Atiyah-J\"anich theorem for $\sigma$-C*-algebras

TL;DR

This paper extends the Atiyah-Jänich theorem to $\sigma$-C*-algebras, showing that the space of Fredholm modular operators with coefficients in such algebras represents a specific K-theory functor.

Contribution

It demonstrates that the space of Fredholm modular operators for $\sigma$-C*-algebras represents the representable K-theory functor, generalizing classical results to a broader algebraic context.

Findings

The space of Fredholm modular operators represents the K-theory functor for $\sigma$-C*-algebras.

The result applies to countably compactly generated spaces.

It generalizes the Atiyah-Jänich theorem to $\sigma$-C*-algebras.

Abstract

K-theory for $ \sigma$-C*-algebras (countable inverse limits of C*-algebras) has been investigated by N. C. Phillips [{\it K-Theory} {\bf 3} (1989), 441--478]. We use his representable K-theory to show that the space of Fredholm modular operators with coefficients in an arbitrary unital $ \sigma$-C*-algebra $A$, represents the functor $X \mapsto {\rm RK}_{0}(C(X, A))$ from the category of countably compactly generated spaces to the category of abelian groups.