Gauge groups of E_0-semigroups obtained from Powers weights

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.