arXiv subjects

Fernando Lledó

Publications and source records attributed to Fernando Lledó.

30 records · Page 2Linked to original sources

arXiv2013

Følner sequences in operator theory and operator algebras

The present article is a review of recent developments concerning the notion of Følner sequences both in operator theory and operator algebras. We also give a new direct proof that any essentially normal operator has an increasing Følner sequence $\{P_n\}$ of non-zero finite rank projections that strongly converges to 1. The proof is based on Brown-Douglas-Fillmore theory. We use Følner sequences to analyze the class of finite operators introduced by Williams in 1970. In the second part of this article we examine a procedure of approximating any amenable trace on a unital and separable C*-algebra by tracial states $\mathrm{Tr}(\cdot P_n)/\mathrm{Tr}(P_n)$ corresponding to a Følner sequence and apply this method to improve spectral approximation results due to Arveson and Bédos. The article concludes with the analysis of C*-algebras admitting a non-degenerate representation which has a Følner sequence or, equivalently, an amenable trace. We give an abstract characterization of these algebras in terms of unital completely positive maps and define Følner C*-algebras as those unital separable C*-algebras that satisfy these equivalent conditions. This is analogous to Voiculescu's abstract characterization of quasidiagonal C*-algebras.

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arXiv2012

On spectral approximation, Følner sequences and crossed products

In this article we study Foelner sequences for operators and mention their relation to spectral approximation problems. We construct a canonical Foelner sequence for the crossed product of a discrete amenable group $Γ$ with a concrete C*-algebra A with a Foelner sequence. We also state a compatibility condition for the action of $Γ$ on A. We illustrate our results with two examples: the rotation algebra (which contains interesting operators like almost Mathieu operators or periodic magnetic Schrödinger operators on graphs) and the C*-algebra generated by bounded Jacobi operators. These examples can be interpreted in the context of crossed products. The crossed products considered can be also seen as a more general frame that included the set of generalized band-dominated operators.

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arXiv2012

F\olner sequences and finite operators

This article analyzes F\olner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by Williams in the 70ies. We prove that each essentially hyponormal operator has a proper F\olner sequence (i.e. a F\olner sequence of projections strongly converging to 1). In particular, any quasinormal, any subnormal, any hyponormal and any essentially normal operator has a proper F\olner sequence. Moreover, we show that an operator is finite if and only if it has a proper F\olner sequence or if it has a non-trivial finite dimensional reducing subspace. We also analyze the structure of operators which have no F\olner sequence and give examples of them. For this analysis we introduce the notion of strongly non-F\olner operators, which are far from finite block reducible operators, in some uniform sense, and show that this class coincides with the class of non-finite operators.

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arXiv2009

Modular Theory by example

The present article contains a short introduction to Modular Theory for von Neumann algebras with a cyclic and separating vector. It includes the formulation of the central result in this area, the Tomita-Takesaki theorem, and several of its consequences. We illustrate this theory through several elementary examples. We also present more elaborate examples and compute modular objects for a discrete crossed product and for the algebra of canonical anticommutation relations (CAR-algebra) in a Fock representation.

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arXiv2009

Operator algebras: an informal overview

In this article we give a short and informal overview of some aspects of the theory of C*- and von Neumann algebras. We also mention some classical results and applications of these families of operator algebras.

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arXiv2008

Realization of minimal C*-dynamical systems in terms of Cuntz-Pimsner algebras

In the present paper we study tensor C*-categories with non-simple unit realised as C*-dynamical systems (F,G,β) with a compact (non-Abelian) group G and fixed point algebra A := F^G. We consider C*-dynamical systems with minimal relative commutant of A in F, i.e. A' \cap F = Z, where Z is the center of A which we assume to be nontrivial. We give first several constructions of minimal C*-dynamical systems in terms of a single Cuntz-Pimsner algebra associated to a suitable Z-bimodule. These examples are labelled by the action of a discrete Abelian group (which we call the chain group) on Z and by the choice of a suitable class of finite dimensional representations of G. Second, we present a construction of a minimal C*-dynamical system with nontrivial Z that also encodes the representation category of G. In this case the C*-algebra F is generated by a family of Cuntz-Pimsner algebras, where the product of the elements in different algebras is twisted by the chain group action. We apply these constructions to the group G = SU(N).

