On coproducts of operator $\mathcal{A}$-systems
Alexandros Chatzinikolaou
TL;DR
This work develops a coherent theory of coproducts in the category of operator $\mathcal{A}$-systems, proving existence for faithful objects and providing two concrete realizations as subsystems of amalgamated free products or as quotients by operator-system kernels. It introduces the universal $\boldsymbol{C}^{*}$-algebra $C^{*}_{u,\mathcal{A}}(\mathcal{S})$ and shows coproducts align with amalgamated free products of universal algebras, while hyperrigidity ensures the $\boldsymbol{C}^{*}$-envelopes behave compatibly via $C^{*}_{e}(\mathcal{S}_{1} \oplus_{\mathcal{A}} \mathcal{S}_{2}) \cong C^{*}_{e}(\mathcal{S}_{1}) *_{\mathcal{A}} C^{*}_{e}(\mathcal{S}_{2})$. The paper also analyzes graph operator systems, demonstrates that coproducts of dual operator $\mathcal{A}$-systems remain dual, and proves that coproducts are stable under inductive limits, thereby providing a robust framework for coproducts in this enriched category.
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.