Quantum Functions

TL;DR

This paper shows that quantum functions between von Neumann algebras are equivalent to normal unital *-homomorphisms, providing a reformulation within the framework of quantum relations and von Neumann modules.

Contribution

It establishes the equivalence between quantum functions and *-homomorphisms, connecting quantum relation concepts with classical algebraic structures.

Findings

Quantum functions coincide with normal unital *-homomorphisms.

Reformulation aligns quantum relations with known algebraic morphisms.

Builds on and clarifies previous results in von Neumann algebra theory.

Abstract

Weaver has recently defined the notion of a quantum relation on a von Neumann algebra. We demonstrate that the corresponding notion of a quantum function between two von Neumann algebras coincides with that of a normal unital $*$-homomorphism in the opposite direction. This is essentially a reformulation of a previously known result from the theory of Hilbert von Neumann modules.