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.

Keywords: $E_0$-semigroup, completely positive map, $q$-positive map

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