Analogues of composition operators in the setting of non-commutative symmetric spaces

Symmetric operator spaces are generalizations of symmetric function spaces such as the classical (commutative) Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>p$$\end{document}-spaces, Orlicz spaces, Lorentz spaces and Banach function spaces. In this setting of (potentially) non-commutative symmetric operator spaces we investigate analogues of composition operators, which are also called quantum composition operators. In particular, we provide sufficient conditions under which a Jordan & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-homomorphism induces a quantum composition operator between non-commutative symmetric spaces and we characterize those bounded operators between non-commutative symmetric spaces that are quantum composition operators. Furthermore, compactness conditions of quantum composition operators are investigated.