Ideal structure and simplicity of the C*-algebras generated by Hilbert bimodules

TL;DR

This paper investigates conditions under which C*-algebras generated by Hilbert bimodules are simple or have specific ideal structures, introducing (I)-freeness and (II)-freeness as key criteria.

Contribution

It introduces (I)-freeness and (II)-freeness conditions for Hilbert bimodules, linking them to simplicity and ideal structure of the generated C*-algebras.

Findings

(I)-freeness ensures independence of generators in O_X when A is faithfully represented.

If X is (I)-free and A is X-simple, then O_X is simple.

(II)-freeness) determines the ideal structure of O_X.

Abstract

Pimsner introduced the C*-algebra O_X generated by a Hilbert bimodule X over a C*-algebra A. We look for additional conditions that X should satisfy in order to study simplicity and, more generally, the ideal structure of O_X when X is finite projective. We introduce two conditions: `(I)-freeness' and `(II)-freeness', stronger than the former, in analogy with [J. Cuntz, W. Krieger, Invent. Math. 56, 251-268] and [J. Cuntz, Invent. Math. 63, 25-40] respectively. (I)-freeness comprehend the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite dimension and the case of `Cuntz-Krieger bimodules'. If X satisfies this condition the C*-algebra O_X does not depend on the choice of the generators when A is faithfully represented. As a consequence, if X is (I)-free and A is X-simple, then O_X is simple. In the case of Cuntz-Krieger algebras, X-simplicity corresponds to irreducibility of the defining matrix. If A is simple and p.i. then O_X is p.i., if A is nonnuclear then O_X is nonnuclear. We therefore provide examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore if X is (II)-free, we determine the ideal structure of O_X.