Continuous C_0(X)-algebra - Imported Knowlpedia Content

Definition

Let $X$ be a locally compact Hausdorff space and let $A$ be a $C_0(X)$ -algebra. For $x\in X$ , write $A_x$ for its quotient fiber and $a(x)$ for the image of $a\in A$ . The algebra $A$ is a continuous $C_0(X)$ -algebra when

x\longmapsto\lVert a(x)\rVert

is continuous on $X$ for every $a\in A$ . These functions already vanish at infinity and are upper semicontinuous for an arbitrary $C_0(X)$ -algebra; continuity is the additional axiom. The specified central nondegenerate action of $C_0(X)$ , and not merely the abstract $C^*$ -algebra $A$ , is part of this definition.

Bundle characterization

The canonical bundle $\bigsqcup_x A_x\to X$ associated with any $C_0(X)$ -algebra is upper semicontinuous. It is a continuous field of $C^*$ -algebras exactly when $A$ is continuous in the sense above. Under this correspondence, $A$ is recovered as the algebra of continuous sections vanishing at infinity Nilsen, bundle realization and continuity criteria.

Continuity is checked element by element. It neither requires local triviality nor forces all fibers to be isomorphic.

Examples and non-examples

For any $C^*$ -algebra $B$ , $C_0(X,B)$ is continuous: the fiber of $f$ at $x$ is $f(x)$ , and $x\mapsto\lVert f(x)\rVert$ is continuous. Section algebras of locally trivial $C^*$ -bundles give further examples.

A general upper-semicontinuous bundle can have a section $a$ whose norm drops discontinuously at a point. Its section algebra remains a $C_0(X)$ -algebra but is not continuous.

References

M. Nilsen, “ $C^*$ -Bundles and $C_0(X)$ -Algebras,” Indiana University Mathematics Journal 45 (1996), 463–477. DOI record. Relevant: quotient fibers, sectional representation, and criteria for continuity.
É. Blanchard, “Déformations de $C^*$ -algèbres de Hopf,” Bulletin de la Société Mathématique de France 124 (1996), 141–215. DOI record. Relevant: continuous $C(X)$ -algebras and their field interpretation.