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 * -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. A x, y · z = x · y, z A for all x, y, z ∈ X. An equivalence A-bimodule is a Hilbert bimodule which is full over left and right. We say that two Hilbert A-bimodules X, Y are Morita equivalent if there is an equivalence A-bimodule E such that X ⊗ A E ∼ = Y ⊗ A E. Remark 1.3. A product system X is (left) essential if each C

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

“…In special cases such structures were considered in [19,Example 3.2], [22,Subsection 3.1]. We give a detailed analysis of this example in [20] where we also explain how the results of the present paper give a new insight to C * -algebras associated with X. Another somewhat trivial but important and instructive example is when P = G is a group.…”

“…Plainly, if K is a ⊗1-invariant ideal in a right-tensor C * -precategory L, then K is itself a righttensor C * -precategory, and K is well-aligned both in L and in K. The condition in (3.2) is a generalization of the notion of compact alignment for product systems of C * -correspondences from [14, Definition 5.7], cf. [6], and see [20] for details. The next lemma shows that (3.2) captures more than just diagonal fibres K(p, p) for p ∈ P .…”

“…Then K X = {K(X p , X q )} p,q∈P is an essential ideal in L X . By [20,Proposition 2.8] we have a natural isomorphism T L X (K X ) ∼ = T (X) where T (X) is the Toeplitz algebra of X defined in [14]. Assume that P is a right LCM semigroup.…”

“…Then K X is well-aligned if and only if X is compactly aligned. In this case, [20,Proposition 2.10] gives…”

“…By [21], if B e is separable or of Type I, apperiodicity of B is equivalent to topological freeness of the dual partial action. Non-trivial examples where the aperiodicity condition holds come for instance from wreath products, see [20].…”