Generated C*-subalgebra

Let $A$ be a $C^*$ -algebra and $S\subseteq A$ . The $C^*$ -subalgebra generated by $S$ , denoted $C^*(S)$ , is the intersection of all $C^*$ -subalgebras of $A$ that contain $S$ . Equivalently, it is the norm closure in $A$ of the algebraic $*$ -subalgebra generated by $S$ . If generated subalgebras are required to contain $1_A$ , one instead takes the closure of the unital $*$ -algebra generated by $S$ , commonly written $C^*(S,1_A)$ . Thus the ambient unital convention is part of the notation.

Universal property

The inclusion $S\subseteq C^*(S)$ is minimal: any $C^*$ -subalgebra $B\subseteq A$ containing $S$ also contains $C^*(S)$ . Consequently, a $*$ -homomorphism defined on $A$ is determined on $C^*(S)$ by its values on $S$ . This is an internal universal property relative to $A$ , not the universal $C^*$ -algebra on abstract generators and relations.

Functional calculus and examples

For a normal element $a\in A$ , the generated algebra $C^*(a,1_A)$ is commutative and contains every element obtained from $a$ by continuous functional calculus. The diagonal matrix units generate the diagonal algebra in $M_n(\mathbb C)$ , whereas all matrix units $e_{ij}$ generate $M_n(\mathbb C)$ . A merely algebraic $*$ -algebra generated by a set may fail to be norm closed and is then strictly smaller than $C^*(S)$ .

Conventions and scope

When $A$ is nonunital, $C^*(S)$ is never made unital by adjoining an external identity unless this is explicitly stated. When $A$ is unital, authors differ on whether “subalgebra” means “unital subalgebra”; the notations $C^*(S)$ and $C^*(S,1_A)$ separate these conventions.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.1 on generated $C^*$ -subalgebras and unit conventions.
Kenneth R. Davidson, $C^*$ -Algebras by Example, American Mathematical Society, 1996. AMS record. Relevant: Chapter I, §1 on concrete $C^*$ -algebras and generated subalgebras.