Journal of Operator Theory

Volume 69, Issue 1, Winter 2013 pp. 233-256.

A family of non-cocycle conjugate

$E_0$ -semigroups obtained from boundary weight doubles

Authors: Christopher Jankowski
Author institution: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva, 84105 Israel

Summary: Let

$\rho \in M_n(\C)^*$ and

$\rho' \in M_{n'}(\C)^*$ be states, and define unital

$q$ -positive maps

$\phi$ and

$\psi$ by

$\phi(A)=\rho(A)I_n$ and

$\psi(D) = \rho'(D)I_{n'}$ for all

$A \in M_n(\C)$ and

$D \in M_{n'}(\C)$ . We show that if

$\nu$ and

$\eta$ are type II Powers weights, then the boundary weight doubles

$(\phi, \nu)$ and

$(\psi, \eta)$ induce non-cocycle conjugate

$E_0$ -semigroups if

$\rho$ and

$\rho'$ have different eigenvalue lists. We then classify the

$q$ -corners and hyper maximal

$q$ -corners from

$\phi$ to

$\psi$ , finding that if

$\nu$ is a type II Powers weight of the form

$\nu(\sqrt{I - \Lambda(1)} B \sqrt{I - \Lambda(1)})=(f,Bf)$ , where

$\Lambda(1) \in B(L^2(0, \infty))$ is the operator of multiplication by

$\mathrm e^{-x}$ , then the

$E_0$ -semigroups induced by

$(\phi, \nu)$ and

$(\psi, \nu)$ are cocycle conjugate if and only if

$n=n'$ and

$\phi$ and

$\psi$ are conjugate.

DOI: http://dx.doi.org/10.7900/jot.2010oct06.1889
Keywords:

$E_0$ -semigroup, completely positive map,

$q$ -positive map

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