Continuous field of C*-algebras

Let $X$ be a locally compact Hausdorff space. A continuous field of $C^*$ -algebras over $X$ consists of $C^*$ -algebras $A_x$ and a $*$ -algebra $\Gamma\subseteq\prod_{x\in X}A_x$ of sections such that: each evaluation $\Gamma\to A_x$ has dense range; $x\mapsto\lVert s(x)\rVert$ is continuous for every $s\in\Gamma$ ; and any section locally uniformly approximable by members of $\Gamma$ belongs to $\Gamma$ . With the standard topology generated by these sections, this is equivalently an upper-semicontinuous $C^*$ -bundle whose norm is continuous. This knowl uses “continuous $C^*$ -bundle” for that equivalent bundle presentation.

Field and bundle presentations

The section axioms determine a topology on the disjoint union $\bigsqcup_x A_x$ . Fiberwise algebraic operations become continuous, and the norm is continuous on the total space. Conversely, the continuous sections of a continuous $C^*$ -bundle recover a field satisfying the local approximation axiom. This equivalence is the reason the two terminologies are often interchanged.

The sections in $\Gamma$ whose norm functions vanish at infinity form the section algebra $\Gamma_0(X,\mathcal A)$ , which remembers the field as a $C_0(X)$ -algebra. The continuity requirement is stronger than the upper-semicontinuity available for a general $C_0(X)$ -algebra.

Examples and non-examples

For a fixed $C^*$ -algebra $B$ , the trivial field $A_x=B$ has section algebra $C_0(X,B)$ . A locally trivial bundle with fiber $B$ is another continuous field, although continuous fields need not be locally trivial and their fibers may change isomorphism type.

An upper-semicontinuous $C^*$ -bundle for which some section has a discontinuous norm function is not a continuous field. It still has a valid section $C^*$ -algebra, but fails the defining continuity condition.

References

J. M. G. Fell, “The Structure of Algebras of Operator Fields,” Acta Mathematica 106 (1961), 233–280. DOI record. Relevant: operator-field axioms and the passage between fields and section algebras.
M. Nilsen, “ $C^*$ -Bundles and $C_0(X)$ -Algebras,” Indiana University Mathematics Journal 45 (1996), 463–477. DOI record. Relevant: bundle models, section algebras, and continuity criteria.