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In mathematics, the '''Dixmier trace''', introduced by {{harvs |txt |authorlink=Jacques Dixmier |first=Jacques |last=Dixmier |year=1966}}, is a non-normal trace on a space of [[linear operator]]s on a [[Hilbert space]] larger than the space of [[trace class operator]]s.  Dixmier traces are examples of [[singular trace]]s.
As a company owner or as an entrepreneur, you will<br><br>


Some applications of Dixmier traces to [[noncommutative geometry]] are described in {{harv|Connes|1994}}.
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==Definition==
If ''H'' is a Hilbert space, then ''L''<sup>1,∞</sup>(''H'') is the space of compact linear operators ''T'' on ''H'' such that the norm
 
:<math>\|T\|_{1,\infty} = \sup_N\frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}</math>
 
is finite, where the numbers ''&mu;''<sub>''i''</sub>(''T'') are the eigenvalues of |''T''| arranged in decreasing order. Let
:<math>a_N = \frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}</math>.
The Dixmier trace Tr<sub>''&omega;''</sub>(''T'') of ''T'' is defined for positive operators ''T'' of ''L''<sup>1,∞</sup>(''H'') to be
 
:<math>\operatorname{Tr}_\omega(T)= \lim_\omega a_N</math>
 
where lim<sub>''&omega;''</sub> is a scale-invariant positive "extension" of the usual limit, to all bounded sequences.  In other words, it has the following properties:
*lim<sub>''&omega;''</sub>(''&alpha;''<sub>''n''</sub>) ≥ 0 if all ''&alpha;''<sub>''n''</sub> ≥ 0 (positivity)
*lim<sub>''&omega;''</sub>(''&alpha;''<sub>''n''</sub>) = lim(''&alpha;''<sub>''n''</sub>) whenever the ordinary limit exists
*lim<sub>''&omega;''</sub>(''&alpha;''<sub>1</sub>, ''&alpha;''<sub>1</sub>, ''&alpha;''<sub>2</sub>, ''&alpha;''<sub>2</sub>, ''&alpha;''<sub>3</sub>, ...) = lim<sub>ω</sub>(''&alpha;''<sub>''n''</sub>) (scale invariance)
 
There are many such extensions (such as a [[Banach limit]] of  ''&alpha;''<sub>1</sub>, ''&alpha;''<sub>2</sub>, ''&alpha;''<sub>4</sub>, ''&alpha;''<sub>8</sub>,...) so there are many different Dixmier traces.
As the Dixmier trace is linear, it extends by linearity to all operators of ''L''<sup>1,∞</sup>(''H'').
If the Dixmier trace of an operator is independent of the choice of lim<sub>''&omega;''</sub> then the operator is called '''measurable'''.
 
==Properties==
*Tr<sub>''&omega;''</sub>(''T'') is linear in ''T''.
*If ''T'' ≥ 0 then Tr<sub>''&omega;''</sub>(''T'') ≥ 0
*If ''S'' is bounded then Tr<sub>ω</sub>(''ST'') = Tr<sub>''&omega;''</sub>(''TS'')
*Tr<sub>ω</sub>(''T'') does not depend on the choice of inner product on ''H''.
*Tr<sub>''&omega;''</sub>(''T'') = 0 for all trace class operators ''T'', but there are compact operators for which it is equal to 1.
 
A trace ''&phi;'' is called '''normal''' if ''&phi;''(sup ''x''<sub>α</sub>) = sup&nbsp;''&phi;''( ''x''<sub>''&alpha;''</sub>) for every bounded increasing directed family of positive operators.  Any normal trace on <math>L^{1,\infty}(H)</math> is equal to the usual trace, so the Dixmier trace is an example of a non-normal trace.
 
==Examples==
A compact self-adjoint operator with eigenvalues 1, 1/2, 1/3, ... has Dixmier trace equal to 1.
 
If the eigenvalues μ<sub>i</sub> of the positive operator ''T'' have the property that  
:<math>\zeta_T(s)= \operatorname{Tr}(T^s)= \sum{\mu_i^s}</math>
converges for Re(''s'')>1 and extends to a meromorphic function near ''s''=1 with at most a simple pole at ''s''=1, then the Dixmier trace
of ''T'' is the residue at ''s''=1 (and in particular is independent of the choice of ω).
 
