论文标题
关于操作员$ \ mathcal {a} $的副本 - 系统
On coproducts of operator $\mathcal{A}$-systems
论文作者
论文摘要
给定一个Unital $ \ boldsymbol {c}^{*} $ - algebra $ \ Mathcal {a} $，我们证明存在两个忠实的操作员$ \ Mathcal {a} $ - 系统。我们表明，我们可以将其视为$ \ boldsymbol {c}^{*} $代数的合并免费产品的子系统，或者是操作员系统内核的商。我们介绍了一个通用$ \ boldsymbol {c}^{*} $ - 操作员$ \ Mathcal {a} $ - 系统的代数 - 在两个操作员$ \ Mathcal {a} $ - 系统的共同体中，这是对$ \ Mathcal coffermative y Mathcal fromive ats y Mathcal a的coproduct of and Mathcal {a} $ { $ \ boldsymbol {c}^{*} $ - 代数。同样，在操作员系统的高量假设下，我们可以识别$ \ boldsymbol {c}^{*} $ - 与$ \ boldsymbol {c}^{**} $的$ \ boldsymbol {c}^{*} $ coproduct的自由产品的信封。我们将图形运算符系统视为运算符$ \ MATHCAL {A} $ - 系统的示例，并证明存在图形操作员系统的coproduct不是图形操作员系统，但是它是双重操作员$ \ MATHCAL {A a} $ - 系统。更一般而言，双运算符$ \ MATHCAL {a} $ - 系统始终是双运算符$ \ Mathcal {a} $ - 系统。我们表明，对操作员系统的归纳限制，相关行为表现良好。
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.
