Enveloping C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be an involutive algebra whose maximal $C^*$ -seminorm

\|a\|_{\max}=\sup_\pi\|\pi(a)\|

is finite for every $a\in A$ , where $\pi$ ranges over its bounded Hilbert-space $*$ -representations. The enveloping $C^*$ -algebra $C^*(A)$ is the completion of $A/\ker\|\cdot\|_{\max}$ in this norm. Its canonical $*$ -homomorphism $\iota:A\to C^*(A)$ has dense image, and every bounded $*$ -representation of $A$ factors canonically and uniquely through a representation of $C^*(A)$ .

Universal property

If $\pi:A\to B(H)$ is a bounded $*$ -representation, there is a unique $*$ -representation $\widetilde\pi:C^*(A)\to B(H)$ satisfying $\pi=\widetilde\pi\circ\iota$ . This property determines $C^*(A)$ up to canonical $*$ -isomorphism. The kernel of $\iota$ is the intersection of the kernels of all bounded $*$ -representations, so $\iota$ need not be injective Blackadar, §II.8.

Existence and examples

For the algebraic group algebra $\mathbb C[\Gamma]$ of a discrete group, unitarity of the group generators supplies uniform bounds, and the enveloping completion is the full group $C^*$ -algebra $C^*(\Gamma)$ . For a general involutive algebra, the supremum can be infinite for some element; then this construction does not produce an enveloping $C^*$ -algebra. If all bounded representations vanish, the completion is the zero algebra.

Terminology

The enveloping $C^*$ -algebra is sometimes called the universal $C^*$ -completion. It must not be confused with the $C^*$ -envelope of an operator system or nonself-adjoint operator algebra, which is a minimal boundary construction rather than the maximal completion over representations. Nor is it the universal enveloping algebra of a Lie algebra, which is purely algebraic.

References

Jacques Dixmier, $C^*$ -Algebras, North-Holland, 1977. Publisher record. Relevant: §2.7 on enveloping $C^*$ -algebras.
Bruce Blackadar, Operator Algebras: Theory of $C^*$ -Algebras and von Neumann Algebras, Springer, 2006. DOI record. Relevant: §II.8 on maximal seminorms and universal completions.