Operator system structures on the unital direct sum of $C^*$-algebras

Abstract

This work is motivated by R\u{a}dulescu's result~\cite{R˘ad04} on the comparison of $C^*$-tensor norms on $C^*(\F_n)\otimes C^*(\F_n)$.

For unital $C^*$-algebras $A$ and $B$, there are natural inclusions of $A$ and $B$ into the unital free product $A\ast_1 B$, the maximal tensor product $A\otimes_{\max} B$ and the minimal tensor product $A\otimes_{\min} B$. These inclusions define three operator system structures on the internal sum $A+B$. Partly using ideas from quantum entanglement theory, we prove various interrelations between these three operator systems. As an application, the present results yield a significant improvement over R\u{a}dulescu's bound. At the same time, this tight comparison is so general that it cannot be regarded as evidence for the QWEP conjecture.