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An´ alise funcional geom´ etrica e a¸c˜ oes de grupos Universidade Federal de Santa Catarina Florian´ opolis, SC, Brazil 19–21 de Fevereiro 2014 Conferˆ encia ´ e apoiada em parte pelo projeto “Pesquisador Visitante Especial” do programa “Ciˆ encia sem Fronteiras” (Linha 1 - CAPES), processo 085/2012

Análise funcional geométrica e aç˜oes de grupos

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Page 1: Análise funcional geométrica e aç˜oes de grupos

Analise funcional geometrica e acoes de grupos

Universidade Federal de Santa CatarinaFlorianopolis, SC, Brazil

19–21 de Fevereiro 2014

Conferencia e apoiada em parte pelo projeto “Pesquisador Visitante Especial” do programa

“Ciencia sem Fronteiras” (Linha 1 - CAPES), processo 085/2012

Page 2: Análise funcional geométrica e aç˜oes de grupos

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Sumario

Horario das palestras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiResumos das palestras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Abdelmoubine Amar Henni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Alcides Buss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Arkady Leiderman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Artur Hideyuki Tomita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Bernhard Burgstaller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Brice Mbombo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Camila Fabre Sehnem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Christina Brech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Cleon S. Barroso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Dana Bartosova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Daniel Pellegrino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Eduardo Scarparo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Eliezer Batista . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Giuliano Boava . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Ivan Pontual Costa e Silva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Johan Oinert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Maria Inez Cardoso Goncalves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Martin Weilandt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Nilson C. Bernardes Jr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Ruy Exel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Tristan Bice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Valentin Burcea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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Horario das palestras

quarta quinta sexta19 fev. 20 fev. 21 fev.

9:15-9:45Nilson C. Bernardes Jr.

Generic Dynamics of

the Cantor space

Cleon BarrosoOn the approximate fixed

point property in abstract spaces

and applications

Artur TomitaCountably compact groups

with convergent sequences

10:00-10:30Giuliano Boava

Descricao das Algebras de Cuntz-Li

como Produtos Cruzados Parciais

Alcides BussWeakly proper actions,

imprimitivity theorems and

duality

Eduardo ScarparoQuasi-lattice ordered groups

and C∗-algebras

10:45-11:15Abdelmoubine Amar Henni

Instanton bundles on

projective spaces

Daniel PellegrinoAs desigualdades de

Bohnenblust-Hille

Camila Fabre SehnemUma classificacao de

fibrados de Fell estaveis

11:30-12:00

Bernhard BurgstallerThe universal property of

inverse semigroup equivariant

KK-theory

Tristan BiceAnnihilators in C∗-Algebras

Maria Inez Cardoso GoncalvesIsometries of a generalized

numerical radius

12:00-2:00 Pausa para o almoco

2:00-2:30

Ivan Pontual Costa e SilvaMarginally outer trapped surfaces

and geodesic incompleteness of

spacetimes

Ruy ExelK-theory of

Cuntz-Pimsner algebras

2:45-3:15Martin Weilandt

Isospectral Alexandrov spaces

Christina BrechUniversal Banach spaces

3:30-4:00

Arkady LeidermanOpen G-bases and compact

resolutions in topological groups

and locally convex spaces

Dana BartosovaLelek fan and

projective Fraısse theory

4:15-4:45

Eliezer BatistaPartial actions and

partial representations of

Hopf algebras

Johan OinertOuterness of partial actions

5:00-5:30Brice Mbombo Dempowo

Amenability test spaces

for Polish groups

Valentin BurceaCR Singularities in

Codimension 2: Normal Forms

A conferencia tera lugar na sala do LAED do Departamento de Matematica (a do Coloquio, noandar terreo).

Organizadores:

Daniel Goncalves (daniel.goncalves arroba ufsc ponto br),Ivan Pontual Costa e Silva (pontual.ivan arroba ufsc ponto br),Vladimir Pestov (vpest283 arroba uottawa ponto ca).

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Resumos das palestras

Instanton bundles on projective spacesAbdelmoubine Amar [email protected]

I’ll present an introduction to the study of instanton bundles using complexes called Monadsand which allow us to study families of instantons and in particular construct a ”moduli space”.

Weakly proper actions, imprimitivity theorems and dualityAlcides BussUniversidade Federal de Santa [email protected]

Coauthors: Siegfried Echterhoff

Abstract: In this talk I will give an introduction to the theory of weakly proper groupactions on C∗-algebras and show how this class of actions provides a unified approach to prove(and in many cases improve and/or generalize) all known imprimitivity and duality theoremsinvolving groups (co)actions and their crossed products, like for instance Green’s ImprimitivityTheorem, or more generally the Symmetric Imprimitivity Theorem, Mansfield’s ImprimitivityTheorem, and the Quigg-Landstad Duality Theorem for group coactions. This is joint workwith Siegfried Echterhoff.

