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In the setting of product systems over group-embeddable monoids, we consider nuclearity of the associated Toeplitz C*-algebra in relation to nuclearity of the coefficient algebra. Our work goes beyond the known cases of single correspondences and compactly aligned product systems over right LCM monoids. Specifically, given a product system over a submonoid of a group, we show, under technical assumptions, that the fixed-point algebra of the gauge action is nuclear iff the coefficient algebra is nuclear; when the group is amenable, we conclude that this happens iff the Toeplitz algebra itself is nuclear. Our main results imply that nuclearity of the Toeplitz algebra is equivalent to nuclearity of the coefficient algebra for every full product system of Hilbert bimodules over abelian monoids, over $ax+b$-monoids of integral domains and over Baumslag-Solitar monoids $BS^+(m,n)$ that admit an amenable embedding, which we provide for $m$ and $n$ relatively prime.

Given a C$^*$-algebra $A$ with an almost periodic time evolution $\sigma$, we define a new C$^*$-algebra $A_c$, which we call the crystal of $(A,\sigma)$, that represents the zero temperature limit of $(A, \sigma)$. We prove that there is a one-to-one correspondence between the ground states of $(A,\sigma)$ and the states on $A_c$, justifying the name. In order to investigate further the relation between low temperature equilibrium states on $A$ and traces on $A_c$, we define a Fock module $\mathcal F$ over the crystal and construct a vacuum representation of $A$ on $\mathcal F$. This allows us to show, under relatively mild assumptions, that for sufficiently large inverse temperatures $\beta$ the $\sigma$-KMS$_\beta$-states on $A$ are induced from traces on $A_c$ by means of the Fock module. In the second part, we compare the K-theoretic structures of $A$ and $A_c$. Previous work by various authors suggests that they have (rationally) isomorphic K-groups. We analyze this phenomenon in detail, confirming it under favorable conditions, but showing that, in general, there is apparently no easy way to relate these groups. As examples, we discuss in particular Exel's results on semi-saturated circle actions, and recent results of Miller on the K-theory of inverse semigroup C$^*$-algebras. In relation to the latter, we introduce the notion of a scale $N$ on an inverse semigroup $I$ and define a new inverse semigroup $I_c$, which we call the crystal of $(I,N)$., Comment: 27 pages: v4: minor corrections, shortened last section, to appear in CIMP; v3: minor fixes, references added; v2: minor fixes

We introduce the notion of relative topological principality for a family $\{H_\alpha\}$ of open subgroupoids of a Hausdorff \'etale groupoid $G$. The C*-algebras $C^*_r(H_\alpha)$ of the groupoids $H_\alpha$ embed in $ C^*_r(G)$ and we show that if $G$ is topologically principal relative to $\{H_\alpha\}$ then a representation of $C^*_r(G)$ is faithful if and only if its restriction to each of the subalgebras $C^*_r(H_\alpha)$ is faithful. This variant of the ideal intersection property potentially involves several subalgebras, and gives a new method of verifying injectivity of representations of reduced groupoid C*-algebras. As applications we prove a uniqueness theorem for Toeplitz C*-algebras of left cancellative small categories that generalizes a recent result of Laca and Sehnem for Toeplitz algebras of group-embeddable monoids, and we also discuss and compare concrete examples arising from integer arithmetic., Comment: 17 pages. A few comments and references added

We use a groupoid model for the spin algebra to introduce boundary conditions on quantum spin systems via a Poisson point process representation. We can describe KMS states of quantum systems by means of a set of equations resembling the standard DLR equations of classical statistical mechanics. We introduce a notion of quantum specification which recovers the classical DLR measures in the particular case of classical interactions. Our results are in the same direction as those obtained recently by Cha, Naaijkens, and Nachtergaele, differently somehow from the predicted by Fannes and Werner., Comment: Very Preliminary version. Comments are welcome

We generalise recent results of Afsar, Larsen and Neshveyev for product systems over quasi-lattice orders by showing that the equilibrium states of quasi-free dynamics on the Nica-Toeplitz $C^*$-algebras of product systems over right LCM monoids must satisfy a positivity condition encoded in a system of inequalities satisfied by their restrictions to the coefficient algebra. We prove that the reduction of this positivity condition to a finite subset of inequalities is valid for a wider class of monoids that properly includes finite-type Artin monoids, answering a question left open in their work. Our main technical tool is a combinatorially generated tree modelled on a recent construction developed by Boyu Li for dilations of contractive representations. We also obtain a reduction of the positivity condition to inequalities arising from a certain minimal subset that may not be finite but has the advantage of holding for all Noetherian right LCM monoids, and we present an example, arising from a finite-type Artin monoid, that exhibits a gap in its inverse temperature space., Comment: Version with a correction to Corollary 6.3 and some minor changes

