Non-cocycle-conjugate $E_0$-semigroups on factors

TL;DR

This paper explores $E_0$-semigroups on various factors beyond type I, constructing uncountably many mutually non-cocycle-conjugate examples on type II and type III factors using tensoring and CCR representations.

Contribution

It introduces new methods to construct uncountably many non-cocycle-conjugate $E_0$-semigroups on type II and type III factors, expanding the understanding of their invariants.

Findings

Constructed uncountable families of non-cocycle-conjugate $E_0$-semigroups on type II$_ty$ factors.

Produced uncountable families on all type III$_ u$ factors using CCR representations.

Analyzed invariants like product systems and super product systems for these semigroups.

Abstract

We investigate $E_0-$semigroups on general factors, which are not necessarily of type I, and analyse associated invariants like product systems, super product systems etc. By tensoring $E_0-$semigroups on type I factors with $E_0-$semigroups on type II$_1$ factor, we produce several families (both countable and uncountable), consisting of mutually non-cocycle-conjugate of $E_0-$semigroups on the hyperfinite II$_\infty$ factor. Using CCR representations associated with quasi-free states, we construct for the first time, uncountable families consisting of mutually non-cocycle-conjugate $E_0-$semigroups on all type III$_\lambda$ factors, for $\lambda \in (0,1]$.