Topological freeness for Hilbert bimodules

TL;DR

This paper demonstrates that topological freeness in the context of Hilbert bimodules ensures the isomorphism of certain $C^*$-algebras to crossed products, providing ideal lattice descriptions and simplicity criteria.

Contribution

It establishes a link between topological freeness of Rieffel's induced functor and the structure of $C^*$-algebras generated by Hilbert bimodules, including isomorphism and simplicity conditions.

Findings

Topological freeness implies isomorphism to crossed product $A\rtimes_X \mathbb{Z}$.

Provides an ideal lattice description for the crossed product.

Offers a simplicity criterion for the crossed product.

Abstract

It is shown that topological freeness of Rieffel's induced representation functor implies that any $C^*$-algebra generated by a faithful covariant representation of a Hilbert bimodule $X$ over a $C^*$-algebra $A$ is canonically isomorphic to the crossed product $A\rtimes_X \mathbb{Z}$. An ideal lattice description and a simplicity criterion for $A\rtimes_X \mathbb{Z}$ are established.