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 group built from rational numbers. 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., Comment: To appear in the memorial volume for Vaughan Jones in the New Zealand Journal of Mathematics

The C*-algebra generated by the left-regular representation of $\mathbb{N}^n$ twisted by a $2$-cocycle is a Toeplitz extension of an $n$-dimensional noncommutative torus, on which each vector $r \in [0,\infty)^n$ determines a one-parameter subgroup of the gauge action. We show that the equilibrium states of the resulting C*-dynamical system are parametrised by tracial states of the noncommutative torus corresponding to the restriction of the cocycle to the vanishing coordinates of $r$. These in turn correspond to probability measures on a classical torus whose dimension depends on a certain degeneracy index of the restricted cocycle. Our results generalise the phase transition on the Toeplitz noncommutative tori used as building blocks in recent work of Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims., Comment: 29 pages; minor changes in version 2

We study the structure of C*-algebras associated with compactly aligned product systems over group embeddable right LCM-semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original cores to those of the controlling pair, and we combine with applications of the C*-envelope theory for cosystems of nonselfadjoint operator algebras recently produced. We derive several applications of these methods that generalize results on single C*-correspondences. First we show that if the controlling group is exact then the co-universal C*-algebra of the product system coincides with the quotient of the Fock C*-algebra by the ideal of strong covariance relations. We show that if the controlling group is amenable then the product system is amenable. In particular if the controlling group is abelian then the co-universal C*-algebra is the C*-envelope of the tensor algebra. Secondly we give necessary and sufficient conditions for the Fock C*-algebra to be nuclear and exact. When the controlling group is amenable we completely characterize nuclearity and exactness of any equivariant injective Nica-covariant representation of the product system. Thirdly we consider controlled maps that enjoy a saturation property. In this case we induce a compactly aligned product system over the controlling pair that shares the same Fock representation, and preserves injectivity. By using co-universality, we show that they share the same reduced covariance algebras. If in addition the controlling pair is a total order then the fixed point algebra of the controlling group induces a super-product system that has the same reduced covariance algebra and is moreover reversible., Comment: 40 pages

We study two classes of operator algebras associated with a unital subsemigroup $P$ of a discrete group $G$: one related to universal structures, and one related to co-universal structures. First we provide connections between universal C*-algebras that arise variously from isometric representations of $P$ that reflect the space $\mathcal{J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation $\mathrm{C}^*_\lambda(P)$ and we consider the boundary quotient $\partial \mathrm{C}^*_\lambda(P)$ related to the minimal boundary space. We show that $\partial \mathrm{C}^*_\lambda(P)$ is co-universal in two different classes: (a) with respect to the equivariant constructible isometric representations of $P$; and (b) with respect to the equivariant C*-covers of the reduced nonselfadjoint semigroup algebra $\mathcal{A}(P)$. If $P$ is an Ore semigroup, or if $G$ acts topologically freely on the minimal boundary space, then $\partial \mathrm{C}^*_\lambda(P)$ coincides with the usual C*-envelope $\mathrm{C}^*_{\text{env}}(\mathcal{A}(P))$ in the sense of Arveson. This covers total orders, finite type and right-angled Artin monoids, the Thompson monoid, multiplicative semigroups of nonzero algebraic integers, and the $ax+b$-semigroups over integral domains that are not a field. In particular, we show that $P$ is an Ore semigroup if and only if there exists a canonical $*$-isomorphism from $\partial \mathrm{C}^*_\lambda(P)$, or from $\mathrm{C}^*_{\text{env}}(\mathcal{A}(P))$, onto $\mathrm{C}^*_\lambda(G)$. If any of the above holds, then $\mathcal{A}(P)$ is shown to be hyperrigid., Comment: 22 pages. Theorem 4.5 to $ax+b$-semigroups of rings has a more general application than originally stated, leading to the updated Remark 4.8(vi)

This paper examines actions of right LCM semigroups by endomorphisms of C*-algebras that encode an additional structure of the right LCM semigroup. We define contractive covariant representations for these semigroup dynamical systems and prove a generalized Stinespring's dilation theorem showing that these representations can be dilated if and only if the map on the C*-algebra is unital and completely positive. This generalizes earlier results about dilations of right LCM semigroups of contractions. In addition, we also give sufficient conditions under which a contractive covariant representation of a right LCM system can be dilated to an isometric representation of the boundary quotient., Comment: 34 pages

