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

Abstract

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

arXiv comment13 pages