Equivariant extensions of *-algebras

TL;DR

This paper introduces a new bivariant functor for *-algebras with group actions, extending classical $Ext$-theory to a $G$-equivariant setting and analyzing invertibility via Toeplitz operators.

Contribution

It defines a $G$-equivariant bivariant functor for *-algebras and operator ideals, extending $Ext$-theory with a focus on equivariance and invertibility.

Findings

The functor relates to the classical $Ext$-functor for $C^*$-algebras.

Invertibility is characterized using Toeplitz operators with abstract symbols.

The framework incorporates group actions into the theory of algebra extensions.

Abstract

A bivariant functor is defined on a category of *-algebras and a category of operator ideals, both with actions of a second countable group $G$, into the category of abelian monoids. The element of the bivariant functor will be $G$-equivariant extensions of a *-algebra by an operator ideal under a suitable equivalence relation. The functor is related with the ordinary $Ext$-functor for $C^*$-algebras defined by Brown-Douglas-Fillmore. Invertibility in this monoid is studied and characterized in terms of Toeplitz operators with abstract symbol.