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Separable neighbourhood of identity in C$^{\ast}$-algebras

Mizanur Rahaman, Mateusz Wasilewski

Abstract

We study the structure of separable elements in bipartite C$^{\ast}$-algebras, focusing on the existence and size of a separable neighbourhood around the identity element. While this phenomenon is well understood in the finite-dimensional setting, its extension to general C$^{\ast}$-algebras presents additional challenges. We show that the problem of determining such a neighbourhood can be reduced to estimating the completely bounded norm of contractive positive maps. This approach allows us to characterize the size of such neighbourhoods in terms of structural properties of the algebra, notably its rank. As a consequence, we also resolve a recent conjecture of Musat and Rørdam.

Separable neighbourhood of identity in C$^{\ast}$-algebras

Abstract

We study the structure of separable elements in bipartite C-algebras, focusing on the existence and size of a separable neighbourhood around the identity element. While this phenomenon is well understood in the finite-dimensional setting, its extension to general C-algebras presents additional challenges. We show that the problem of determining such a neighbourhood can be reduced to estimating the completely bounded norm of contractive positive maps. This approach allows us to characterize the size of such neighbourhoods in terms of structural properties of the algebra, notably its rank. As a consequence, we also resolve a recent conjecture of Musat and Rørdam.

Paper Structure

This paper contains 8 sections, 27 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The matrix algebra case
  4. Horodeckis criterion for von Neumann algebras
  5. The C$^*$-algebra case
  6. Relation with the rank of subhomogeneous algebras
  7. A question of Musat-Rørdam in MusatRordam
  8. Acknowledgments

Key Result

Theorem 1

For two unital C$^*$-algebras $A, B$, let If both have infinite rank, then $\gamma(A, B)=0$. In the case where at least one has a finite rank we have where $\eta(A, B) :=\sup\{\|\Phi\|_{cb}: \Phi: A\rightarrow B, \ \text{contractive positive map}\}.$

Theorems & Definitions (61)

  • Theorem
  • Definition 2.1
  • Lemma 2.2: BrownOzawa
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3: MR1421501
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • ...and 51 more