Nadia S. Larsen

CO-UNIVERSAL ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS, AND GAUGE-INVARIANT UNIQUENESS THEOREMS

Let X be a product system over a quasi-lattice ordered group. Under mild hypotheses, we associate to X a C � -algebra which is co-universal for injective Nica covariant Toeplitz representations of X which preserve the gauge coaction. Under appropriate amenability criteria, this co-universalC � -algebra coincides with the Cuntz- Nica-Pimsner algebra introduced by Sims and Yeend. We prove two key uniqueness theorems, and indicate how to use our theorems to realise a number of reduced crossed products as instances of our co-universal algebras. In each case, it is an easy corollary that the Cuntz-Nica-Pimsner algebra is isomorphic to the corresponding full crossed product.

Crossed products by abelian semigroups via transfer operators

We propose a generalisation of Exels crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowlers theory of Toeplitz and Cuntz-Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz-Pimsner algebras of product systems.

Partial actions and KMS states on relative graph C⁎-algebras ☆

Abstract The relative graph C ⁎ -algebras introduced by Muhly and Tomforde are generalizations of both graph algebras and their Toeplitz extensions. For an arbitrary graph E and a subset R of the set of regular vertices of E we show that the relative graph C ⁎ -algebra C ⁎ ( E , R ) is isomorphic to a partial crossed product for an action of the free group generated by the edge set on the relative boundary path space. Given a time evolution on C ⁎ ( E , R ) induced by a function on the edge set, we characterize the KMS β states and ground states using an abstract result of Exel and Laca. Guided by their work on KMS states for Toeplitz–Cuntz–Krieger type algebras associated to infinite matrices, we obtain complete descriptions of the convex sets of KMS states of finite type and of KMS states of infinite type whose associated measures are supported on recurrent infinite paths. This allows us to give a complete concrete description of the convex set of all KMS states for a big class of graphs which includes all graphs with finitely many vertices.

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.

On C*-algebras associated to right LCM semigroups

We initiate the study of the internal structure of C*-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroups or semigroups that satisfy Cliffords condition. Our main findings are results about uniqueness of the full semigroup C*-algebra. We build our analysis upon a rich interaction between the group of units of the semigroup and the family of constructible right ideals. As an application we identify algebraic conditions on S under which C*(S) is purely infinite and simple.

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.

Representations of Hecke Algebras and Dilations of Semigroup Crossed Products

We consider a family of Hecke C*-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.

KMS STATES ON NICA–TOEPLITZ ALGEBRAS OF PRODUCT SYSTEMS

We investigate KMS states of Fowlers Nica-Toeplitz algebra

Crossed Products by Semigroups of Endomorphisms and Groups of Partial Automorphisms

\mathcal{NT}(X)

Equilibrium states on right LCM semigroup C*-algebras

associated to a compactly aligned product system