A of the uniformly bounded continuous functions on G. By construction the algebra is invariant under left translation. By amenability there is an invariant state φ on A. It follows that

(x,y)0=φ(fx,y)

is a new inner product on H satisfying

M1xx0Mx

where

M=supgTg<.

So there is a positive invertible operator P such that

(x,y)0=(Px,y).

By construction

(Tgx,Tgy)0=(x,y)0.

Let S be the unique positive square root of P. Then

(STgx,STgy)=(PTgx,Tgy)=(Px,y)=(Sx,Sy).

Applying S–1 to x and y, it follows that

(STgS1x,STgS1y)=(x,y).

Since the operators

Ug=STgS1

are invertible, it follows that they are unitary.

Examples of non-unitarizable representations

SL(2,R)

The complentary series of irreducible unitary representations of SL(2,R) was introduced by Template:Harvtxt. These representations can be realized on functions on the circle or on the real line: the Cayley transform provides the unitary equivalence between the two realizations.[1]

In fact for 0 < σ < 1/2 and f, g continuous functions on the circle define

(f,g)σ=14π2ππππf(s)g(t)kσ(st)dsdt,

where

kσ(s)=(1coss)σ1/2.

Since the function kσ is integrable, this integral converges. In fact

(f,g)σfg,

where the norms are the usual L2 norms.

The functions

fm(t)=eimt

are orthogonal with

(fm,fm)σ=i=1|m|i1/2σi1/2+σ=Γ(1/2+σ)Γ(|m|+1/2σ)Γ(1/2σ)Γ(m+1/2+σ).

Since these quantities are positive, (f,g)σ defines an inner product. The Hilbert space completion is denoted by Hσ.

For F, G continuous functions of compact support on R, define

(F,G)σ=F(x)G(y)|xy|2σ1dxdy.

Since, regarded as distributions, the Fourier transform of |x|2σ – 1 is Cσ|t|–2σ for some positive constant Cσ, the above expression can be rewritten:

(F,G)σ=CσF^(t)G^(t)|t|2σdt.

Hence it is an inner product. Let H'σ denote its Hilbert space completion.

The Cayley transform gives rise to an operator U:

Uf(x)=2σ/23/4π1|x+i|12σf(xix+i).

U extends to a isometry of Hσ onto H 'σ. Its adjoint is given by

U*F(eit)=23/4σ/2π|1eit|12σF(1+eit1eit).

The Cayley transform exchanges the actions by Möbius transformations of SU(1,1) on S1 and of SL(2, R) on R.

The operator U interwtines corresponding actions of SU(1,1) on Hσ and SL(2,R) on H 'σ.

For g in SU(1,1) given by

g=(αββα),

with

|α|2|β|2=1,

and f continuous, set

πσ(g1)f(z)=|βz+α|12σf(αz+ββz+α).

For g' in SL(2,R) given by

g=(abcd),

with adbc = 1, set

πσ((g)1)F(x)=|cx+d|12σF(ax+bcx+d).

If g ' corresponds to g under the Cayley transform then

Uπσ(g)U*=πσ(g).

Polar decomposition shows that SL(2,R) = KAK with K = SO(2) and A the subgroup of positive diagonal matrices. K corresponds to the diagonal matrices in SU(1,1). Since evidently K acts unitarily on Hσ and A acts unitarily on H 'σ, both representations are unitary. The representations are irreducible because the action of the Lie algebra on the basis vectors fm is irreducible. This family of irreducible unitary representations is called the complementary series.

Template:Harvtxt constructed an analytic continuation of this family of representations as follows.[2] If s = σ + iτ, g lies in SU(1,1) and f in Hσ, define

πs(g1)f(z)=|βz+α|12sf(αz+ββz+α).

Similarly if g ' lies in SL(2,R) and F in H 'σ, define

πs((g)1)F(x)=|cx+d|12sF(ax+bcx+d).

As before the unitary U intertwines these two actions. K acts unitarily on Hσ and A by a uniformly bounded representation on H 'σ. The action of the standard basis of the complexification Lie algebra on this basis can be computed:[3]

πs(L0)fm=mfm,πs(L1)fm=(m+1/2+s)fm+1,πs(L1)fm=(m1/2s)fm1.

If the representation were unitarizable for τ ≠ 0, then the similarity operator T on Hσ would have to commute with K, since K preserves the original inner product. The vectors Tfm would therefore still be orthogonal for the new inner product and the operators

Li=TLiT1

would satisfy the same relations for

fm=Tfm=λmfm.

In this case

[Lm,Ln]=(mn)Lm+n,(Li)*=Li.

