Cohomology for spatial superproduct systems

Abstract

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