To each monoid $P$ that embeds in a group we associate a universal Toeplitz C*-algebra $T_u(P)$ defined via generators and relations; $T_u(P)$ is a quotient of Li's semigroup C*-algebra $C^*(P)$ and they are isomorphic if and only if $P$ satisfies independence. We give a partial crossed product realization of $T_u(P)$ and show that several key results known for $C^*(P)$ when $P$ satisfies independence are also valid for $T_u(P)$ when independence fails. At the level of the reduced semigroup C*-algebra $T_r(P)$, we show that nontrivial ideals have nontrivial intersection with the reduced crossed product of the diagonal subalgebra by the action of the group of units of $P$, generalizing a result of Li for monoids with trivial unit group. We also characterize when the action of the group of units is topologically free and we show that in this case a representation of $T_r(P)$ is faithful iff it is jointly proper. This yields a uniqueness theorem for C*-algebras generated by semigroups of isometries that unifies several classical results. We provide a presentation for the covariance algebra of the product system over $P$ with one-dimensional fibers in terms of a notion of foundation sets of constructible ideals that generalizes that of Sims and Yeend for quasi-lattice orders. The covariance algebra is a full, or universal, analogue of the boundary quotient. We give purely algebraic sufficient conditions on $P$ for the boundary quotient to be purely infinite simple, which reduce to Starling's conditions in the case of right LCM monoids. We discuss applications of our results to examples that include a numerical semigroup and the $ax+b$-monoid of an integral domain. This is particularly interesting in the case of nonmaximal orders in number fields, for which we show independence always fails. In addition, we simplify and generalize results for right LCM monoids., Comment: Added two references, former Proposition 6.18 became Theorem 6.18 with same statement and proof, minor typos were corrected. Final version, to appear in Transactions of the AMS. 50 pages, 1 table

A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify to the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras., Comment: 25 pages, accepted version, Advances in Mathematics, to appear

We prove a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in the sense of Nica, then so are its submonoids. As applications, we complete the classification of Artin monoids with respect to Nica amenability by showing that only the right-angled ones are amenable in the sense of Nica and we show that the Nica amenability of a graph product of right-LCM semigroups is inherited by the factors., Comment: 12 pages

We study the phase transition of KMS states for the C*-algebras of $ax+b$-semigroups of algebraic integers in which the multiplicative part is restricted to a congruence monoid, as in recent work of Bruce generalizing earlier work of Cuntz, Deninger, and Laca. Here we realize the extremal low-temperature KMS states as generalized Gibbs states by constructing concrete representations induced from extremal traces of certain group C*-algebras. We use these representations to compute the Murray--von Neumann type of extremal KMS states and we determine explicit partition functions for the type I factor states. The collection of partition functions that arise this way is an invariant under $\mathbb{R}$-equivariant isomorphism of C*-dynamical systems, which produces further invariants through the analysis of the topological structure of the KMS state space. As an application we characterize several features of the underlying number field and congruence monoid in terms of these invariants. In most cases our systems have infinitely many type I factor KMS states and at least one type II factor KMS state at the same inverse temperature and there are infinitely many partition functions. In order to deal with this multiplicity, we establish, in the context of general C*-dynamical systems, a precise way to associate partition functions to extremal KMS states that are of type I, and we then show that for our systems these partition functions depend only on connected components in the KMS simplex. The discussion of partition functions of general C*-dynamical systems may be of interest by itself and is likely to have applications in other contexts, so we include it in a self-contained initial section that is partly expository and is independent of the number-theoretic background and of the technical results about congruence monoids., Comment: final version, accepted for publication in Comm. Math. Phys.; 38 pages

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 generalised 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., Comment: 9 pages; pictures drawn using TiKZ

