Quantum L_p and Orlicz spaces

TL;DR

This paper explores how quantum Orlicz spaces can be utilized to analyze quantum dynamical systems, extending traditional $L_p$ space techniques to more general non-commutative settings.

Contribution

It introduces the construction of quantum Orlicz spaces and discusses their application in studying quantum dynamical systems beyond classical $L_p$ frameworks.

Findings

Quantum Orlicz spaces generalize $L_p$ spaces for non-commutative analysis.

Techniques for lifting dynamical systems to quantum Orlicz spaces are developed.

The approach enhances the analysis of quantum dynamical systems in operator algebras.

Abstract

Let $\A$ ($\cM$) be a $C^*$-algebra (a von Neumann algebra respectively). By a quantum dynamical system we shall understand the pair $({\A}, T)$ ($({\cM}, T)$) where $T : {\A} \to {\A}$ ($T : {\cM} \to {\cM}$) is a linear, positive (normal respectively), and identity preserving map. In our lecture, we discuss how the techniques of quantum Orlicz spaces may be used to study quantum dynamical systems. To this end, we firstly give a brief exposition of the theory of quantum dynamical systems in quantum $L_p$ spaces. Secondly, we describe the Banach space approach to quantization of classical Orlicz spaces. We will discuss the necessity of the generalization of $L_p$-space techniques. Some emphasis will be put on the construction of non-commutative Orlicz spaces. The question of lifting dynamical systems defined on von Neumann algebra to a dynamical system defined in terms of quantum Orlicz space will be discussed.