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| {{Unreferenced|date=December 2009}}
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| In [[mathematics]], more specifically in the theory of [[C*-algebra]]s, a '''universal C*-algebra''' is one characterized by a universal property.
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| 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.
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| 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]] *
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| such that
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| * <math> e^* = e, \quad </math>
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| * <math> (x^*)^* = x,\quad </math>
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| * <math>(x y)^* = y^* x^*.\quad</math>
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| Define
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| :<math>\ell^1(S) = \{\varphi:S \rightarrow \mathbb{C}: \|\varphi\| = \sum_{x \in S}|\varphi(x)| < \infty\}.</math>
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| ''l''<sup>1</sup>(''S'') is a [[Banach space]], and becomes an [[algebra over a field|algebra]] under ''convolution'' defined as follows:
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| :<math> [\varphi \star \psi](x) = \sum_{\{u,v: u v = x\}} \varphi(u) \psi(v)</math>
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| ''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
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| :<math> \varphi^*(x) = \overline{\varphi(x^*)}</math>
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| '''Theorem'''. ''l''<sup>1</sup>(''S'') is a [[B-star-algebra|B*-algebra]] with identity.
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| 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
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| :<math> \|\varphi\| = \sup_{f} \|\pi_f(\varphi)\| </math>
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| 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.
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| ==References==
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| *{{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}}
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| {{DEFAULTSORT:Universal C*-Algebra}}
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| [[Category:C*-algebras]]
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I like Rock climbing. Appears boring? Not!
I try to learn Portuguese in my free time.
Feel free to surf to my web site :: Hostgator Coupon Codes - dawonls.dothome.co.kr -