Nica-Toeplitz algebras associated with product systems over right LCM semigroups

Abstract

We prove uniqueness of representations of Nica-Toeplitz algebras associated to product systems of $C^{\ast }$-correspondences over right LCM semigroups by applying our previous abstract uniqueness results developed for $C^{\ast }$-precategories. Our results provide an interpretation of conditions identified in work of Fowler and Fowler-Raeburn, and apply also to their crossed product twisted by a product system, in the new context of right LCM semigroups, as well as to a new, Doplicher-Roberts type $C^{\ast }$-algebra associated to the Nica-Toeplitz algebra. As a derived construction we develop Nica-Toeplitz crossed products by actions with completely positive maps. This provides a unified framework for Nica-Toeplitz semigroup crossed products by endomorphisms and by transfer operators. We illustrate these two classes of examples with semigroup $C^{\ast }$-algebras of right and left semidirect products.

Title changed from "Nica-Toeplitz algebras associated with right tensor C*-precategories over right LCM semigroups: part II examples". The manuscript accepted in J. Math. Anal. Appl

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Cited by in corpus (6)

• C*-envelopes of tensor algebras of product systems
• Co-universality and controlled maps on product systems over right LCM-semigroups
• Product-system models for twisted $C^{\ast }$-algebras of topological higher-rank graphs
• Topological freeness for $C^{\ast }$-correspondences
• Zappa-Szép actions of groups on product systems
• Product systems and their representations: an approach using Fock spaces and Fell bundles