Iterating the Cuntz-Nica-Pimsner construction for compactly aligned product systems

Abstract

In this article we study how decompositions of a quasi-lattice ordered group $(G,P)$ relate to decompositions of the Nica-Toeplitz algebra ${\mathcal{N}\mathcal{T}}_{\mathbf{X}}$ and Cuntz-Nica-Pimsner algebra ${\mathcal{N}\mathcal{O}}_{\mathbf{X}}$ of a compactly aligned product system $\mathbf{X}$ over $P$. In particular, we are interested in the situation where $(G,P)$ may be realised as the semidirect product of quasi-lattice ordered groups. Our results generalise Deaconu's work on iterated Toeplitz and Cuntz-Pimsner algebras - we show that the Nica-Toeplitz algebra and Cuntz-Nica-Pimsner algebra of a compactly aligned product system over ${\mathbb{N}}^k$ may be realised as $k$-times iterated Toeplitz and Cuntz-Pimsner algebras respectively.

63 pages, added Subsection 5.3: Examples

References in corpus (3)

• C*-algebras associated to product systems of Hilbert bimodules
• Iterating the Pimsner construction
• Product-system models for twisted $C^{\ast }$-algebras of topological higher-rank graphs

Cited by in corpus (3)

• Toeplitz extensions in noncommutative topology and mathematical physics
• Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras
• Product systems and their representations: an approach using Fock spaces and Fell bundles