Homology for operator algebras I: Spectral homology for reflexive algebras

TL;DR

This paper introduces a stable homology theory for certain operator algebras, linking spectral homology to algebra automorphisms and defining a new cohomology concept to analyze algebraic structures.

Contribution

It develops a homology framework for CSL algebras, connecting spectral homology with automorphism properties and introducing essential Hochschild cohomology.

Findings

Spectral homology groups are computable and meaningful.

Trivial first spectral homology implies Schur automorphisms are quasispatial.

Essential Hochschild cohomology is defined via point weak star closure.

Abstract

A stable homology theory is defined for completely distributive CSL algebras in terms of the point-neighbourhood homology of the partially ordered set of meet-irreducible elements of the invariant projection lattice. This specialises to the simplicial homology of the underlying simplicial complex in the case of a digraph algebra. These groups are computable and useful. In particular it is shown that if the first spectral homology group is trivial then Schur automorphisms are automatically quasispatial. This motivates the introduction of essential Hochschild cohomology which we define by using the point weak star closure of coboundaries in place of the usual coboundaries.