Homology for operator algebras II : Stable homology for non-self-adjoint algebras

TL;DR

This paper introduces a new homology theory for non-self-adjoint operator algebras, linking it to K-theory and applications in classifying subalgebras and limit algebras.

Contribution

It defines a novel homology based on partial isometries, connecting it to K-theory and providing tools for classifying operator algebra substructures.

Findings

Zeroth order group matches K_0 group of generated C*-algebra

Identifies applications in classifying regular subalgebras

Shows significance of stable homology in operator algebra classification

Abstract

A new homology is defined for a non-self-adjoint operator algebra and distinguished masa which is based upon cycles and boundaries associated with complexes of partial isometries in the stable algebra. Under natural hypotheses the zeroth order group coincides with the $\ K_0$\ group of the generated C*-algebra. Several identifications and applications are given, and in particular it is shown how stable homology is significant for the classification of regular subalgebras and regular limit algebras.