A Note on the Quantum Family of Maps

TL;DR

This paper extends the concept of quantum mapping spaces from finite-dimensional C*-algebras to general compact Hausdorff spaces, incorporating topology and introducing a free product construction.

Contribution

It proposes a modified framework for quantum maps that applies to continuous spaces, broadening the scope beyond finite sets.

Findings

Introduces a new notion of quantum maps for $C(X)$ with topology.

Defines a free product of C*-algebras indexed by compact spaces.

Establishes functorial properties of the new construction.

Abstract

The notion and theory of the quantum space of all maps from a quantum space pioneered by So{\l}tan have been mainly focused on finite-dimensional C*-algebras which are matrix algebra bundles over a finite set $S$. We propose a modification of this notion to cover the case of $C\left( X\right) $ for general compact Hausdorff spaces $X$ instead of finite sets $S$ while taking into account of the topology of $X$. A notion of free product of copies of a unital C*-algebra topologically indexed by a compact Hausdorff space arises naturally, and satisfies some desired functoriality.