We consider the Hecke pair consisting of the group $P^+_K$ of affine transformations of a number field $K$ that preserve the orientation in every real embedding and the subgroup $P^+_O$ consisting of transformations with algebraic integer coefficients. The associated Hecke algebra $C^*(P^+_K,P^+_O)$ has a natural time evolution $\sigma$, and we describe the corresponding phase transition for KMS$_\beta$-states and for ground states. From work of Yalkinoglu and Neshveyev it is known that a Bost-Connes type system associated to $K$ has an essentially unique arithmetic subalgebra. When we import this subalgebra through the isomorphism of $C^*(P^+_K,P^+_O)$ to a corner in the Bost-Connes system established by Laca, Neshveyev and Trifkovic, we obtain an arithmetic subalgebra of $C^*(P^+_K,P^+_O)$ on which ground states exhibit the `fabulous' property with respect to an action of the Galois group $Gal(K^{ab}/H_+(K))$, where $H_+(K)$ is the narrow Hilbert class field. In order to characterize the ground states of the $C^*$-dynamical system $(C^*(P^+_K,P^+_O),\sigma)$, we obtain first a characterization of the ground states of a groupoid $C^*$-algebra, refining earlier work of Renault. This is independent from number theoretic considerations, and may be of interest by itself in other situations., Comment: 21 pages; v2: minor changes and corrections

In recent work, Cuntz, Deninger and Laca have studied the Toeplitz type C*-algebra associated to the affine monoid of algebraic integers in a number field, under a time evolution determined by the absolute norm. The KMS equilibrium states of their system are parametrized by traces on the C*-algebras of the semidirect products $J \rtimes O^*$ resulting from the multiplicative action of the units $O^*$ on integral ideals $J$ representing each ideal class. At each fixed inverse temperature $\beta > 2$, the extremal equilibrium states correspond to extremal traces of $C^*(J\rtimes O^*)$. Here we undertake the study of these traces using the transposed action of $O^*$ on the duals $\hat J$ of the ideals and the recent characterization of traces on transformation group C*-algebras due to Neshveyev. We show that the extremal traces of $C^*(J\rtimes O^*)$ are parametrized by pairs consisting of an ergodic invariant measure for the action of $O^*$ on $\hat{J}$ together with a character of the isotropy subgroup associated to the support of this measure. For every ideal the dual group $\hat {J}$ is a d-torus on which $O^*$ acts by linear toral automorphisms. Hence, the problem of classifying all extremal traces is a generalized version of Furstenberg's celebrated $\times 2$ $\times 3$ conjecture. We classify the results for various number fields in terms of ideal class group, degree, and unit rank, and we point along the way the trivial, the intractable, and the conjecturally classifiable cases. At the topological level, it is possible to characterize the number fields for which infinite $O^*$-invariant sets are dense in $\hat{J} $, thanks to a theorem of Berend; as an application we give a description of the primitive ideal space of $C^*(J\rtimes O^*)$ for those number fields., Comment: 17 pages

Each multiplicative real-valued homomorphism on a quasi-lattice ordered monoid gives rise to a quasi-periodic dynamics on the associated Toeplitz C*-algebra; here we study the KMS equilibrium states of the resulting C*-dynamical system. We show that, under a nondegeneracy assumption on the homomorphism, there is a critical inverse temperature $\beta_c$ such that at each inverse temperature $\beta \geq \beta_c$ there exists a unique KMS state. Strictly above $\beta_c$, the KMS states are generalised Gibbs states with density operators determined by analytic extension to the upper half-plane of the unitaries implementing the dynamics. These are faithful Type~I states. The critical value $\beta_c$ is the largest real pole of the partition function of the system and is related to the clique polynomial and skew-growth function of the monoid, relative to the degree map given by the logarithm of the multiplicative homomorphism. Motivated by the study of equilibrium states, we give a proof of the inversion formula for the growth series of a quasi-lattice ordered monoid in terms of the clique polynomial as in recent work of Albenque--Nadeau and McMullen for the finitely generated case, and in terms of the skew-growth series as in recent work of Saito. Specifically, we show that $e^{-\beta_c}$ is the smallest pole of the growth series and thus is the smallest positive real root of the clique polynomial. We use this to show that equilibrium states in the subcritical range can only occur at inverse temperatures that correspond to roots of the clique polynomial in the interval $(e^{-\beta_c},1)$, but we are not aware of any examples in which such roots exist., Comment: 15 pages

We consider self-similar actions of groupoids on the path spaces of finite directed graphs, and construct examples of such self-similar actions using a suitable notion of graph automaton. Self-similar groupoid actions have a Cuntz-Pimsner algebra and a Toeplitz algebra, both of which carry natural dynamics lifted from the gauge actions. We study the equilibrium states (the KMS states) on the resulting dynamical systems. Above a critical inverse temperature, the KMS states on the Toeplitz algebra are parametrised by the traces on the full $C^*$-algebra of the groupoid, and we describe a program for finding such traces. The critical inverse temperature is the logarithm of the spectral radius of the incidence matrix of the graph, and at the critical temperature the KMS states on the Toeplitz algebra factor through states of the Cuntz-Pimsner algebra. Under a verifiable hypothesis on the self-similar action, there is a unique KMS state on the Cuntz-Pimsner algebra. We discuss an explicit method of computing the values of this KMS state, and illustrate with examples., Comment: 48 pages

