A Markov dilation for self-adjoint Schur multipliers

Abstract

Abstract. We give a formula for Markov dilation in the sense of Anantharaman-Delaroche for real positive Schur multipliers on B(H).The classical theory of semigroups has many applications and connections with ergodic theory, martingales and probability (see [12]). The recent developments of noncommutative integration in von Neumann provide analogues of these notions ([1], [5], [6]). For instance, classical Markov semigroups on a probability space are generalized to semigroups of unital completely positive maps preserving a given faithful state. It is natural to try to adapt techniques from the commutative theory to the noncommutative one. Dealing with C *

BMO-estimates for non-commutative vector valued Lipschitz functions

“…In what follows, this fact, together with Theorem 2.5 allows us to interpolate between respective BMO−space and L 2 −space. The crucial part of the argument is similar to that of Ricard [Ric08]. Proof.…”

Harmonic analysis and BMO-spaces of free Araki--Woods factors

“…In this Section we show that the radial semi-group on free Araki-Woods factors has a good reversed Markov dilation. The first step in the proof of Proposition 5.10 is due to Ricard (see the final remarks of [Ric08]). We need to find a suitable analogue for semi-groups which we do by an ultraproduct argument.…”

Matrix N-dilations of quantum channels

“…The question of whether the closure of the set of matrix factorizable correlation matrices is the same as the set of factorizable correlation matrices is equivalent to Connes' embedding conjecture [7]. One wide class of Schur product channels that are known to be factorizable is the set of Schur product channels arising from correlation matrices with all real entries [5] [3]. Such a channel always admits a factorization by means of trace-orthogonal, anti-commuting unitaries.…”