Quantum Heisenberg Manifolds as Twisted Groupoid $C^*$-Algebras

TL;DR

This paper demonstrates that quantum Heisenberg manifolds can be represented as twisted groupoid $C^*$-algebras, providing a new perspective on their structure and linking them to groupoid theory.

Contribution

It introduces a novel realization of quantum Heisenberg manifolds as twisted groupoid $C^*$-algebras, expanding their theoretical framework.

Findings

Quantum Heisenberg manifolds are realizable as twisted groupoid $C^*$-algebras.

Provides a new perspective connecting noncommutative manifolds and groupoid $C^*$-algebras.

Enhances understanding of the structure and classification of quantum Heisenberg manifolds.

Abstract

The quantum Heisenberg manifolds are noncommutive manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds as the generalized fixed-point algebras of certain crossed product $C^*$-algebras, and they also can be realized as crossed products of $C(\mathbb{T}^2)$ by Hilbert $C^*$-bimodules in the sense of Abadie et al. In this paper, we describe how the quantum Heisenberg manifolds can also be realized as twisted groupoid $C^*$-algebras.