Generalized asymptotic algebras and $\mathrm{E}$-theory for non-separable $\mathrm{C}^*$-algebras

TL;DR

This paper introduces a generalized form of E-theory for non-separable C*-algebras that captures complex asymptotic behaviors, ensuring the existence of all desired algebraic structures and extending applicability to infinite-dimensional index theory.

Contribution

It defines a new E-theory framework incorporating complex asymptotics, applicable to non-separable C*-algebras, with all algebraic properties preserved.

Findings

Provides a generalized E-theory with all long exact sequences

Ensures the existence of composition products in the new framework

Applicable to equivariant settings for arbitrary discrete groups

Abstract

In previous definition of $\mathrm{E}$-theory, separability of the $\mathrm{C}^*$-algebras is needed either to construct the composition product or to prove the long exact sequences. Considering the latter, the potential failure of the long exact sequences can be traced back to the fact that these $\mathrm{E}$-theory groups accommodate information about asymptotic processes in which one real parameter goes to infinity, but not about more complicated asymptotics parametrized by directed sets. We propose a definition for $\mathrm{E}$-theory which also incorporates this additional information by generalizing the notion of asymptotic algebras. As a consequence, it not only has all desirable products but also all long exact sequences, even for non-separable $\mathrm{C}^*$-algebras. More precisely, our construction yields equivariant $\mathrm{E}$-theory for $\mathbb{Z}_2$-graded $G$-$\mathrm{C}^*$-algebras for arbitrary discrete groups $G$. We suspect that our model for $\mathrm{E}$-theory could be the right entity to investigate index theory on infinite dimensional manifolds.