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

TL;DR

This paper investigates the structure of separable elements near the identity in bipartite C*-algebras, linking the size of neighborhoods to algebraic properties and resolving a recent conjecture.

Contribution

It introduces a method to estimate the size of separable neighborhoods using the completely bounded norm, extending finite-dimensional insights to general C*-algebras.

Findings

Characterized the size of separable neighborhoods in terms of algebraic rank.

Reduced the problem to estimating the completely bounded norm of positive maps.

Resolved a recent conjecture of Musat and Rordam.

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 R{\o}rdam.