We investigate the factor types of the extremal KMS states for the preferred dynamics on the Toeplitz algebra and the Cuntz--Krieger algebra of a strongly connected finite $k$-graph. For inverse temperatures above 1, all of the extremal KMS states are of type I$_\infty$. At inverse temperature 1, there is a dichotomy: if the $k$-graph is a simple $k$-dimensional cycle, we obtain a finite type I factor; otherwise we obtain a type III factor, whose Connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph., Comment: 16 pages; 1 picture prepared using TikZ. Version 2: this version to appear in Math. Ann

We study the KMS states of the C*-algebra of a strongly connected finite k-graph. We find that there is only one 1-parameter subgroup of the gauge action that can admit a KMS state. The extreme KMS states for this preferred dynamics are parameterised by the characters of an abelian group that captures the periodicity in the infinite-path space of the graph. We deduce that there is a unique KMS state if and only if the k-graph C*-algebra is simple, giving a complete answer to a question of Yang. When the k-graph C*-algebra is not simple, our results reveal a phase change of an unexpected nature in its Toeplitz extension.

We consider the dynamics on the C*-algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single KMS state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex matrix, and for inverse temperatures at and below the critical value. We prove a general result which describes all the KMS states at a fixed inverse temperature, and then apply this theorem to a variety of examples. We find that there can be many patterns of phase transition, depending on the behaviour of paths in the underlying graph., Comment: Publication version. To appear in Ergodic Theory and Dynamical Systems

We consider a family of Cuntz-Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems. We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz-Pimsner algebra; if the self-similar group is contracting, then the Cuntz-Pimsner algebra has only one KMS state. We apply these results to a number of examples, including the self-similar group actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group., Comment: The paper has been updated to agree with the published version

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 \beta, the simplex of KMS_\beta states 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 critical inverse temperature \beta_c: for \beta larger than \beta_c, there is a d-dimensional simplex of KMS states; when \beta=\beta_c and the one-parameter subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz-Krieger algebra. As in previous studies for k=1, our main tool is the Perron-Frobenius theory for irreducible nonnegative matrices, though here we need a version of the theory for commuting families of matrices.

We consider a finite directed graph E, and the gauge action on its Toeplitz-Cuntz-Krieger algebra, viewed as an action of R. For inverse temperatures larger than a critical value \beta_c, we give an explicit construction of all the KMS_{\beta} states. If the graph is strongly connected, then there is a unique KMS_{\beta_c} state, and this state factors through the quotient map onto the C*-algebra C*(E) of the graph. Our approach is direct and relatively elementary.

We give a complete description of the primitive ideal space of the C*-algebra associated to the ring of integers R in a number field K as considered in a recent paper by Cuntz, Deninger and Laca.

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra $\cT[R]$. It is an extension of the ring C*-algebra $\cA[R]$ studied previously by the first named author in collaboration with X.Li. In contrast to $\cA[R]$, it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$-semigroup $R\rtimes R^\times$ on $\ell^2 (R\rtimes R^\times)$. The algebra $\cT[R]$ carries a natural one-parameter automorphism group $(\sigma_t)_{t\in\Rz}$. We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The "partition functions" are partial Dedekind $\zeta$-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\zeta$-functions, which we then use to show uniqueness of the $\beta$-KMS state for each inverse temperature $\beta\in(1,2]$., Comment: 38 pages

An integer matrix $A\in M_d(\Z)$ induces a covering $\sigma_A$ of $\T^d$ and an endomorphism $\alpha_A:f\mapsto f\circ \sigma_A$ of $C(\T^d)$ for which there is a natural transfer operator $L$. In this paper, we compute the KMS states on the Exel crossed product $C(\T^d)\rtimes_{\alpha_A,L}\N$ and its Toeplitz extension. We find that $C(\T^d)\rtimes_{\alpha_A,L}\N$ has a unique KMS state, which has inverse temperature $\beta=\log|\det A|$. Its Toeplitz extension, on the other hand, exhibits a phase transition at $\beta=\log|\det A|$, and for larger $\beta$ the simplex of KMS$_\beta$ states is isomorphic to the simplex of probability measures on $\T^d$.

