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arXiv:1411.0577 · Infinite Dimensional Analysis Quantum Probability and Related Topics

The algebraic structure of quantum partial isometries

Teodor Banica

Published Mar 1, 2016• 5 citations
arXivHTMLPDF

Abstract

The partial isometries of [Formula: see text] form compact semigroups [Formula: see text]. We discuss here the liberation question for these semigroups, and for their discrete versions [Formula: see text]. Our main results concern the construction of half-liberations [Formula: see text] and of liberations [Formula: see text]. We include a detailed algebraic and probabilistic study of all these objects, justifying our “half-liberation” and “liberation” claims.

DOI
10.1142/s021902571650003x

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The partial isometries of [Formula: see text] form compact semigroups [Formula: see text]. We discuss here the liberation question for these semigroups, and for their discrete versions [Formula: see text]. Our main results concern the construction of half-liberations [Formula: see text] and of liberations [Formula: see text]. We include a detailed algebraic and probabilistic study of all these objects, justifying our “half-liberation” and “liberation” claims.
In this study, we first deal with the canonical anti-commutation relation algebra in which the Bernoulli creation and annihilation operators generate the structure of ultra hyperfinite C[Formula: see text]-algebra. This observation simplifies the construction of the minimal quantum Markov semigroup construct from quantum Bernoulli noises by taking a filter through the family gamma of finite subsets of the natural numbers. As a result, we obtain a Feller quantum Markov semigroup defined on the C[Formula: see text]-algebra, which in turn is extended to the entire algebra of bounded operators. Then, one condition for us to obtain a faithful normal invariant state of this semigroup is an equilibrium state (it satisfies the detailed balance conditions). Finally, we use the quantum Wasserstein distance to discuss the W-entropy rate of this invariant state.
The problem of characterizing GKLS-generators and CP-maps with an invariant von Neumann algebra [Formula: see text] appeared in different guises in the literature. We prove two unifying results, which hold even for weakly closed *-algebras: first, we show how to construct a normal form for [Formula: see text]-invariant GKLS-generators, if a normal form for [Formula: see text]-invariant CP-maps is known — rendering the two problems essentially equivalent. Second, we provide a normal form for [Formula: see text]-invariant CP-maps if [Formula: see text] is atomic (which includes the finite-dimensional case). As an application we reproduce several results from the literature as direct consequences of our characterizations and thereby point out connections between different fields.
This paper deals with the asymptotic behavior of a quantum dynamical semigroup [Formula: see text] acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a generator which can be written in a well-known form named after Lindblad and Davies. If the semigroup has a faithful normal stationary state ρ, necessary and sufficient conditions are derived for the w*-convergence of [Formula: see text] to [Formula: see text], where [Formula: see text] is the conditional expectation of an element X onto the subalgebra of fixed points. Our main results are expressed in terms of the Lindblad–Davies generator .