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arXiv · 1706.04951

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

Bartosz K. Kwasniewski·Nadia S. Larsen
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Abstract

We prove uniqueness of representations of Nica-Toeplitz algebras associated to product systems of $C^*$-correspondences over right LCM semigroups by applying our previous abstract uniqueness results developed for $C^*$-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 $C^*$-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 $C^*$-algebras of right and left semidirect products.

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Bartosz K. Kwasniewski, Nadia S. Larsen. 2018-10-11. Nica-Toeplitz algebras associated with product systems over right LCM semigroups. https://doi.org/10.1016/j.jmaa.2018.10.020

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