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

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In this article we study how decompositions of a quasi-lattice ordered group $(G,P)$ relate to decompositions of the Nica-Toeplitz algebra NTX and Cuntz-Nica-Pimsner algebra NOX of a compactly aligned product system 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 Nᵏ may be realised as k-times iterated Toeplitz and Cuntz-Pimsner algebras respectively.