Relations amongst the distances between $C^{*}$-subalgebras and some canonically associated operator algebras

Abstract

We prove that the Christensen distance (resp., the Kadison-Kastler distance) between two $C^*$-subalgebras $\mathcal{A}$ and $\mathcal{B}$ of a $C^*$-algebra $\mathcal{C}$ is equal to that between their enveloping von Neumann algebras $\mathcal{A}^{**}$ and $\mathcal{B}^{**}$ (resp., the tensor product algebras $\mathcal{A} \otimes^{\min} \mathcal{D}$ and $\mathcal{B} \otimes^{\min} \mathcal{D}$, for any unital commutative $C^*$-algebra $\mathcal{D}$).