C*-subalgebra

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.

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.

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.

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