C_0(X)-algebra - Imported Knowlpedia Content

Definition

Let $X$ be a locally compact Hausdorff space. A $C_0(X)$ -algebra is a $C^*$ -algebra $A$ together with a nondegenerate $*$ -homomorphism

\Phi\colon C_0(X)\longrightarrow ZM(A),

where $ZM(A)$ is the center of the multiplier algebra. Thus $\Phi(f)a=a\Phi(f)$ and the closed linear span of $C_0(X)A$ is $A$ . One usually writes $fa$ for $\Phi(f)a$ . The pair $(A,\Phi)$ , rather than $A$ alone, is the $C_0(X)$ -algebra.

Geometric meaning

The central action says where an element of $A$ is supported over $X$ . Multiplying by $f\in C_0(X)$ localizes that element to the region where $f$ is nonzero. Nondegeneracy ensures that this localization sees all of $A$ , including when $A$ is nonunital.

The structure map is part of the data: the same underlying $C^*$ -algebra can carry inequivalent $C_0(X)$ -algebra structures.

Fibers

For $x\in X$ , let

I_x=\overline{\Phi(\{f\in C_0(X):f(x)=0\})A}.

This is a closed two-sided ideal, and the quotient

A_x=A/I_x

is the fiber of $A$ at $x$ . Each $a\in A$ determines a section $x\mapsto a(x)$ , where $a(x)=a+I_x$ .

Morphisms and examples

The basic example is $A=C_0(X,B)$ , with $\Phi(f)g(x)=f(x)g(x)$ ; every fiber is $B$ . More generally, the section algebra of an upper-semicontinuous $C^*$ -bundle is a $C_0(X)$ -algebra.

A homomorphism between $C_0(X)$ -algebras is $C_0(X)$ -linear when it respects the given actions. An ordinary $*$ -homomorphism need not do so.

References

May Nilsen, “C-Bundles and $C_0(X)$ -Algebras,” Indiana University Mathematics Journal* 45 (1996), 463–477. DOI record. Relevant: the definition and bundle realization of $C_0(X)$ -algebras.
Dana P. Williams, Crossed Products of C-Algebras*, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: Appendix C on $C_0(X)$ -algebras.