Quantum L_p and Orlicz spaces

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.