Symmetric operator spaces are generalizations of symmetric function spaces such as the classical (commutative) \(L^p\) -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 \(*\) -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.
