On type II_0 E_0-semigroups induced by boundary weight doubles

TL;DR

This paper introduces a new method to construct a large class of non-cocycle conjugate E_0-semigroups using boundary weight doubles involving q-positive maps and boundary weights, expanding the understanding of their structure.

Contribution

It defines boundary weight maps via boundary weight doubles with q-positive maps, leading to the construction of uncountably many non-cocycle conjugate E_0-semigroups.

Findings

Constructed uncountably many non-cocycle conjugate E_0-semigroups for each n>1.

Established a new framework for inducing E_0-semigroups from boundary weight doubles.

Extended the class of known E_0-semigroups through this novel construction.

Abstract

Powers has shown that each spatial E_0-semigroup can be obtained from the boundary weight map of a CP-flow acting on B(K \otimes L^2(0, \infty)) for some separable Hilbert space K. In this paper, we define boundary weight maps through boundary weight doubles (\phi, \nu), where \phi: M_n(\C) \to M_n(\C) is a q-positive map and \nu is a boundary weight over L^2(0, \infty). These doubles induce CP-flows over K for 1<dim(K)<\infty which then minimally dilate to E_0-semigroups by a theorem of Bhat. Through this construction, we obtain uncountably many mutually non-cocycle conjugate E_0-semigroups for each n>1, n \in \mathbb{N}.