There has recently been much interest in the $C^*$-algebras of directed graphs. Here we consider product systems $E$ of directed graphs over semigroups and associated $C^*$-algebras $C^*(E)$ and $\mathcal{T}C^*(E)$ which generalise the higher-rank graph algebras of Kumjian-Pask and their Toeplitz analogues. We study these algebras by constructing from $E$ a product system $X(E)$ of Hilbert bimodules, and applying recent results of Fowler about the Toeplitz algebras of such systems. Fowler's hypotheses turn out to be very interesting graph-theoretically, and indicate new relations which will have to be added to the usual Cuntz-Krieger relations to obtain a satisfactory theory of Cuntz-Krieger algebras for product systems of graphs; our algebras $C^*(E)$ and $\mathcal{T}C^*(E)$ are universal for families of partial isometries satisfying these relations.
Our main result is a uniqueness theorem for $\mathcal{T}C^*(E)$ which has particularly interesting implications for the $C^*$-algebras of non-row-finite higher-rank graphs. This theorem is apparently beyond the reach of Fowler's theory, and our proof requires a detailed analysis of the expectation onto the diagonal in $\mathcal{T}C^*(E)$.
For a row-finite higher-rank graph Λ, we construct a higher-rank graph TΛ such that the Toeplitz algebra of Λ is isomorphic to the Cuntz-Krieger algebra of TΛ. We then prove that the higher-rank graph TΛ is always aperiodic and use this fact to give another proof of a uniqueness theorem for the Toeplitz algebras of higher-rank graphs.
We consider a family of operator-algebraic dynamical systems involving the Toeplitz algebras of higher-rank graphs. We explicitly compute the KMS states (equilibrium states) of these systems built from small graphs with up to four connected components.
To a higher rank directed graph $(Λ, d)$, in the sense of Kumjian and Pask, one can associate natural noncommutative analytic Toeplitz algebras, both weakly closed and norm closed. We introduce methods for the classification of these algebras in the case of single vertex graphs.
We consider a family of higher-dimensional noncommutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori, and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz noncommutative solenoids are direct limits of the Toeplitz extensions of noncommutative tori. We consider natural dynamics on these Toeplitz algebras, and compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrised by the probability measures on an (ordinary) solenoid.
The Toeplitz algebra $\mathcal{T}C^{*}(Λ)$ for a finite $k$-graph $Λ$ is equipped with a continuous one-parameter group $α^{r}$ for each $ r\in \mathbb{R}^{k}$, obtained by composing the map $\mathbb{R} \ni t \to (e^{itr_{1}}, \dots , e^{itr_{k}}) \in \mathbb{T}^{k}$ with the gauge action on $\mathcal{T}C^{*}(Λ)$. In this paper we give a complete description of the $β$-KMS states for the $C^{*}$-dynamical system $(\mathcal{T}C^{*}(Λ), α^{r})$ for all finite $k$-graphs $Λ$ and all values of $β\in \mathbb{R}$ and $r\in \mathbb{R}^{k}$.
The Toeplitz algebra of a finite graph of rank $k$ carries a natural action of the torus ${\mathbb T}^k$, and composing with an embedding of ${\mathbb R}$ in ${\mathbb T}^k$ gives a dynamics on the Toeplitz algebra. For inverse temperatures larger than a critical value, the KMS states for this dynamics are well-understood, and this analysis is the first step in our program. At the critical inverse temperature, much less is known, and the second step in our program is an analysis of the KMS states at the critical value. This is the main technical contribution of the present paper. The third ste…
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 cr…
Fowler introduced the notion of a product system: a collection of Hilbert bimodules $\mathbf{X}=\left\{\mathbf{X}_p:p\in P\right\}$ indexed by a semigroup $P$, endowed with a multiplication implementing isomorphisms $\mathbf{X}_p\otimes_A \mathbf{X}_q\cong \mathbf{X}_{pq}$. When $P$ is quasi-lattice ordered, Fowler showed how to associate a $C^*$-algebra $\mathcal{NT}_\mathbf{X}$ to $\mathbf{X}$, generated by a universal representation satisfying some covariance condition. In this article we prove a uniqueness theorem for these so called Nica-Toeplitz algebras.
We investigate KMS states of Fowler's Nica-Toeplitz algebra $\mathcal{NT}(X)$ associated to a compactly aligned product system $X$ over a semigroup $P$ of Hilbert bimodules. This analysis relies on restrictions of these states to the core algebra which satisfy appropriate scaling conditions. The concept of product system of finite type is introduced. If $(G, P)$ is a lattice ordered group and $X$ is a product system of finite type over $P$ satisfying certain coherence properties, we construct KMS$_β$ states of $\NT(X)$ associated to a scalar dynamics from traces on the coefficient algebra of t…
We consider C*-algebras generated by a single Hilbert bimodule (Pimsner-Toeplitz algebras) and by a product systems of Hilbert bimodules. We give a new proof of a theorem of Pimsner, which states that any representation of the generating bimodule gives rise to a representation of the Pimsner-Toeplitz algebra. Our proof does not make use of the conditional expectation onto the subalgebra invariant under the dual action of the circle group. We then prove the analogous statement for the case of product systems, generalizing a theorem of Arveson from the case of product systems of Hilbert spaces.