We consider a Hecke algebra naturally associated with the affine group with totally positive multiplicative part over an algebraic number field K and we show that the C*-algebra of the Bost-Connes system for K can be obtained from our Hecke algebra by induction, from the group of totally positive principal ideals to the whole group of ideals. Our Hecke algebra is therefore a full corner, corresponding to the narrow Hilbert class field, in the Bost-Connes C*-algebra of K; in particular, the two algebras coincide if and only if K has narrow class number one. Passing the known results for the Bost-Connes system for K to this corner, we obtain a phase transition theorem for our Hecke algebra. In another application of induction we consider an extension L/K of number fields and we show that the Bost-Connes system for L embeds into the system obtained from the Bost-Connes system for K by induction from the group of ideals in K to the group of ideals in L. This gives a C*-algebraic correspondence from the Bost-Connes system for K to that for L. Therefore the construction of Bost-Connes systems can be extended to a functor from number fields to C*-dynamical systems with equivariant correspondences as morphisms. We use this correspondence to induce KMS-states and we show that for beta>1 certain extremal KMS_beta-states for L can be obtained, via induction and rescaling, from KMS_{[L:K]beta}-states for K. On the other hand, for 0

We complete the analysis of KMS-states of the Toeplitz algebra of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature beta in the critical interval [1,2], the unique KMS_beta-state is of type III_1. We prove this by reducing the type classification from the Toeplitz algebra to that of the symmetric part of the Bost-Connes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of the Toeplitz algebra in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of the affine semigroup on the Nica spectrum, we can also recover all the KMS-states originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMS_beta-states for beta>2 that does not ostensibly come from an action on the C*-algebra., Comment: 15 pages, AMS-LaTeX

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 states on the two quotients. We then show that $\TT(\nxnx)$, $\qn$ and our new quotients are all interesting new examples for Larsen's theory of Exel crossed products by semigroups.

We show that the group ${\mathbb Q \rtimes \mathbb Q^*_+}$ of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup ${\mathbb N \rtimes \mathbb N^\times}$. The associated Toeplitz $C^*$-algebra ${\mathcal T}({\mathbb N \rtimes \mathbb N^\times})$ is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of this Toeplitz algebra in terms of generators and relations, and use this to show that the $C^*$-algebra ${\mathcal Q_\mathbb N}$ recently introduced by Cuntz is the boundary quotient of $({\mathbb Q \rtimes \mathbb Q^*_+}, {\mathbb N \rtimes \mathbb N^\times})$ in the sense of Crisp and Laca. The Toeplitz algebra ${\mathcal T}({\mathbb N \rtimes \mathbb N^\times})$ carries a natural dynamics $\sigma$, which induces the one considered by Cuntz on the quotient ${\mathcal Q_\mathbb N}$, and our main result is the computation of the KMS$_\beta$ (equilibrium) states of the dynamical system $({\mathcal T}({\mathbb N \rtimes \mathbb N^\times}), {\mathbb R},\sigma)$ for all values of the inverse temperature $\beta$. For $\beta \in [1, 2]$ there is a unique KMS$_\beta$ state, and the KMS$_1$ state factors through the quotient map onto ${\mathcal Q_\mathbb N}$, giving the unique KMS state discovered by Cuntz. At $\beta =2$ there is a phase transition, and for $\beta>2$ the KMS$_\beta$ states are indexed by probability measures on the circle. There is a further phase transition at $\beta=\infty$, where the KMS$_\infty$ states are indexed by the probability measures on the circle, but the ground states are indexed by the states on the classical Toeplitz algebra ${\mathcal T}(\mathbb N)$., Comment: 38 pages

We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and the stabilizer of an end relative to a vertex stabilizer, assuming that the actions are sufficiently transitive. We focus on identifying the structure of the resulting Hecke algebras, give explicit multiplication tables of the canonical generators and determine whether the Hecke algebra has a universal C*-completion. The paper unifies past algebraic and analytic approaches by focusing on the common geometric thread.The results have implications for the general theory of totally disconnected locally compact groups.

