en>Ale2006: /* Further reading */ add RSA Key Extraction via Low-Bandwidth Acoustic Cryptanalysis

2014-10-15T17:14:02Z

Further reading: add RSA Key Extraction via Low-Bandwidth Acoustic Cryptanalysis

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	~~{{Unreferenced\|date=December 2009}}~~		I like Rock climbing. Appears boring? Not!<br>I try to learn Portuguese in my free time.<br><br>Feel free to surf to my web site :: Hostgator Coupon Codes - [http://dawonls.dothome.co.kr/db/?document_srl=382462 dawonls.dothome.co.kr] -
	~~In [[mathematics]], more specifically in the theory of [[C-algebra]]s, a '''universal C-algebra''' is one characterized by a universal property~~.

	A universal C-algebra can be expressed as a presentation, in terms of generators and relations. One requires that the generators must be realizable as bounded operators on a Hilbert space, and that the relations must prescribe a uniform bound on the norm of each generator. For example, the universal C-algebra generated by a unitary element ''u'' has presentation <~~''u'' \| ''uu'' = ''uu'' = 1~~>. By the functional calculus, this C-algebra is the continuous functions on the unit circle in the complex plane. Any C-algebra generated by a unitary element is the homomorphic image of this universal C*-algebra.

	~~We next describe a general framework for defining a large class of these algebras.~~ ~~Let ''S'' be a [[countable]] [[semigroup]] (~~in which we denote the operation by juxtaposition) with [[identity function\|identity]] ''e'' and with an [[Involution (mathematics)\|involution]] *
	~~such that~~

	* <math> e^* = e, \quad </math>

	* <math> (x^)^ = x,\quad </math>

	~~* <math>(x y)^* = y^* x^*~~.~~\quad~~<~~/math~~>

	~~Define~~

	:<~~math~~>~~\ell^1(S) = \{\varphi~~:~~S \rightarrow \mathbb{C}~~: ~~\\|\varphi\\| = \sum_{x \in S}\|\varphi(x)\| < \infty\}.</math>~~

	~~''l''<sup>1</sup>(''S'') is a [~~[~~Banach space]], and becomes an [[algebra over a field\|algebra]] under ''convolution'' defined as follows~~:

	~~:<math> [\varphi \star \psi](x) = \sum_{\{u,v: u v = x\}} \varphi(u) \psi(v)<~~/~~math>~~

	~~''l''<sup>1</sup>(''S'') has a multiplicative identity, viz, the function δ<sub>''e''</sub> which is zero except at ''e'', where it takes the value 1. It has the involution~~
	~~:<math> \varphi^(x) = \overline{\varphi(x^)}<~~/~~math>~~

	~~'''Theorem'''~~. ~~''l''<sup>1</sup>(''S'') is a [[B-star-algebra\|B*-algebra]] with identity~~.

	~~The universal C-algebra of contractions generated by ''S'' is the C-enveloping algebra of ''l''<sup>1</sup>(''S'')~~. ~~We can describe it as follows: For every state ''f'' of ''l''<sup>1<~~/~~sup>(''S''), consider the [[Gelfand–Naimark–Segal construction\|cyclic representation]] π<sub>''f''<~~/~~sub> associated to ''f''. Then~~
	~~:<math> \\|\varphi\\|~~ = ~~\sup_{f} \\|\pi_f(\varphi)\\| </math>~~
	~~is a C-seminorm on ''l''<sup>1</sup>(''S''), where the supremum ranges over states ''f'' of ''l''<sup>1</sup>(''S'')~~. ~~Taking the quotient space of ''l''<sup>1</sup>(''S'') by the two-sided ideal of elements of norm 0, produces a normed algebra which satisfies the C-property~~. ~~Completing with respect to this norm, yields a C*-algebra~~.

	~~==References==~~

	{{citation \|first=T.\|last=Loring \|title=Lifting Solutions to Perturbing Problems in C-Algebras \|volume=8 \|series=[[Fields Institute Monographs]] \| year=1997 \|publisher=[[American Mathematical Society]] ~~\|isbn=0~~-~~8218-0602-5}}~~

	~~{{DEFAULTSORT:Universal C*-Algebra}}~~
	~~[[Category:C*-algebras]]~~

en>Rmehtany: /* General */

2014-01-15T23:39:45Z

General

New page

{{Unreferenced|date=December 2009}}
In [[mathematics]], more specifically in the theory of [[C*-algebra]]s, a '''universal C*-algebra''' is one characterized by a universal property.

A universal C*-algebra can be expressed as a presentation, in terms of generators and relations. One requires that the generators must be realizable as bounded operators on a Hilbert space, and that the relations must prescribe a uniform bound on the norm of each generator. For example, the universal C*-algebra generated by a unitary element ''u'' has presentation <''u'' | ''u*u'' = ''uu*'' = 1>. By the functional calculus, this C*-algebra is the continuous functions on the unit circle in the complex plane. Any C*-algebra generated by a unitary element is the homomorphic image of this universal C*-algebra.

We next describe a general framework for defining a large class of these algebras. Let ''S'' be a [[countable]] [[semigroup]] (in which we denote the operation by juxtaposition) with [[identity function|identity]] ''e'' and with an [[Involution (mathematics)|involution]] *
such that

* <math> e^* = e, \quad </math>

* <math> (x^*)^* = x,\quad </math>

* <math>(x y)^* = y^* x^*.\quad</math>

Define

:<math>\ell^1(S) = \{\varphi:S \rightarrow \mathbb{C}: \|\varphi\| = \sum_{x \in S}|\varphi(x)| < \infty\}.</math>

''l''1(''S'') is a [[Banach space]], and becomes an [[algebra over a field|algebra]] under ''convolution'' defined as follows:

:<math> [\varphi \star \psi](x) = \sum_{\{u,v: u v = x\}} \varphi(u) \psi(v)</math>

''l''1(''S'') has a multiplicative identity, viz, the function δ''e'' which is zero except at ''e'', where it takes the value 1. It has the involution
:<math> \varphi^*(x) = \overline{\varphi(x^*)}</math>

'''Theorem'''. ''l''1(''S'') is a [[B-star-algebra|B*-algebra]] with identity.

The universal C*-algebra of contractions generated by ''S'' is the C*-enveloping algebra of ''l''1(''S''). We can describe it as follows: For every state ''f'' of ''l''1(''S''), consider the [[Gelfand–Naimark–Segal construction|cyclic representation]] π''f'' associated to ''f''. Then
:<math> \|\varphi\| = \sup_{f} \|\pi_f(\varphi)\| </math>
is a C*-seminorm on ''l''1(''S''), where the supremum ranges over states ''f'' of ''l''1(''S''). Taking the quotient space of ''l''1(''S'') by the two-sided ideal of elements of norm 0, produces a normed algebra which satisfies the C*-property. Completing with respect to this norm, yields a C*-algebra.

==References==

*{{citation |first=T.|last=Loring |title=Lifting Solutions to Perturbing Problems in C*-Algebras |volume=8 |series=[[Fields Institute Monographs]] | year=1997 |publisher=[[American Mathematical Society]] |isbn=0-8218-0602-5}}

{{DEFAULTSORT:Universal C*-Algebra}}
[[Category:C*-algebras]]

Side channel attack - Revision history

en>Ale2006: /* Further reading */ add RSA Key Extraction via Low-Bandwidth Acoustic Cryptanalysis

en>Rmehtany: /* General */