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

Abstract Symmetric operator spaces are generalizations of symmetric function spaces such as the classical (commutative) $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -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 $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> -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.