Universal C*-algebra

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 containing 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

* $e^*\; =\; e,\; quad$

* $(x^*)^*\; =\; x,quad$

* $(x\; y)^*\; =\; y^*\; x^*.quad$

Define

:

"l"¹("S") is a Banach space, and becomes an algebra under "convolution" defined as follows:

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

"l"¹("S") has a multiplicative identity, viz, the function &delta;_"e" which is zero except at "e", where it takes the value 1. It has the involution :$varphi^*(x)\; =\; overline\{varphi(x^*)\}$

Theorem. "l"¹("S") is a 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 cyclic representation &pi;_"f" associated to "f". Then:$|varphi|\; =\; sup\_\{f\}\; |pi\_f(varphi)|$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.