It is elementary to verify that infinitesimally such a representation cannot exist if τ ≠ 0.[4]

Indeed let v0 = f '0 and set

v1=L1v0.

Then

L1v1=cv0

for some constant c. On the other hand

v12=(L1v0,v1)=(v0,L1v1)=cv02.

Thus c must be real and positive. The formulas above show that

c=14s2=14σ2+τ22iστ,

so the representation πs is unitarizable only if τ = 0.

Free group on two generators

The group G = SL(2,R) contains the discrete group Γ = SL(2,Z) as a closed subgroup of finite covolume, since this subgroup acts on the upper half plane with a fundamental domain of finite hyperbolic area.[5] The group SL(2,Z) contains a subgroup of index 12 isomorphic to F2 the free group on two generators.[6] Hence G has a subgroup Γ1 of finite covolume, isomorphic to F2. If L is a closed subgroup of finite covolume in a locally compact group G, and π is non-unitarizable uniformly bounded representation of G on a Hilbert space L, then its restriction to L is uniformly bounded and non-unitarizable. For if not, applying a bounded invertible operator, the inner product can be made invariant under L; and then in turn invariant under G by redefining

(x,y)1=HG(gx,gy)dg.

As in the previous proof, uniform boundedess guarantees that the norm defined by this inner product is equivalent to the original inner product. But then the original representation would be unitarizable on G, a contradiction. The same argument works for any discrete subgroup of G of finite covolume. In particular the surface groups, which are cocompact subgroups, have uniformly bounded representations that are not unitarizable.

There are more direct constructions of uniformly bounded representations of free groups that are non-unitarizable: these are surveyed in Template:Harvtxt. The first such examples are described in Template:Harvtxt, where an analogue of the complementary series is constructed.

Later Template:Harvtxt gave a related but simpler construction, on the Hilbert space H = 2(F2), of a holomorphic family of uniformly bounded representations πz of F2 for |z| < 1; these are non-unitarizable when 1/√3 < |z| < 1 and z is not real. Let L(g) denote the reduced word length on F2 for a given set of generators a, b. Let T be the bounded operator defined on basis elements by

Te1=0,Teg=eg,

where g ' is obtained by erasing the last letter in the expression of g as a reduced word; identifying F2 with the vertices of its Cayley graph, a rooted tree,[7] this corresponds to passing from a vertex to the next closest vertex to the origin or root. For |z| < 1

πz(g)=(IzT)1λ(g)(IzT)

is well-defined on finitely supported functions. Template:Harvtxt had earlier proved that it extends to a uniformly bounded representation on H satisfying

πz(g)1+|z|1|z|.

In fact it is easy to check that the operator λ(g)Tλ(g)–1T has finite rank, with rangeVg, the finite-dimensional space of functions supported on the set of vertices joining g to the origin. For on any function vanishing on this finite set, T and λ(g)Tλ(g)–1 are equal; and they both leave invariant Vg, on which they acts as contractions and adjoints of each other. Hence if f has finite support and norm 1,

πz(g)f=λ(g)f+n=0zn+1Tn[T,λ(g)]f1+2n=0n|z|n+1=1+|z|1|z|.

For |z| < 1/√3, these representations are all similar to the regular representation λ. If on the other hand 1/√3 < |z| <1, then the operator

D=πz(a)+πz(a1)+πz(b)+πz(b1)

satisfies

Df=(3z+z1)f

where f in H is defined by

f(1)=1,f(g)=34(3z)L(g)(g1).

Thus, if z is not real, D has an eigenvalue which is not real. But then πz cannot be unitarizable, since otherwise D would be similar to a self-adjoint operator.

The Dixmier Problem

Jacques Dixmier asked in 1950 whether amenable groups are characterized by unitarizability, i.e. the property that all their uniformly bounded representations are unitarizable. This problem remains open to this day.

An elementary induction argument shows that a subgroup of a unitarizable group remains unitarizable. Therefore, the von Neumann conjecture would have implied a positive answer to Dixmier's problem, had it been true. In any case, it follows that a counter-example to Dixmier's conjecture could only be a non-amenable group without free subgroups. In particular, Dixmier's conjecture is true for all linear groups by the Tits alternative.

Considerable progress has been made by Pisier who linked unitarizability to a notion of factorization length. This allowed him to solve a modified form of the Dixmier problem.

The potential gap between unitarizability and amenability can be further illustrated by the following open problems, all of which become elementary if "unitarizable" were replaced by "amenable":

  • Is a directed union of unitarizable groups unitarizable?
  • If G contains a normal amenable subgroup N such G/N is unitarizable, does it follow that G is unitarizable? (It is elementary that G is unitarizable if N is so and G/N is amenable.)

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010