We give a complete description of the phase transition of the Bost-Connes type systems for number fields recently introduced by Connes-Marcolli-Ramachandran and Ha-Paugam. We also introduce a notion of K-lattices and discuss an interpretation of these systems in terms of 1-dimensional K-lattices., Comment: 12 pages; small corrections, references added

We develop a general framework for analyzing KMS-states on C*-algebras arising from actions of Hecke pairs. We then specialize to the system recently introduced by Connes and Marcolli and classify its KMS-states for inverse temperatures \beta\ne 0,1. In particular, we show that for each \beta\in(1,2] there exists a unique KMS_\beta-state., Comment: 23 pages; minor corrections, final version

We study a C*-dynamical system arising from the ring inclusion of the 2\times 2 integer matrices in the rational ones. The orientation preserving affine groups of these rings form a Hecke pair that is closely related to a recent construction of Connes and Marcolli; our dynamical system consists of the associated reduced Hecke C*-algebra endowed with a canonical dynamics defined in terms of the determinant function. We show that the Schlichting completion also consists of affine groups of matrices, over the finite adeles, and we obtain results about the structure and induced representations of the Hecke C*-algebra. In a somewhat unexpected parallel with the one dimensional case studied by Bost and Connes, there is a group of symmetries given by an action of the finite integral ideles, and the corresponding fixed point algebra decomposes as a tensor product over the primes. This decomposition allows us to obtain a complete description of a natural class of equilibrium states which conjecturally includes all KMS_\beta-states for \beta\ne 0,1., Comment: 30 pages; minor corrections, final version

We study the quotients of the Toeplitz C*-algebra of a quasi-lattice ordered group (G,P), which we view as crossed products by a partial actions of G on closed invariant subsets of a totally disconnected compact Hausdorff space, the Nica spectrum of (G,P). Our original motivation and our main examples are drawn from right-angled Artin groups, but many of our results are valid for more general quasi-lattice ordered groups. We show that the Nica spectrum has a unique minimal closed invariant subset, which we call the boundary spectrum, and we define the boundary quotient to be the crossed product of the corresponding restricted partial action. The main technical tools used are the results of Exel, Laca, and Quigg on simplicity and ideal structure of partial crossed products, which depend on amenability and topological freeness of the partial action and its restriction to closed invariant subsets. When there exists a generalised length function, or controlled map, defined on G and taking values in an amenable group, we prove that the partial action is amenable on arbitrary closed invariant subsets. Our main results are obtained for right-angled Artin groups with trivial centre, that is, those with no cyclic direct factor; they include a presentation of the boundary quotient in terms of generators and relations that generalises Cuntz's presentation of O_n, a proof that the boundary quotient is purely infinite and simple, and a parametrisation of the ideals of the Toeplitz C*-algebra in terms of subsets of the standard generators of the Artin group., Comment: 26 pages

We associate a canonical Hecke pair of semidirect product groups to the ring inclusion of the algebraic integers $\oo$ in a number field $\kk$, and we construct a C*-dynamical system on the corresponding Hecke C*-algebra, analogous to the one constructed by Bost and Connes for the inclusion of the integers in the rational numbers. We describe the structure of the resulting Hecke C*-algebra as a semigroup crossed product and then, in the case of class number one, analyze the equilibrium (KMS) states of the dynamical system. The extreme KMS$_\beta$ states at low-temperature exhibit a phase transition with symmetry breaking that strongly suggests a connection with class field theory. Indeed, for purely imaginary fields of class number one, the group of symmetries, which acts freely and transitively on the extreme KMS$_\infty$ states, is isomorphic to the Galois group of the maximal abelian extension over the field. However, the Galois action on the restrictions of extreme KMS$_\infty$ states to the (arithmetic) Hecke algebra over $\kk$, as given by class-field theory, corresponds to the action of the symmetry group if and only if the number field $\kk$ is $\Q$.

A continuous one-parameter group of unitary isometries of a right Hilbert C*-bimodule induces a quasi-free dynamics on the Cuntz-Pimsner C*-algebra of the bimodule and on its Toeplitz extension. The restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial, and the corresponding KMS states of the Toeplitz-Cuntz-Pimsner and Cuntz-Pimsner C*-algebras are characterized in terms of traces on the algebra of coefficients. This generalizes and sheds light onto various earlier results about KMS states of the gauge actions on Cuntz algebras, Cuntz-Krieger algebras, and crossed products by endomorphisms. We also obtain a more general characterization, in terms of KMS weights, for the case in which the inducing isometries are not unitary, and accordingly, the restriction of the quasi-free dynamics to the algebra of coefficients is nontrivial.

