Section algebra of a C*-bundle - Imported Knowlpedia Content

Definition

Let $p\colon\mathcal A\to X$ be an upper-semicontinuous $C^*$ -bundle over a locally compact Hausdorff space. Its section algebra is

\Gamma_0(X,\mathcal A) =\{s\colon X\to\mathcal A:p\circ s=\operatorname{id}_X,\ s \text{ is continuous, and }\lVert s(x)\rVert\to0\text{ at infinity}\}.

Pointwise algebraic operations, pointwise involution, and the norm $\lVert s\rVert=\sup_x\lVert s(x)\rVert$ make this a $C^*$ -algebra. Completeness follows from the bundle axioms and the uniform norm on sections.

Vanishing at infinity

The norm function of a section vanishes at infinity when, for every $\varepsilon>0$ , the set

\{x\in X:\lVert s(x)\rVert\geq\varepsilon\}

is compact. Compactly supported continuous sections therefore belong to $\Gamma_0(X,\mathcal A)$ . The subscript $0$ records this condition; bounded continuous sections need not vanish at infinity.

The C_0(X)-action

For $f\in C_0(X)$ and $s\in\Gamma_0(X,\mathcal A)$ , define

(fs)(x)=f(x)s(x).

This gives a central nondegenerate action of $C_0(X)$ , so the section algebra is a $C_0(X)$ -algebra. Evaluation at $x$ maps a section to $s(x)$ and identifies the corresponding quotient fiber with $\mathcal A_x$ .

Examples and reconstruction

For the trivial bundle $X\times B\to X$ , one obtains

\Gamma_0(X,X\times B)\cong C_0(X,B).

For a general upper-semicontinuous bundle, the section algebra retains enough information to reconstruct both its fibers and total-space topology. This is the bundle-to-algebra direction of the $C_0(X)$ -algebra bundle correspondence.

References

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