Tensor products of weak hyperrigid sets

Abstract

In this article, we show that concerning the spatial tensor product of W∗-algebras, the tensor product of two weak hyperrigid operator systems is weak hyperrigid. We prove this result by demonstrating unital completely positive maps have unique extension property for operator systems if and only if the tensor product of two unital completely positive maps has unique extension property for the tensor product of operator systems. Consequently, we prove as a corollary that the tensor product of two boundary representations for operator systems is boundary representation for the tensor product of operator systems. The corollary is an analogue result of Hopenwasser’s [9] in the setting of W∗-algebras.