Maximal C*-seminorm - Knowlpedia

Let $A$ be an involutive algebra. For $a\in A$ , set

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

where $\pi$ ranges over all bounded Hilbert-space $*$ -representations of $A$ . If this supremum is finite for every $a$ , it is the maximal $C^*$ -seminorm on $A$ : it satisfies $\|a^*a\|_{\max}=\|a\|_{\max}^2$ and dominates the seminorm arising from every such representation. It can have a nonzero kernel, so “universal $C^*$ -norm” is literally a norm only after the appropriate quotient.

Enveloping completion

When $\|\cdot\|_{\max}$ is finite, first form $A/\ker\|\cdot\|_{\max}$ , then complete it. The resulting $C^*$ -algebra $C^*(A)$ has the universal property that every bounded $*$ -representation of $A$ factors uniquely through a representation of $C^*(A)$ . The quotient and completion are separate steps and should not be folded into the seminorm's definition.

Existence and failure

The supremum is automatically finite when algebraic relations impose uniform bounds on the images of generators, as happens for unitary generators. For a general involutive algebra it may be infinite on some element; then no enveloping $C^*$ -algebra with all bounded $*$ -representations has been defined by this formula. If there are no nonzero bounded representations, the seminorm is identically zero.

Maximality and terminology

Every $C^*$ -seminorm $p$ on $A$ is bounded above by $\|\cdot\|_{\max}$ : represent the $C^*$ -completion of $A/\ker p$ faithfully on a Hilbert space and compare norms. Some authors say “universal $C^*$ -seminorm”; the adjective “maximal” emphasizes the supremum over all representations, not a chosen faithful one.

References

Bruce Blackadar, Operator Algebras: Theory of $C^*$ -Algebras and von Neumann Algebras, Encyclopaedia of Mathematical Sciences 122, Springer, 2006. DOI record. Relevant: section II.8 on universal $C^*$ -algebras and maximal seminorms.
Jacques Dixmier, $C^*$ -Algebras, North-Holland, 1977. Publisher record. Relevant: section 2.7 on enveloping $C^*$ -algebras.