Commutative C*-algebra - Imported Knowlpedia Content

Definition

A commutative $C^*$ -algebra, also called an abelian $C^*$ -algebra, is a $C^*$ -algebra $A$ whose multiplication satisfies

ab=ba\qquad\text{for all }a,b\in A.

Commutativity concerns the algebra product, not merely commutation of a distinguished family of elements. The algebra may be unital or nonunital. Its involution, norm, and completeness are still part of the $C^*$ -structure, so a commutative $C^*$ -algebra is more than an abstract commutative $*$ -algebra.

Function-algebra model

If $X$ is a locally compact Hausdorff space, the continuous complex-valued functions vanishing at infinity form a commutative $C^*$ -algebra $C_0(X)$ , with pointwise operations, complex conjugation, and the supremum norm. It is unital exactly when $X$ is compact. Familiar special cases include $C(X)$ for compact $X$ , $c_0$ for a countable discrete space, and $\mathbb C^n$ for a finite discrete space.

Gelfand duality

For a commutative $C^*$ -algebra $A$ , let $\widehat A$ be its space of nonzero characters with the weak-star topology. The Gelfand transform

a\longmapsto\widehat a,\qquad \widehat a(\chi)=\chi(a),

is an isometric $*$ -isomorphism $A\cong C_0(\widehat A)$ . Conversely, evaluation at points recovers the character space of $C_0(X)$ . This is the commutative Gelfand--Naimark theorem Murphy, Chapter 2.

Subalgebras and generated examples

A family of commuting normal operators on a Hilbert space generates a commutative concrete $C^*$ -algebra. The continuous functional calculus for one normal element $a$ identifies the unital algebra it generates with continuous functions on the spectrum of $a$ . By contrast, an algebra generated by noncommuting operators is generally noncommutative even when each generator is normal.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: Chapter 2 on the Gelfand transform and commutative $C^*$ -algebras.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 1 on basic $C^*$ -algebra structure and commutative examples.