Extensions of Hilbert bimodules and associated Cuntz-Pimsner algebras

TL;DR

This paper generalizes the concept of extensions from Hilbert modules to Hilbert bimodules and explores how these extensions influence the structure of associated Cuntz-Pimsner algebras, including multiplier bimodules.

Contribution

It introduces a new framework for extensions of Hilbert bimodules and demonstrates their impact on the structure of Cuntz-Pimsner algebras, including realizations as restricted direct-sum bimodules.

Findings

Extensions of Hilbert bimodules induce extensions of Cuntz-Pimsner algebras.

The Cuntz-Pimsner algebra of the multiplier bimodule can be characterized.

Extensions can be realized as restricted direct-sum bimodules.

Abstract

We extend the definition of an extension of a right Hilbert module to the setting of Hilbert bimodules and show that an extension of Hilbert bimodules induces an extension of Cuntz-Pimsner algebras. We also study the Cuntz-Pimsner algebra associated to the multiplier bimodule and show that an extension can be realised as a restricted direct-sum bimodule.