On biduals of 𝐶*-tensor products

Abstract

Huruya [4] has proven that, for C ∗ {C^*} -algebras A 1 {A_1} and A 2 {A_2} , \[ ( A 1 ⊗ A 2 ) ∗ ∗ = A 1 ∗ ∗ ⊗ ¯ A 2 ∗ ∗ {({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**} \] for every A 2 {A_2} if and only if A 1 {A_1} is scattered. We strengthen this by proving that ( A 1 ⊗ A 2 ) ∗ ∗ = A 1 ∗ ∗ ⊗ ¯ A 2 ∗ ∗ {({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**} if and only if A 1 {A_1} or A 2 {A_2} is scattered. We discuss ramifications to representation theory and related questions regarding normal representations of W ∗ {W^*} -tensor products.