Universal C*-algebra

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In mathematics, more specifically in the theory of C*-algebras, 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 e and with an involution * such that

$e^{*}=e,\quad$

$(x^{*})^{*}=x,\quad$

$(xy)^{*}=y^{*}x^{*}.\quad$

Define

$\ell ^{1}(S)=\{\varphi :S\rightarrow {\mathbb {C}}:\|\varphi \|=\sum _{{x\in S}}|\varphi (x)|<\infty \}.$

l¹(S) is a Banach space, and becomes an algebra under convolution defined as follows:

$[\varphi \star \psi ](x)=\sum _{{\{u,v:uv=x\}}}\varphi (u)\psi (v)$

l¹(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

$\varphi ^{*}(x)=\overline {\varphi (x^{*})}$

Theorem. l¹(S) is a B*-algebra with identity.

The universal C*-algebra of contractions generated by S is the C*-enveloping algebra of l¹(S). We can describe it as follows: For every state f of l¹(S), consider the cyclic representation π_f associated to f. Then

$\|\varphi \|=\sup _{{f}}\|\pi _{f}(\varphi )\|$

is a C*-seminorm on l¹(S), where the supremum ranges over states f of l¹(S). Taking the quotient space of l¹(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

Loring, T. (1997), Lifting Solutions to Perturbing Problems in C*-Algebras, Fields Institute Monographs 8, American Mathematical Society, ISBN 0-8218-0602-5

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