Hilbert C*-bimodules of finite index and approximation properties of C*-algebras

TL;DR

This paper demonstrates that finite index Hilbert bimodules between C*-algebras ensure shared approximation properties like WEP, QWEP, and LLP, and explores their stability under Morita equivalence and examples involving crossed products.

Contribution

It establishes the link between finite index bimodules and the transfer of approximation properties, and analyzes stability under Morita equivalence and non-finite index cases.

Findings

Finite index bimodules ensure shared approximation properties.

Stability of WEP, QWEP, LLP under Morita equivalence.

Examples of bimodules not of finite index sharing properties.

Abstract

Let $A$ and $B$ be arbitrary $C^*$-algebras, we prove that the existence of a Hilbert $A$-$B$-bimodule of finite index ensures that the WEP, QWEP, and LLP along with other finite-dimensional approximation properties such as CBAP and (S)OAP are shared by $A$ and $B$. For this, we first study the stability of the WEP, QWEP and LLP under Morita equivalence of $C^*$-algebras. We present examples of Hilbert $A$-$B$-bimodules which are not of finite index, while such properties are shared between $A$ and $B$. To this end, we study twisted crossed products by amenable discrete groups.