We consider group-subgroup pairs in which the group is a semidirect product and the subgroup is contained in the normal part. We give conditions for the pair to be a Hecke pair and we show that the enveloping Hecke algebra and Hecke C*-algebra are canonically isomorphic to semigroup crossed products, generalizing earlier results of Arledge, Laca and Raeburn and of Brenken., Comment: 9 pages

Given a countably infinite 0-1 matrix A without identically zero rows, let O_A be the Cuntz-Krieger algebra recently introduced by the authors and T_A be the Toeplitz extension of O_A, once the latter is seen as a Cuntz-Pimsner algebra, as recently shown by Szymanski. We study the KMS equilibrium states of C*-dynamical systems based on O_A and T_A, with dynamics satisfying \sigma_t(s_x) = N_x^{it} s_x for the canonical generating partial isometries s_x and arbitrary real numbers N_x > 1. The KMS_\beta states on both O_A and T_A are completely characterized for certain values of the inverse temperature \beta, according to the position of \beta relative to three critical values, defined to be the abscissa of convergence of certain Dirichlet series associated to A and the N(x). Our results for O_A are derived from those for T_A by virtue of the former being a covariant quotient of the latter. When the matrix A is finite, these results give theorems of Olesen and Pedersen for O_n and of Enomoto, Fujii and Watatani for O_A as particular cases., Comment: Plain TeX, 48 pages

When S is a discrete subsemigroup of a discrete group G such that G = S^{-1} S, it is possible to extend circle-valued multipliers from S to G; to dilate (projective) isometric representations of S to (projective) unitary representations of G; and to dilate/extend actions of S by injective endomorphisms of a C*-algebra to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilated action of G. This shows that crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making powerful tools available to study their ideal structure and representation theory. The dilation of the system giving the Bost-Connes Hecke C*-algebra from number theory is constructed explicitly as an application: it is the crossed product corresponding to the multiplicative action of the positive rationals on the additive group of finite adeles., Comment: 11 pages

We compute explicitly the primitive ideal space of the Bost-Connes Hecke C*-algebra by embedding it as a full corner in a transformation group C*-algebra and applying a general theorem of Williams. This requires the computation of the quasi-orbit space for the action of the multiplicative positive rationals on the space of finite adeles. We then carry out a similar computation for the action of the nonzero rationals on the space of full adeles., Comment: 15 pages

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 by Laca and Raeburn. An application of this to right-angled Artin groups gives a uniqueness theorem for the C^*-algebra generated by a collection of isometries such that any two of them either *-commute or else have orthogonal ranges. In contrast, the nonabelian Artin groups of finite type considered by Brieskorn and Saito and Deligne have canonical quasi-lattice orders that are not amenable in the sense of Nica, so their Toeplitz algebras are not universal and the C^*-algebra generated by a collection of isometries satisfying the Artin relations fails to be unique., Comment: 19 pages; nontrivial revision of sections 2 and 4; Thm. 21 is a stronger version of old Cor. 20; other main results are unchanged

We compute the K-theory of the Cuntz-Krieger C^*-algebras associated to infinite matrices., Comment: Final version with substantial improvements and corrections, LaTeX, 14 pages, no figures

Given an arbitrary infinite 0--1 matrix A having no identically zero rows, we define an algebra OA as the universal C*-algebra generated by partial isometries subject to conditions that generalize, to the infinite case, those introduced by Cuntz and Krieger for finite matrices. We realize OA as the crossed product algebra for a partial dynamical system and, based on this description, we extend to the infinite case some of the main results known to hold in the finite case, namely the uniqueness theorem, the classification of ideals, and the simplicity criteria. OA is always nuclear and we obtain conditions for it to be unital and purely infinite., Comment: Plain TeX, 40 pages, no figures, A minor error in Theorem 9.1 was corrected

A collection of partial isometries whose range and initial projections satisfy a specified set of conditions often gives rise to a partial representation of a group. The C*-algebra generated by the partial isometries is thus a quotient of the universal C*-algebra for partial representations of the group, from which it inherits a crossed product structure, of an abelian C*-algebra by a partial action of the group. Questions of faithfulness of representations, simplicity, and ideal structure of these C*-algebras can then be addressed in a unified manner from within the theory of partial actions. We do this here, focusing on two key properties of partial dynamical systems, namely amenability and topological freeness; they are the essential ingredients of our main results in which we characterize faithful representations, simplicity and the ideal structure of crossed products. As applications we consider three situations involving C*-algebras generated by partial isometries: partial representations of groups, Toeplitz algebras of quasi-lattice ordered groups, and Cuntz-Krieger algebras. These C*-algebras share a crossed product structure which we give here explicitly and which we use to study them in terms of the underlying partial actions., Comment: LaTeX, 24 pages