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arXiv2007

Existence of spectral gaps, covering manifolds and residually finite groups

In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $Γ$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings $(X,g_\eps) \to (M,g_\eps)$ such that the spectrum of the Laplacian $Δ_{(X_\eps,g_\eps)}$ has at least a prescribed finite number of spectral gaps provided $\eps$ is small enough. If $Γ$ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that $\spec Δ_{(X,g_\eps)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(λ)$ of components of $\spec {\laplacian {(X,g_\eps)}}$ that intersect the interval $[0,λ]$. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results.

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arXiv2006

On the nuclearity of certain Cuntz-Pimsner algebras

We give a short proof of the nuclearity property of a class of Cuntz-Pimsner algebras associated with a Hilbert A-bimodule M, where A is a separable and nuclear C*-algebra. We assume that the left A-action on the bimodule M is given in terms of compact module operators and that M is direct summand of the standard Hilbert module over A.

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arXiv2005

Generating spectral gaps by geometry

Motivated by the analysis of Schrödinger operators with periodic potentials we consider the following abstract situation: Let $Δ_X$ be the Laplacian on a non-compact Riemannian covering manifold $X$ with a discrete isometric group $Γ$ acting on it such that the quotient $X/Γ$ is a compact manifold. We prove the existence of a finite number of spectral gaps for the operator $Δ_X$ associated with a suitable class of manifolds $X$ with non-abelian covering transformation groups $Γ$. This result is based on the non-abelian Floquet theory as well as the Min-Max-principle. Groups of type I specify a class of examples satisfying the assumptions of the main theorem.

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arXiv2004

Duality of compact groups and Hilbert C-systems for C-algebras with a nontrivial center

In the present paper we prove a duality theory for compact groups in the case when the C*-algebra A, the fixed point algebra of the corresponding Hilbert C*-system (F,G), has a nontrivial center Z and the relative commutant satisfies the minimality condition A.'\cap F = Z as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C*-system is expressed by means of an inclusion of C*-categories T_\c < T, where T_cis a suitable DR-category and T a full subcategory of the category of endomorphisms of A. Both categories have the same objects and the arrows of T can be generated from the arrows of T_cand the center Z. A crucial new element that appears in the present analysis is an abelian group C(G), which we call the chain group of G, and that can be constructed from certain equivalence relation defined on G^, the dual object of G. The chain group, which is isomorphic to the character group of the center of G, determines the action of irreducible endomorphisms of A when restricted to Z. Moreover, C(G) encodes the possibility of defining a symmetry $ε$ also for the larger category T of the previous inclusion.

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arXiv2003

Massless relativistic wave equations and quantum field theory

We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincare' group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and canonical representations do not coincide. We use massless relativistic wave equations as constraint equations reducing the fibre degrees of freedom of the covariant representation. They are characterized by invariant (and in contrast with the massive case non reducing) one-dimensional projections. The definition of one-particle Hilbert space structure that specifies the quantum field uses distinguished elements of the intertwiner space between E(2) (the two-fold cover of the 2-dimensional Euclidean group) and its conjugate. We conclude with a brief comparison between the free nets constructed in Section 3 and a recent alternative construction that uses the notion of modular localization.

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arXiv2002

Twisted duality of the CAR-Algebra

We give a complete proof of the twisted duality property M(q)'= Z M(q^\perp) Z* of the (self-dual) CAR-Algebra in any Fock representation. The proof is based on the natural Halmos decomposition of the (reference) Hilbert space when two suitable closed subspaces have been distinguished. We use modular theory and techniques developed by Kato concerning pairs of projections in some essential steps of the proof. As a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator. This formula can be also applied to fermionic free nets, hence giving a formula of the modular operator for any double cone.

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