Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.

KMS States on Nica-Toeplitz $$\hbox {C}^*$$-algebras

Given a quasi-lattice ordered group (G, P) and a compactly aligned product system X of essential $$\hbox {C}^*$$ C ∗ -correspondences over the monoid P, we show that there is a bijection between the gauge-invariant $$\hbox {KMS}_\beta $$ KMS β -states on the Nica-Toeplitz algebra $$\mathcal {NT}(X)$$ NT ( X ) of X with respect to a gauge-type dynamics, on one side, and the tracial states on the coefficient algebra A satisfying a system (in general infinite) of inequalities, on the other. This strengthens and generalizes a number of results in the literature in several directions: we do not make any extra assumptions on P and X, and our result can, in principle, be used to study KMS-states at any finite inverse temperature $$\beta $$ β . Under fairly general additional assumptions we show that there is a critical inverse temperature $$\beta _c$$ β c such that for $$\beta >\beta _c$$ β > β c all $$\hbox {KMS}_\beta $$ KMS β -states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of $$\hbox {KMS}_\beta $$ KMS β -states in terms of tracial states on A, while at $$\beta =\beta _c$$ β = β c we have a phase transition manifesting itself in the appearance of $$\hbox {KMS}_\beta $$ KMS β -states that are not of Gibbs type. In the case of right-angled Artin monoids we show also that our system of inequalities for traces on A can be reduced to a much smaller system, a finite one when the monoid is finitely generated. Most of our results generalize to arbitrary quasi-free dynamics on $$\mathcal {NT}(X)$$ NT ( X ) .

Reconstructing directed graphs from generalized gauge actions on their Toeplitz algebras

We show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalized gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.

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.