2,265 research outputs found
Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples
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.Comment: After the elemenitation in Version 2 of a false class of examples in
Version 1, we now provide also correct examples for unital CP-maps and Markov
semigroups on M_d for d>2 without invariant masa
Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
We study dynamical optimal transport metrics between density matrices
associated to symmetric Dirichlet forms on finite-dimensional -algebras.
Our setting covers arbitrary skew-derivations and it provides a unified
framework that simultaneously generalizes recently constructed transport
metrics for Markov chains, Lindblad equations, and the Fermi
Ornstein--Uhlenbeck semigroup. We develop a non-nommutative differential
calculus that allows us to obtain non-commutative Ricci curvature bounds,
logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral
gap estimates
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words
E-semigroups Subordinate to CCR Flows
The subordinate E-semigroups of a fixed E-semigroup are in one-to-one
correspondence with local projection-valued cocycles of that semigroup. For the
CCR flow we characterise these cocycles in terms of their stochastic
generators, that is, in terms of the coefficient driving the quantum stochastic
differential equation of Hudson-Parthasarathy type that such cocycles
necessarily satisfy. In addition various equivalence relations and order-type
relations on E-semigroups are considered, and shown to work especially well in
the case of those semigroups subordinate to the CCR flows by exploiting our
characterisation.Comment: 14 pages; to appear in Communications on Stochastic Analysis. Minor
modifications made from version
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