Fiber of a C_0(X)-algebra

Let $A$ be a $C_0(X)$ -algebra with structure map $\Phi$ , and let $x\in X$ . Put

I_x=\overline{\Phi(\{f\in C_0(X):f(x)=0\})A}.

The fiber of $A$ at $x$ is the quotient $C^*$ -algebra

A_x=A/I_x.

For $a\in A$ , its image in $A_x$ is denoted $a(x)$ . This is an algebraic fiber attached to the $C_0(X)$ -structure, not merely a set-theoretic preimage of a map. Equivalently, one quotients out the part of $A$ localized away from $x$ .

Pointwise norm

The quotient norm gives

\lVert a(x)\rVert=\inf_{b\in I_x}\lVert a+b\rVert.

For fixed $a\in A$ , the function $x\mapsto\lVert a(x)\rVert$ is upper semicontinuous and vanishes at infinity. It need not be continuous; continuity for every $a$ is an additional condition on the $C_0(X)$ -algebra.

Exact sequence

Evaluation at $x$ is the quotient map $q_x\colon A\to A_x$ , and it fits into the short exact sequence

0\longrightarrow I_x\longrightarrow A \xrightarrow{q_x}A_x\longrightarrow0.

The fiber may be zero. Two distinct points may also produce isomorphic fibers, because the construction records position over $X$ as well as the fiber’s abstract isomorphism class.

Bundle interpretation

The disjoint union

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

has a canonical topology making it an upper-semicontinuous $C^*$ -bundle. Each $a\in A$ then gives the continuous section $x\mapsto a(x)$ , and these sections determine the topology. The resulting section algebra recovers $A$ .

References

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