Maps on noncommutative Orlicz spaces

Abstract

A generalization of the Pistone-Sempi argument, demonstrating the utility of non-commutative Orlicz spaces, is presented. The question of lifting positive maps defined on von Neumann algebra to maps on corresponding noncommutative Orlicz spaces is discussed. In particular, we describe those Jordan * -morphisms on semifinite von Neumann algebras which in a canonical way induce quantum composition operators on noncommutative Orlicz spaces. Consequently, it is proved that the framework of noncommutative Orlicz spaces is well suited for an analysis of a large class of interesting noncommutative dynamical systems.

Multiplication operators on non-commutative spaces

“…Proposition 2.6. [22] Let ϕ be an Orlicz function and ϕ * its complementary function. Then L ϕ * (τ ), equipped with the norm · 0 ϕ * defined for x ∈ L ϕ * (τ ) by…”

“…[22] Let ϕ be an Orlicz function and x ∈ S(A, τ ). There exists some α > 0 such that ∞ 0 ϕ(αµ x (t))dt < ∞ if and only if there exists some β > 0 such that ϕ(β|x|) ∈ S(A, τ ) and τ (ϕ(β|x|)) < ∞.…”

An Alternative Approach to Weighted Non-commutative Banach Function Spaces

“…The theory of non-commutative Banach function spaces has become a well developed and important area in real analysis. The papers of Fack and Kosaki [5], Dodds, Dodds and de Pagter [2], and, more recently, Labuschagne and Majewski [6] are of particular interest for the context of this paper.…”

“…The authors went on to prove that the weighted non-commutative Banach function spaces are in fact Banach spaces that inject continuously into M [6,Theorem 3.7]. Throughout the proof the map τ x : M → R : a → ∞ 0 µ t (a)µ t (x)dt was used implicitly.…”

“…Once we have established this relationship, we will define a weighted non-commutative decreasing rearrangement based on the map τ x . In [6] the authors assumed x ∈ L 1 ( M, τ ), but we will relax this assumption to x ∈ L 1 ( M, τ ) + M.…”