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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