Cohomology for spatial super-product systems

TL;DR

This paper develops a cohomology theory for spatial super-product systems, computes 2-cocycles for Clifford super-product systems, and distinguishes certain E0-semigroups up to isomorphism, revealing new classification results.

Contribution

It introduces a cohomology framework for super-product systems and applies it to classify Clifford super-product systems and related E0-semigroups.

Findings

Clifford super-product systems are distinguished by their 2-cocycles.

CAR flows are non-cocycle-conjugate for different ranks.

Automorphism groups of Clifford super-product systems are computed.

Abstract

We introduce a cohomology theory for spatial super- product systems and compute the $2-$cocycles for some basic examples called as Clifford super-product systems, thereby distinguish them up to isomorphism. This consequently proves that a family of $E_0-$semigroups on type III factors, which we call as CAR flows, are non-cocycle-conjugate for different ranks. Similar results follows for the even CAR flows as well. We also compute the automorphsim group of the Clifford super-product systems.