C*-subalgebra - Knowlpedia

Let $A$ be a $C^*$ -algebra. A $C^*$ -subalgebra of $A$ is a complex linear subspace $B\subseteq A$ that is closed under multiplication and involution and is closed in the norm of $A$ . With the inherited operations and norm, $B$ is itself a $C^*$ -algebra. If $A$ is unital, the definition does not by itself require $1_A\in B$ ; that extra condition defines a unital $C^*$ -subalgebra in the unit-preserving convention.

Generated

C^*

-subalgebras

For a subset $S\subseteq A$ , the $C^*$ -subalgebra generated by $S$ , written $C^*(S)$ , is the intersection of all $C^*$ -subalgebras containing $S$ . Equivalently, it is the norm closure of the algebraic $*$ -algebra generated by $S$ . In a unital context, $C^*(S,1_A)$ records that the ambient identity is also adjoined. This notation prevents an otherwise common ambiguity about whether generated subalgebras are required to be unital.

Automatic completeness and spectral behavior

Norm closedness makes $B$ complete. Moreover, a $C^*$ -subalgebra has the same spectral behavior for each of its elements as the ambient algebra after units are handled consistently; this is spectral permanence. Consequently continuous functional calculus performed in $A$ for a normal element $b\in B$ remains inside the unital $C^*$ -subalgebra generated by $b$ . These facts would fail for a merely algebraic $*$ -subalgebra.

Ideals and examples

Every closed two-sided ideal is a $C^*$ -subalgebra, but most $C^*$ -subalgebras are not ideals. Diagonal matrices form a commutative $C^*$ -subalgebra of $M_n(\mathbb C)$ but are not an ideal when $n>1$ . The compact operators form both a $C^*$ -subalgebra and a two-sided ideal in $B(\mathcal H)$ . Polynomial functions inside $C([0,1])$ are an algebraic $*$ -subalgebra but not a $C^*$ -subalgebra because they are not norm closed.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.1 on $C^*$ -subalgebras, generated algebras, and spectral permanence.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.1 on subalgebra and unit conventions.