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

TL;DR

This paper analyzes how two canonical distances between $C^*$-subalgebras, the Kadison-Kastler distance $d_{KK}$ and Christensen distance $d_0$, behave under passing to enveloping von Neumann algebras and under minimal tensoring with a commutative $C^*$-algebra. It proves that $d_{0}(\mathcal{A},\mathcal{B}) = d_{0}(\mathcal{A}^{**},\mathcal{B}^{**})$ and $d_{KK}(\mathcal{M},\mathcal{N}) = d_{KK}(\mathcal{M}^{**},\mathcal{N}^{**})$ for von Neumann subalgebras, affirming a tight relation between subalgebra perturbations and their biduals. For tensoring, it shows that $d_{0}(\mathcal{A} \otimes^{\min} \mathcal{D}, \mathcal{B} \otimes^{\min} \mathcal{D})$ equals $d_{0}(\mathcal{A},\mathcal{B})$ when $\mathcal{D}$ is commutative and unital (and $\le d_{0}(\mathcal{A},\mathcal{B})$ in general), with analogous results for $d_{KK}$; further, when $\mathcal{D}$ is commutative, $d_{KK}(\mathcal{A} \otimes^{\min} \mathcal{D}, \mathcal{B} \otimes^{\min} \mathcal{D}) \le d_{KK}(\mathcal{A},\mathcal{B})$ and equality holds if $\mathcal{D}$ is unital. The paper also treats the case of scattered $\mathcal{D}$, showing $d_{0}(\mathcal{A},\mathcal{B}) \le d_{0}(\mathcal{A} \otimes^{\min} \mathcal{D}, \mathcal{B} \otimes^{\min} \mathcal{D})$ with equality when $\mathcal{D}$ is commutative, and uses the structure $(\mathcal{C} \otimes^{\min} \mathcal{D})^{**} \cong \mathcal{C}^{**} \bar{\otimes} \mathcal{D}^{**}$ for scattered $\mathcal{D}$ to drive the results.

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 $\mathcal{D}$).

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Lemma 2.2
• Remark 2.3
• Proposition 2.4
• Proposition 2.5
• Remark 2.6