Marcelo Laca

The *-algebras of infinite graphs

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ON THE TOEPLITZ ALGEBRAS OF RIGHT-ANGLED AND FINITE-TYPE ARTIN GROUPS

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. 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. The associated Toeplitz algebras have a universal property, and their representations are faithful if the generating isometries satisfy a joint properness condition. When applied to right-angled Artin groups this yields 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. The analogous result fails to hold for the nonabelian Artin groups of finite type considered by Brieskorn and Saito, and Deligne.

EQUILIBRIUM STATES ON THE CUNTZ-PIMSNER ALGEBRAS OF SELF-SIMILAR ACTIONS

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.

BOUNDARY QUOTIENTS OF THE TOEPLITZ ALGEBRA OF THE AFFINE SEMIGROUP OVER THE NATURAL NUMBERS

We study the Toeplitz algebra 𝒯(ℕ⋊ℕ × ) and three quotients of this algebra: the C * -algebra 𝒬 ℕ recently 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⋊ℕ × 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 𝒯(ℕ⋊ℕ × ) to describe the KMS states on the two quotients. We then show that 𝒯(ℕ⋊ℕ × ), 𝒬 ℕ and our new quotients are all interesting new examples for Larsen’s theory of Exel crossed products by semigroups.

THE K-THEORY OF CUNTZ-KRIEGER ALGEBRAS FOR INFINITE MATRICES

We compute the K-theory groups of the Cuntz-Krieger C -algebra OA associ- ated to an infinite matrix A of zeros and ones.

Hecke algebras of semidirect products

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.

Phase transitions on Hecke C*-algebras and class-field theory over ℚ

Abstract We associate a canonical Hecke pair of semidirect product groups to the ring inclusion of the algebraic integers 𝒪 in a number field 𝒦, 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 KMSβ 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∞ 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∞ states to the (arithmetic) Hecke algebra over 𝒦, as given by class-field theory, corresponds to the action of the symmetry group if and only if the number field 𝒦 is ℚ.

EXTENDING MULTIPLIERS FROM SEMIGROUPS

A multiplier on a normal subsemigroup of a group can be extended to a multiplier on the group. This is used to show that normal cancellative semigroups have the same second cohomology as the group they generate, gen- eralising earlier results of Arveson, Chernoff, and Dinh. The main tool is a dilation theorem for isometric multiplier representations of semigroups.

Phase transition in the Connes–Marcolli GL2-system

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 6 0; 1. In particular, we show that for each 2 (1; 2) there exists a unique KMS -state.

Continuous Fell bundles associated to measurable twisted actions

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.