Universal C*-algebra
|
|
In the theory of C*-algebras a universal C*-algebra is one characterized by a universal property. We 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
- <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>
l1(S) is a Banach space, and becomes an algebra under convolution defined as follows:
- <math> [\varphi \star \psi](x) = \sum_{\{u,v: u v = x\}} \varphi(u) \psi(v)<math>
l1(S) has a multiplicative identity, viz, the function δe which is zero except at e, where it takes the value 1. It has the involution
- <math> \varphi^*(x) = \overline{\varphi(x^*)}<math>
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 πf associated to f. Then
- <math> \|\varphi\| = \sup_{f} \|\pi_f(\varphi)\| <math>
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.