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 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

\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: u v = x\}} \varphi(u) \psi(v)

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

\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 πf 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.