On coproducts of operator $\mathcal{A}$-systems

TL;DR

This paper establishes the existence and properties of coproducts of faithful operator -systems, linking them to free products of -algebras, and explores their structure, examples, and behavior under limits.

Contribution

It introduces a universal --system, characterizes coproducts via free products, and analyzes their structure and examples, including graph operator systems.

Findings

Coproducts of faithful operator -systems exist and relate to free products.

The --system coproduct is isomorphic to an amalgamated free product.

Coproducts of dual operator -systems are always dual operator -systems.

Abstract

Given a unital $\boldsymbol{C}^{*}$-algebra $\mathcal{A}$, we prove the existence of the coproduct of two faithful operator $\mathcal{A}$-systems. We show that we can either consider it as a subsystem of an amalgamated free product of $\boldsymbol{C}^{*}$-algebras, or as a quotient by an operator system kernel. We introduce a universal $\boldsymbol{C}^{*}$-algebra for operator $\mathcal{A}$-systems and prove that in the case of the coproduct of two operator $\mathcal{A}$-systems, it is isomorphic to the amalgamated over $\mathcal{A}$, free product of their respective universal $\boldsymbol{C}^{*}$-algebras. Also, under the assumptions of hyperrigidity for operator systems, we can identify the $\boldsymbol{C}^{*}$-envelope of the coproduct with the amalgamated free product of the $\boldsymbol{C}^{*}$-envelopes. We consider graph operator systems as examples of operator $\mathcal{A}$-systems and prove that there exist graph operator systems whose coproduct is not a graph operator system, it is however a dual operator $\mathcal{A}$-system. More generally, the coproduct of dual operator $\mathcal{A}$-systems is always a dual operator $\mathcal{A}$-system. We show that the coproducts behave well with respect to inductive limits of operator systems.