Given a finitely aligned $k$-graph $Λ$, we let $Λ^i$ denote the $(k-1)$-graph formed by removing all edges of degree $e_i$ from $Λ$. We show that the Toeplitz-Cuntz-Krieger algebra of $Λ$, denoted by $\mathcal{T}C^*(Λ)$, may be realised as the Toeplitz algebra of a Hilbert $\mathcal{T}C^*(Λ^i)$-bimodule. When $Λ$ is locally-convex, we show that the Cuntz-Krieger algebra of $Λ$, which we denote by $C^*(Λ)$, may be realised as the Cuntz-Pimsner algebra of a Hilbert $C^*(Λ^i)$-bimodule. Consequently, $\mathcal{T}C^*(Λ)$ and $C^*(Λ)$ may be viewed as iterated Toeplitz and iterated Cuntz-Pimsner al…
In this article we study how decompositions of a quasi-lattice ordered group $(G,P)$ relate to decompositions of the Nica-Toeplitz algebra $\mathcal{NT}_\mathbf{X}$ and Cuntz-Nica-Pimsner algebra $\mathcal{NO}_\mathbf{X}$ of a compactly aligned product system $\mathbf{X}$ over $P$. In particular, we are interested in the situation where $(G,P)$ may be realised as the semidirect product of quasi-lattice ordered groups. Our results generalise Deaconu's work on iterated Toeplitz and Cuntz-Pimsner algebras - we show that the Nica-Toeplitz algebra and Cuntz-Nica-Pimsner algebra of a compactly align…
In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution $\diamond$ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if $F_j, G_j$ ($1\leq j\leq N$) are polynomials of $z$ and $\bar{z}$ then $\sum_{j=1}^{N}T_{F_j}T_{G_j}-T_{H}$ is a finite rank operator for some $L^{1}$-function $H$ if and only if $\sum_{j=1}^{N}F_j\diamond G_j$ belongs to $L^1$ and $H=\sum_{j=1…
In this paper, we consider the Toeplitz algebra associated to actions of Ore semigroups on $C^{*}$-algebras. In particular, we consider injective and surjective actions of such semigroups. We use the theory of groupoid dynamical systems to represent the Toeplitz algebra as a groupoid crossed product. We also discuss the K-theory of the Toeplitz algebra in some examples. For instance, we show that for the semigroup of positive matrices, the K-theory of the associated Toeplitz algebra vanishes.
We consider the semigroup crossed product of the additive natural numbers by the multiplicative natural numbers. We study its Toeplitz C*-algebra generated by the right-regular representation, 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. We identify the Crisp-Laca boundary quotient as the C*-algebra of the corresponding g…
We study the Toeplitz algebra $\TT(\N\rtimes\N^\times)$ and three quotients of this algebra: the $C^*$-algebra $\qn$ recntly introduced by Cuntz, and two new ones, which we call the additive and multiplicative boundary quotients. These quotients are universal for Nica-covariant representations of $\N\rtimes\N^\times$ satisfying extra relations, and can be realised as partial crossed products. We use the structure theory for partial crossed products to prove a uniqueness theorem for the additive boundary quotient, and use the recent analysis of KMS states on $\TT(\nxnx)$ to describe the KMS sta…
In this paper we study the product of Toeplitz operators on the harmonic Bergman space of the unit disk of the complex plane C. Mainly, we discuss when the product of two quasihomogeneous Toeplitz operators is also a Toeplitz operator, and when such operators commute.
We define partial product systems over N. They generalise product systems over N and Fell bundles over Z. We define Toeplitz C*-algebras and relative Cuntz-Pimsner algebras for them and show that the section C*-algebra of a Fell bundle over Z is a relative Cuntz-Pimsner algebra. We describe the gauge-invariant ideals in the Toeplitz C*-algebra.
We introduce the notion of a topological higher-rank graph, a unified generalization of the higher-rank graph and the topological graph. Using groupoid techniques, we define the Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs, and show that the $C^*$-algebras defined are coherent with the existing theory.
Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras O_n, and the Cuntz-Krieger algebras O_B. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger algebras of directed graphs, which includes Cuntz's uniqueness theorem for O_\infty.
Let R be a finite Blaschke product of degree at least two with R(0)=0. Then there exists a relation between the associated composition operator C_R on the Hardy space and the C*-algebra associated with the complex dynamical system on the Julia set of R. We study the C*-algebra generated by both the composition operator C_R and the Toeplitz operator T_z to show that the quotient algebra by the ideal of the compact operators is isomorphic to the C*-algebra associated with the complex dynamical system, which is simple and purely infinite.
