C_0(X)-algebras and upper-semicontinuous C*-bundles

Statement

Let $X$ be a locally compact Hausdorff space. Every $C_0(X)$ -algebra $A$ is $C_0(X)$ -linearly $*$ -isomorphic to the section algebra

\Gamma_0(X,\mathcal A)

of an upper-semicontinuous $C^*$ -bundle $p\colon\mathcal A\to X$ . Its fiber at $x$ is canonically the quotient $A_x$ . Conversely, the section algebra of every such bundle is a $C_0(X)$ -algebra, and reconstructing its bundle recovers $\mathcal A$ up to bundle isomorphism.

Construction from an algebra

For each $x\in X$ , form $A_x=A/I_x$ and the disjoint union

\mathcal A=\bigsqcup_{x\in X}A_x.

There is a unique upper-semicontinuous bundle topology compatible with the quotient norms and making every canonical section

\widehat a(x)=a+I_x

continuous. The map $a\mapsto\widehat a$ is then an isometric $C_0(X)$ -linear $*$ -isomorphism from $A$ onto $\Gamma_0(X,\mathcal A)$ .

Construction from a bundle

Starting with $p\colon\mathcal A\to X$ , scalar multiplication of sections by functions in $C_0(X)$ defines the central nondegenerate structure map

C_0(X)\longrightarrow ZM(\Gamma_0(X,\mathcal A)).

The quotient of the section algebra at $x$ evaluates onto $\mathcal A_x$ . These evaluation maps assemble to the original bundle, including its topology.

Scope

The theorem uses upper-semicontinuous bundles, not only continuous or locally trivial ones. Restricting to continuous bundles would omit $C_0(X)$ -algebras for which some norm function $x\mapsto\lVert a(x)\rVert$ is discontinuous. The correspondence is relative to the chosen $C_0(X)$ -action; forgetting that action loses the base-space information.

References

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