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

TL;DR

This paper establishes that the Christensen and Kadison-Kastler distances between $C^*$-subalgebras are preserved when passing to their associated von Neumann algebras or tensor products with commutative $C^*$-algebras, revealing fundamental relations among these distances.

Contribution

It proves the invariance of these distances under passage to enveloping von Neumann algebras and certain tensor products, clarifying their behavior in operator algebra theory.

Findings

Distances are equal between subalgebras and their enveloping von Neumann algebras.

Distances are preserved under tensoring with unital commutative $C^*$-algebras.

Results unify understanding of distances across different operator algebra contexts.

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 $\mathcal{D}$).