Let R be a finite Blaschke product. We study the C*-algebra TC_R generated by both the composition operator C_R and the Toeplitz operator T_z on the Hardy space. We show that the simplicity of the quotient algebra OC_R by the ideal of the compact operators can be characterized by the dynamics near the Denjoy-Wolff point of R if the degree of R is at least two. Moreover we prove that the degree of finite Blaschke products is a complete isomorphism invariant for the class of OC_R such that R is a finite Blaschke product of degree at least two and the Julia set of R is the unit circle, using the …
The graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute and which do not. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice orders. As a consequence the Toeplitz algebras of these groups are universal for covariant isometric representations on Hilbert space, and their representations are faithful if the isometries satisfy a properness condition given b…
We provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the Cuntz-Krieger algebra.
We provide groupoid models for Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs. Extending the groupoid models used in the theory of graph algebras and topological dynamical systems to our setting, we prove results on essential freeness and amenability of the groupoids which capture the existing theory, and extend results involving group crossed products of graph algebras.
We study the equilibrium or KMS states of the Toeplitz C*-algebra of a finite higher-rank graph which is reducible. The Toeplitz algebra carries a gauge action of a higher-dimensional torus, and a dynamics arises by choosing an embedding of the real numbers in the torus. Here we use an embedding which leads to a dynamics which has previously been identified as "preferred", and we scale the dynamics so that 1 is a critical inverse temperature. As with 1-graphs, we study the strongly connected components of the vertices of the graph. The behaviour of the KMS states depends on both the graphical …
We use product systems of $C^*$-correspondences to introduce twisted $C^*$-algebras of topological higher-rank graphs. We define the notion of a continuous $\mathbb{T}$-valued $2$-cocycle on a topological higher-rank graph, and present examples of such cocycles on large classes of topological higher-rank graphs. To every proper, source-free topological higher-rank graph $Λ$, and continuous $\mathbb{T}$-valued $2$-cocycle $c$ on $Λ$, we associate a product system $X$ of $C_0(Λ^0)$-correspondences built from finite paths in $Λ$. We define the twisted Cuntz--Krieger algebra $C^*(Λ,c)$ to be the C…
Consider the pair $(G,P)$ consisting of an abelian lattice-ordered discrete group $G$ and its positive cone $P$. Let $α$ be an action of $P$ by extendible endomorphisms of a $C^*$-algebra $A$. We show that the Nica-Toeplitz algebra $\mathcal{T}_{\textrm{cov}}(A\times_α P)$ is a full corner of a group crossed product $\mathcal{B}\rtimes_βG$, where $\mathcal{B}$ is a subalgebra of $\ell^{\infty}(G,A)$ generated by a collection of faithful copies of $A$, and the action $β$ on $\mathcal{B}$ is given by the shift on $\ell^{\infty}(G,A)$. By using this realization, we show that the ideal $\mathcal{I…
We find necessary and sufficient conditions for the product of two truncated Toeplitz operators on a model space to itself be a truncated Toeplitz operator, and as a result find a characterization for the maximal algebras of bounded truncated Toeplitz operators.
We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmos's Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamse's Theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two analytic functions), whose co-analytic part has a "coprime decomposition," is normal or analytic. We also prove that the coprime decomposition condition is essential. Finally, we …
Consider a higher-rank graph of rank k. Both the Cuntz-Krieger algebra and the Toeplitz-Cuntz-Krieger algebra of the graph carry natural gauge actions of the torus T^k, and restricting these gauge actions to one-parameter subgroups of T^k gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures β, the simplex of KMS_βstates on the Toeplitz-Cuntz-Krieger algebra has dimension d one less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a c…
This note is a follow-up to a recent paper by the author. Most of that theory is now realized in a new setting where the vector space of symbols is not necessarily an algebra nor is it equipped with an inner product, although it does have a conjugation. As in the previous paper one does not need to put a measure on this vector space. A Toeplitz quantization is defined and shown to have most of the properties as in the previous paper, including creation and annihilation operators. As in the previous paper this theory is implemented by densely defined Toeplitz operators which act in a Hilbert sp…
We characterize the state space of a Toeplitz-Cuntz algebra TO_n in terms of positive operator matrices $Ω$ on Fock space which satisfy sl(Ω) \le Ω, where sl(Ω) is the operator matrix obtained from Ωby taking the trace in the last variable. Essential states correspond to those matrices Ωwhich are slice-invariant. As an application we show that a pure essential product state of the fixed-point algebra for the action of the gauge group has precisely a circle of pure extensions to TO_n.
The theory of Toeplitz quantization presented in our previous paper is extended and further developed to include diverse and interesting non-commutative realizations of the classical Euclidean plane. This is done using Hilbert spaces of entire functions, where polynomials in one complex variable form a dense subspace. The complex coordinate naturally acts as an unbounded multiplication operator generating, together with its adjoint, a highly non-commutative *-algebra of operators. The Toeplitz operators are then geometrically constructed as special elements from this algebra; they are associat…
We study Toeplitz-type operators with respect to specific wavelets whose Fourier transforms are related to Laguerre polynomials. On the one hand, this choice of wavelets underlines the fact that these operators acting on wavelet subspaces share many properties with the classical Toeplitz operators acting on the Bergman spaces. On the other hand, it enables to study poly-Bergman spaces and Toeplitz operators acting on them from a different perspective. Restricting to symbols depending only on vertical variable in the upper half-plane of the complex plane these operators are unitarily equivalent…