{{harvtxt|Connes|1988}} showed that Wodzicki's [[noncommutative residue]] {{harv|Wodzicki|1984}} of a [[pseudodifferential operator]] on a [[manifold]] is equal to its Dixmier trace.
 
==References==
*Albeverio, S.; Guido, D.; Ponosov, A.; Scarlatti, S.: Singular traces and compact operators. J. Funct. Anal. 137 (1996), no.&nbsp;2, 281—302.
*{{Citation | doi=10.1007/BF01218391 | last1=Connes | first1=Alain | author1-link=Alain Connes | title=The action functional in noncommutative geometry | url=http://projecteuclid.org/euclid.cmp/1104161823 | mr=953826 | year=1988 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=117 | issue=4 | pages=673–683}}
*{{Citation | last1=Connes | first1=Alain | author1-link=Alain Connes | title=Non-commutative geometry | url=ftp://ftp.alainconnes.org/book94bigpdf.pdf | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-185860-5 | year=1994}}
*{{Citation | last1=Dixmier | first1=Jacques | title=Existence de traces non normales | mr=0196508 | year=1966 | journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries a et B | issn=0151-0509 | volume=262 | pages=A1107&ndash;A1108}}
*{{Citation | last1=Wodzicki | first1=M. | title=Local invariants of spectral asymmetry | doi=10.1007/BF01403095 | mr=728144 | year=1984 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=75 | issue=1 | pages=143–177}}
 
== See also ==
 
* [[Singular trace]]
 
[[Category:Von Neumann algebras]]
[[Category:Hilbert space]]
[[Category:Operator theory]]

Latest revision as of 18:55, 12 May 2014

As a company owner or as an entrepreneur, you will

need to document all your income and costs for tax

purposes. Also, it will serve as a very good way to track

your finances and make positive that you are not losing

cash. If you are business owner, you may want to

outsource book keeping for your on the internet enterprise.

Why?

There are really a lot of reasons why you must

outsource book maintaining solutions. The first reason is

that you will be in a position to lower your perform load.

As an alternative of carrying out it yourself, you can have a person do

it for you via outsourcing.

Another fantastic advantage in outsourcing book keeping

jobs for your online enterprise is that it is significantly more

cost powerful than obtaining your personal book keeper in the

office. This indicates that you will be in a position to save more

funds for your organization or decrease your overhead

cost.

Essentially, book keeping includes handling each

financial aspect of your company. Every thing will be

recorded here. From organization revenues, to profits and

losses incurred by your company, as well as other

expenditures, everything will be recorded. This surprising powered by use with has a myriad of surprising warnings for why to provide for it. You have to

contemplate that at the time of tax session, you might get

more than loaded with function as book keeping work can be

fairly time consuming. You will actually have a lot

a lot more function to do and might not have enough time to handle

your business.

And, since of the operate load of book keeping, you

have to don"t forget that a error in this type of

records can incur severe penalties, especially if you

are late in submitting it.

Right now, you will see distinct on the internet book keeping

websites that delivers book maintaining solutions for a fee.

You have to contemplate that outsourcing book keeping

will not only save you a lot of time and money, but it

will also enhance the efficiency of your company and

the way you run it.

Even so, ahead of you get the solutions of a book keeper

and outsource it to a book maintaining service organization,

you have to keep in mind that you require to make positive that

the book maintaining organization you program on outsourcing your

books to ought to only give you with the highest

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that their employees is effectively trained experts who are

knowledgeable in book keeping.

Outsourcing your book keeping jobs is certainly the

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completed in time. It will support you run your company much

smoother, and because of the low the cost of each

book maintaining job, you can be confident that it will be significantly

less expensive to outsource it than obtaining an in home book

keeper.

Outsourcing jobs can support you cut down a huge amount

on your costs. You will see that you will save a

lot more funds in a year by means of outsourcing book

keeping jobs, rather than have your very personal book

keeper functioning for you complete time.

Also, by outsourcing, you can be sure that you will

never ever have to do the book maintaining work yourself. By

performing so, you will be in a position to attend to far more important

issues in your enterprise, which signifies that you will

maximize efficiency and profit of your organization.

So, if you want to save money and run your enterprise

more effectively, you might want to attempt outsourcing

book keeping for your online company.. My family friend learned about partner site by searching Yahoo.

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