Upper-semicontinuous C*-bundle

An upper-semicontinuous $C^*$ -bundle over a topological space $X$ is an open continuous surjection $p\colon\mathcal A\to X$ whose fiber $\mathcal A_x=p^{-1}(x)$ is a $C^*$ -algebra. Fiberwise addition, scalar multiplication, multiplication, and involution are continuous, while $a\mapsto\lVert a\rVert$ is upper semicontinuous. The zero elements satisfy the convergence axiom: if $p(a_i)\to x$ and $\lVert a_i\rVert\to0$ , then $a_i\to0_x$ . These axioms relate the fiber topologies without requiring local product charts or a fixed model fiber.

Upper semicontinuity

Upper semicontinuity of the norm means that

\{a\in\mathcal A:\lVert a\rVert<r\}

is open for every $r>0$ . Equivalently, for a convergent net $a_i\to a$ ,

\limsup_i\lVert a_i\rVert\leq\lVert a\rVert.

The norm need not be continuous. Requiring continuity defines the stricter notion of a continuous $C^*$ -bundle.

Continuous sections

A section is a map $s\colon X\to\mathcal A$ with $p\circ s=\operatorname{id}_X$ . Continuity is measured in the topology of the total space, not by first identifying all fibers with a fixed algebra. For every continuous section, the function $x\mapsto\lVert s(x)\rVert$ is upper semicontinuous.

When $X$ is locally compact Hausdorff, sections vanishing at infinity form the section $C^*$ -algebra $\Gamma_0(X,\mathcal A)$ .

Why local triviality is not assumed

The fibers may vary in isomorphism type and dimension, so an upper-semicontinuous $C^*$ -bundle need not be locally trivial. This flexibility is essential: every $C_0(X)$ -algebra has such a bundle model, whereas continuous or locally trivial bundles describe only special cases.

References

May Nilsen, “C-Bundles and $C_0(X)$ -Algebras,” Indiana University Mathematics Journal* 45 (1996), 463–477. DOI record. Relevant: the bundle construction and 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 on upper-semicontinuous bundles.