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

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.Comment: 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. App

Nica-Toeplitz algebras associated with right-tensor C*-precategories over right LCM semigroups

We introduce and analyze the full [Formula: see text] and the reduced [Formula: see text] Nica–Toeplitz algebra associated to an ideal [Formula: see text] in a right-tensor [Formula: see text]-precategory [Formula: see text] over a right LCM semigroup [Formula: see text]. These [Formula: see text]-algebras unify cross-sectional [Formula: see text]-algebras associated to Fell bundles over discrete groups and Nica–Toeplitz [Formula: see text]-algebras associated to product systems. They also allow a study of Doplicher–Roberts versions of the latter. A new phenomenon is that when [Formula: see text] is not right cancellative then the canonical conditional expectation takes values outside the ambient algebra. Our main result is a uniqueness theorem that gives sufficient conditions for a representation of [Formula: see text] to generate a [Formula: see text]-algebra naturally lying between [Formula: see text] and [Formula: see text]. We also characterize the situation when [Formula: see text]. Unlike previous results for quasi-lattice monoids, [Formula: see text] is allowed to contain nontrivial invertible elements, and we accommodate this by identifying an assumption of aperiodicity of an action of the group of invertible elements in [Formula: see text]. One prominent condition for uniqueness is a geometric condition of Coburn’s type, exploited in the work of Fowler, Laca and Raeburn. Here we shed new light on the role of this condition by relating it to a [Formula: see text]-algebra associated to [Formula: see text] itself

Topological freeness for C*-correspondences

We study conditions that ensure uniqueness theorems of Cuntz–Krieger type for relative Cuntz–Pimsner algebras associated to a ⁎-correspondence X over a ⁎-algebra A. We give general sufficient conditions phrased in terms of a multivalued map acting on the spectrum of A. When is of Type I we construct a directed graph dual to X and prove a uniqueness theorem using this graph. When is liminal, we show that topological freeness of this graph is equivalent to the uniqueness property for , as well as to an algebraic condition which we call J-acyclicity of X. As an application we improve the Fowler–Raeburn uniqueness theorem for the Toeplitz algebra . We give new simplicity criteria for . We generalize and enhance uniqueness results for relative quiver ⁎-algebras of Muhly and Tomforde. We also discuss applications to crossed products by endomorphisms