arXiv subjects

Nadia S. Larsen

Publications and source records attributed to Nadia S. Larsen.

36 records · Page 2Linked to original sources

arXiv2015

von Neuman algebras of strongly connected higher-rank graphs

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.

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arXiv2012

KMS states on Nica-Toeplitz algebras of product systems

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 the product system. Our results were motivated by, and generalize some of the results of Laca and Raeburn obtained for the Toeplitz algebra of the affine semigroup over the natural numbers.

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arXiv2012

Inner coactions, Fell bundles, and abstract uniqueness theorems

We prove gauge-invariant uniqueness theorems with respect to maximal and normal coactions for $C^*$-algebras associated to product systems of $C^*$-correspondences. Our techniques of proof are developed in the abstract context of Fell bundles. We employ inner coactions to prove an essential-inner uniqueness theorem for Fell bundles. As application, we characterise injectivity of homomorphisms on Nica's Toeplitz algebra $\Tt(G, P)$ of a quasi-lattice ordered group $(G, P)$ in the presence of a finite non-trivial set of lower bounds for all non-trivial elements in $P$.

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arXiv2012

The 2-adic ring $C^*$-algebra of the integers and its representations

We study the 2-adic version of the ring $C^*$-algebra of the integers. First, we work out the precise relation between the Cuntz algebra $\cO_2$ and our 2-adic ring $C^*$-algebra in terms of representations. Secondly, we prove a 2-adic duality theorem identifying the crossed product arising from 2-adic affine transformations on the 2-adic numbers with the analogous crossed product algebra over the real numbers. And finally, as an outcome of this duality result, we construct an explicit imprimitivity bimodule and prove that it transports one canonical representation into the other.

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arXiv2011

The Cuntz Algebra Q_N and C*-Algebras of Product Systems

We consider a product system over the multiplicative semigroup N^x of Hilbert bimodules which is implicit in work of S. Yamashita and of the second named author. We prove directly, using universal properties, that the associated Nica-Toeplitz algebra is an extension of the C^*-algebra Q_N introduced recently by Cuntz.

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arXiv2010

Dilations of semigroup crossed products as crossed products of dilations

Laca constructed a minimal automorphic dilation for every semigroup dynamical system arising from an action of an Ore semigroup by injective endomorphisms of a unital $C^*$-algebra. Here we show that the semigroup crossed product with its action by inner endomorphisms given by the implementing isometries has as minimal automorphic dilation the group crossed product of the original dilation. Applications include recent examples studied by Cuntz and the second named author.

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arXiv2009

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-universal C*-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.

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arXiv2008

Direct limits, multiresolution analyses, and wavelets

A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on $L^2(\R^n)$, the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids.

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arXiv2007

On Bost-Connes type systems for number fields

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.

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arXiv2007

Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra

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_β-states for β\ne 0,1.

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arXiv2007

Generalized multiresolution analyses with given multiplicity functions

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space $\H$ that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function $m$ which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space $\H$ is $L^2(\mathbb R^n)$, the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function $m$ satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function $m$.

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arXiv2007

Generalised Hecke algebras and C^*-completions

For a Hecke pair $(G, H)$ and a finite-dimensional representation $σ$ of $H$ on $V_σ$ with finite range we consider a generalised Hecke algebra $\H_σ(G, H)$, which we study by embedding the given Hecke pair in a Schlichting completion $(G_σ, H_σ)$ that comes equipped with a continuous extension $σ$ of $H_σ$. There is a (non-full) projection $p_σ\in C_c(G_σ, {\cc B}(V_σ))$ such that $\H_σ(G, H)$ is isomorphic to $p_σC_c(G_σ, {\cc B}(V_σ))p_σ$. We study the structure and properties of $C^*$-completions of the generalised Hecke algebra arising from this corner realisation, and via Morita-Fell-Rieffel equivalence we identify, in some cases explicitly, the resulting proper ideals of $C^*(G_σ, {\cc B}(V_σ))$. By letting $σ$ vary, we can compare these ideals. The main focus is on the case with $\dimσ=1$ and applications include $ax+b$-groups and the Heisenberg group.

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arXiv2007

Projective multi-resolution analyses arising from direct limits of Hilbert modules

The authors have recently shown how direct limits of Hilbert spaces can be used to construct multi-resolution analyses and wavelets in $L^2(\R)$. Here they investigate similar constructions in the context of Hilbert modules over $C^*$-algebras. For modules over $C(\T^n)$, the results shed light on work of Packer and Rieffel on projective multi-resolution analyses for specific Hilbert $C(\T^n)$-modules of functions on $\R^n$. There are also new applications to modules over $C(C)$ when $C$ is the infinite path space of a directed graph.

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arXiv2006

From filters to wavelets via direct limits

We present a new proof of a theorem of Mallat which describes a construction of wavelets starting from a quadrature mirror filter. Our main innovation is to show how the scaling function associated to the filter can be used to identify a certain direct limit of Hilbert spaces with $L^2(\R)$ in such a way that one can immediately identify the wavelet basis. Our arguments also use a pair of isometries introduced by Bratteli and Jorgensen, and exploit the geometry inherent in the Cuntz relations satisfied by these isometries.

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arXiv2005

Crossed products by abelian semigroups via transfer operators

We propose a generalisation of Exel's 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 Fowler's 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.

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arXiv2003

Subquotients of Hecke C^*-algebras

The Bost-Connes Hecke C^*-algebra can be regarded as a direct limit of subalgebras involving finite sets of primes. Each of these finite-prime analogues of the Bost-Connes algebra is a crossed product by a semigroup N^F, where F is finite. We describe composition series of ideals for these semigroup crossed products in which the intermediate subquotients are finite direct sums of classifiable AT-algebras. We then use similar techniques to identify subquotients of the Bost-Connes algebra as classifiable AT-algebras.

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arXiv2001

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.

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arXiv2001

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.

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