arXiv subjects
Publications and source records attributed to Marcelo Laca.
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 β\ne 0,1. In particular, we show that for each β\in(1,2] there exists a unique KMS_β-state.
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.
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$_β$ 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.
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 σ_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_βstates on both O_A and T_A are completely characterized for certain values of the inverse temperature β, according to the position of βrelative 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.
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.
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.
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.
We compute the K-theory of the Cuntz-Krieger C^*-algebras associated to infinite matrices.
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.
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.
Let F_n be the fixed-point algebra of the gauge action of the circle on the Cuntz algebra O_n. For every pure state ρof F_n and every representation θof C(T) we construct a representation of O_n, and we use the resulting class of representations to parameterize the space of all states of O_n which extend ρ. We show that the gauge group acts transitively on the pure extensions of ρand that the action is p-to-1 with p the period of ρunder the usual shift. We then use the above representations of O_n to construct endomorphisms of B(H) which we classify up to conjugacy in terms of the parameters ρand θ. In particular our construction yields every ergodic endomorphism αwhose tail algebra $\bigcap_kα^k(B(H))$ has a minimal projection, and our results classify these ergodic endomorphisms by an equivalence relation on the pure states of F_n. As examples we analyze the ergodic endomorphisms arising from periodic pure product states of F_n, for which we are able to give a geometric complete conjugacy invariant, generalizing results of Stacey, Laca, and Bratteli-Jorgensen-Price on the shifts of Powers.
Given a measurable twisted action of a second-countable, locally compact group G on a separable C*-algebra A, we prove the existence of a topology on AxG making it a continuous bundle, whose cross sectional C*-algebra is isomorphic to the Busby--Smith--Packer--Raeburn crossed product.