Gauge groups of E_0-semigroups obtained from Powers weights

TL;DR

This paper explicitly computes the gauge groups of certain type II_0 E_0-semigroups derived from boundary weight doubles, revealing new classes not previously classified and clarifying their cocycle conjugacy and conjugacy relations.

Contribution

It provides explicit gauge group calculations for a new family of E_0-semigroups from boundary weight doubles, expanding the classification framework.

Findings

Gauge groups computed explicitly for a new family of E_0-semigroups.

Many E_0-semigroups are not cocycle conjugate to previously known examples.

Classification results up to cocycle conjugacy and conjugacy for specific boundary weight cases.

Abstract

The gauge group is computed explicitly for a family of E_0-semigroups of type II_0 arising from the boundary weight double construction introduced earlier by Jankowski. This family contains many E_0-semigroups which are not cocycle cocycle conjugate to any examples whose gauge groups have been computed earlier. Further results are obtained regarding the classification up to cocycle conjugacy and up to conjugacy for boundary weight doubles $(\phi, \nu)$ in two separate cases: first in the case when $\phi$ is unital, invertible and q-pure and $\nu$ is any type II Powers weight, and secondly when $\phi$ is a unital $q$-positive map whose range has dimension one and $\nu(A) = (f, Af)$ for some function f such that $(1-e^{-x})^{1/2}f(x) \in L^2(0,\infty)$. All E_0-semigroups in the former case are cocycle conjugate to the one arising simply from $\nu$, and any two E_0-semigroups in the latter case are cocycle conjugate if and only if they are conjugate.