Invariants for E_0-semigroups on II_1 factors

TL;DR

This paper introduces four new invariants for classifying E_0-semigroups on II_1 factors, computes them for specific flows, and demonstrates the existence of infinitely many non-cocycle-conjugate examples.

Contribution

It develops new invariants for E_0-semigroups on II_1 factors and applies them to classify Clifford and free flows, revealing their rich structure.

Findings

All studied flows have trivial gauge groups and coupling indices.

Super product systems are computed explicitly for these flows.

Infinitely many mutually non-cocycle-conjugate E_0-semigroups are shown to exist.

Abstract

We introduce four new cocycle conjugacy invariants for E_0-semigroups on II_1 factors: a coupling index, a dimension for the gauge group, a super product system and a C*-semiflow. Using noncommutative It\^o integrals we show that the dimension of the gauge group can be computed from the structure of the additive cocycles. We do this for the Clifford flows and even Clifford flows on the hyperfinite II_1 factor, and for the free flows on the free group factor $L(F_\infty)$. In all cases the index is 0, which implies they have trivial gauge groups. We compute the super product systems for these families and, using this, we show they have trivial coupling index. Finally, using the C*-semiflow and the boundary representation of Powers and Alevras, we show that the families of Clifford flows and even Clifford flows contain infinitely many mutually non-cocycle-conjugate E_0-semigroups.