Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples

TL;DR

This paper investigates the structure of maximal commutative subalgebras invariant under CP-maps and semigroups, providing counterexamples and new insights into their invariance properties in operator algebras.

Contribution

It offers new counterexamples and structural analysis of invariant subalgebras under CP-maps, addressing open questions in operator algebra theory.

Findings

Existence of Markov CP-semigroups on M_d without invariant maximal commutative subalgebras for d>2

Counterexamples to natural questions about invariant subalgebras

Structural results on generators of norm continuous semigroups leaving subalgebras invariant

Abstract

We solve, mainly by counterexamples, many natural questions regarding maximal commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a von Neumann algebra. In particular, we discuss the structure of the generators of norm continuous semigroups on B(G) leaving a maximal commutative subalgebra invariant and show that there exists Markov CP-semigroups on M_d without invariant maximal commutative subalgebras for any d>2.