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On stability of distance under some tensor products and some calculations

Sumit Kumar

TL;DR

This work analyzes how three notions of proximity between subalgebras—Kadison–Kastler distance $d_{KK}$, Christensen distance $d_0$, and Mashood–Taylor distance $d_{MT}$—behave under non-operator tensor products. It proves that $d_{KK}$ and $d_0$ are stable under the Banach-space injective tensor product with a unital commutative C*-algebra (and similarly under related constructions for subspaces), and under the Banach-space projective tensor product with a unital C*-algebra for subalgebras, with equalities when the tensor factor is unital. The paper also provides explicit computations in crossed-product settings, showing that for reduced twisted crossed products and for crossed-product von Neumann algebras, the distances between subalgebras associated to distinct subgroups are maximal, i.e., equal to 1, and that $d_{MT}$ is also maximal in the finite-trace case. Collectively, these results extend perturbation-type stability phenomena from operator algebras to stable behavior under Banach-space tensor products and provide concrete distance calculations in crossed-product constructions, highlighting robustness of these distance notions under natural tensor operations.

Abstract

We prove that the Kadison-Kastler and Christensen distances are stable under the Banach space injective tensor product (resp., the Banach space projective tensor product) of a Banach space with any unital commutative $C^*$-algebra (resp., of a $C^*$-algebra with any unital $C^*$-algebra). Apart from these stability results, we make some explicit calculations of the Kadison-Kastler, Christensen and Mashood-Taylor distances between certain subalgebras of some crossed-product operator algebras.

On stability of distance under some tensor products and some calculations

TL;DR

This work analyzes how three notions of proximity between subalgebras—Kadison–Kastler distance , Christensen distance , and Mashood–Taylor distance —behave under non-operator tensor products. It proves that and are stable under the Banach-space injective tensor product with a unital commutative C*-algebra (and similarly under related constructions for subspaces), and under the Banach-space projective tensor product with a unital C*-algebra for subalgebras, with equalities when the tensor factor is unital. The paper also provides explicit computations in crossed-product settings, showing that for reduced twisted crossed products and for crossed-product von Neumann algebras, the distances between subalgebras associated to distinct subgroups are maximal, i.e., equal to 1, and that is also maximal in the finite-trace case. Collectively, these results extend perturbation-type stability phenomena from operator algebras to stable behavior under Banach-space tensor products and provide concrete distance calculations in crossed-product constructions, highlighting robustness of these distance notions under natural tensor operations.

Abstract

We prove that the Kadison-Kastler and Christensen distances are stable under the Banach space injective tensor product (resp., the Banach space projective tensor product) of a Banach space with any unital commutative -algebra (resp., of a -algebra with any unital -algebra). Apart from these stability results, we make some explicit calculations of the Kadison-Kastler, Christensen and Mashood-Taylor distances between certain subalgebras of some crossed-product operator algebras.
Paper Structure (14 sections, 21 theorems, 71 equations)

This paper contains 14 sections, 21 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two notions of distance between subspaces of normed spaces
  4. Kadison-Kastler distance
  5. Near inclusions and Christensen distance
  6. Mashood-Taylor distance
  7. Distance between subspaces of injective tensor product of certain Banach spaces
  8. Injective tensor product
  9. Distance between subalgebras of Banach space projective tensor product of $C^*$-algebras
  10. Banach space projective tensor product
  11. Some concrete calculations
  12. Distance between subalgebras of reduced twisted crossed product $C^*$-algebras
  13. Reduced twisted crossed product
  14. Distance between subalgebras of crossed-product von Neumann algebras

Key Result

Theorem 1.1

Ch1 Let $\mathcal{C}$ be a $C^*$-algebra with $C^*$-subalgebras $\mathcal{A}$ and $\mathcal{B}$, and let $\mathcal{D}$ be a nuclear $C^*$-algebra. If $\mathcal{A}$ has property $D_{k}$ (for some $k\in (0, \infty))$ and $\mathcal{A} \subseteq_{\gamma} \mathcal{B}$, then $\mathcal{A} \otimes^{\min} \m

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 30 more