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Open G-bases and compact resolutions in topological groups and locally convexspacesArkady LeidermanBen-Gurion University of the Negev, [email protected]

Coauthors: S. Gabriyelyan (Beer Sheva), J. Kakol (Poznan)

A Hausdorff topological group is said to have G-base if it admits a base of neighbourhoodsof the unit {Uα}, where indices α belong to the partially ordered set NN, and the conditionUα ⊂ Uβ , whenever β < α, holds.Each metrizable group has a G-base. One of the main results is

Theorem 1. A topological group is metrizable iff it is Frechet-Urysohn and has a G-base.

Characterizations of metrizability of locally convex spaces, in particular the spaces of continuousreal functions with compact-open topology Cc(X), are given using G-bases.

Theorem 2. If X is a metrizable kω-space, then the free abelian topological group A(X) andthe free locally convex topological space L(X) have a G-base.

It will be explained that G-bases are related naturally to the compact resolutions swallowingthe compact sets Another link is established with sequential properties.We show that any topological group with a G-base which is a k-space has the strong Pytkeevproperty.

Theorem 3. For a Cech-complete X the space Cc(X) has the strong Pytkeev property if andonly if is Lindelof.

The strong Pytkeev property has been investigated for several well known classes of locallyconvex spaces including (DF)-spaces and strict (LM)-spaces. We pose a dozen open questions.

Note that recently T. Banakh proved that the space Cc(X) has the strong Pytkeev propertyfor any separable metrizable X.

Countably compact groups with convergent sequencesArtur Hideyuki TomitaDepartment of Mathematics - Universidade de Sao [email protected]

Coauthors: Ana Carolina Boero (UFABC), Irene Castro Pereira (UFPA)

The construction of countably compact groups without non-trivial convergent sequences wasobtained using some set-theoretic assumption and it is still unknown if there is such an examplein ZFC.

Dikranjan asked whether every abelian group that admits a compatible countably compactgroup topology admits also a countably compact group topology with a non-trivial convergentsequence. For the torsion case it is not difficult to prove this is true. For the non-torsion case,Boero, Garcia Ferreira and Tomita (Topology Appl., 2012) showed that free abelian groupof cardinality continuum admits such a group topology using selective ultrafilters. We willdiscuss this example and sketch the more general result that answer Dikranjan’s question fornon-torsion abelian groups of cardinality at most continuum.

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The universal property of inverse semigroup equivariant KK-theoryBernhard [email protected]

After prework of Cuntz, Higson proved that every homotopy invariant, stable and split-exact functor from the category of separable C∗-algebras to abelian groups factors throughKasparov’s KK-theory. Thomson generalized this result to the G-equivariant setting, where Gdenotes a locally compact group. It seems to us that at a slight adaption of Thomson’s proofgives Higson’s universality result also in the inverse semigroup equivariant setting.

Amenability test spaces for Polish groupsBrice MbomboUniversidade de Sao [email protected]

Coauthors: Yousef Al-Gadid and Vladimir Pestov

A topological group G is amenable if every continuous action of G on a compact space Xadmits an invariant Borel probability measure.A compact space K is an amenability test space for a class C of topological groups if a groupG ∈ C is amenable if and only if every continuous action of G on K admits an invariant Borelprobability measure.De la Harpe and Giordano (C. R. Acad. Sci. Paris 324 (1997), 1255–1258), answering a questionof Grigorchuk, had proved that the Cantor space 2ℵ0 is an amenability test space for discretecountable groups. Bogatyi and Fedorchuk (Topol. Methods Nonlinear Anal. 29 (2007), 383–401) had obtained the same conclusion for the Hilbert cube [0, 1]ℵ0 .In this talk, we will eventualy (If time permits) provide a brief historic introduction to thenotion of amenability and discuss the problem of the existence of test spaces for amenabilityand extreme amenability for all polish groups.

Uma classificacao de fibrados de Fell estaveisCamila Fabre SehnemUniversidade Federal de Santa [email protected]

Coauthors: Ruy Exel

Dada uma acao parcial α de um grupo discreto G sobre uma C∗-algebra A, um fibradode Fell e associado ao C∗-sistema dinamico parcial (A,G, α). No entanto, nem todo fibradode Fell sobre um grupo discreto pode ser obtido a partir de um C∗-sistema dinamico parcial.Quando B e um fibrado de Fell estavel sobre um grupo enumeravel e a algebra da fibra unidadee separavel, R. Exel (Proc. London Math. Soc. 74 (1997), 417-443) provou que existe umaaco parcial torcida do grupo base na algebra da fibra unidade cujo fibrado de Fell associadocoincide com B. Nesta palestra, apresentaremos um progresso neste resultado, mostrando queB pode ser obtido a partir de uma acao parcial nao torcida do grupo base na algebra da fibraunidade.

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Universal Banach spacesChristina BrechUniversidade de Sao [email protected]

Coauthors: Piotr Koszmider

The classical result of Banach and Mazur states that C[0, 1] is a universal separable Banachspace. We consider the question on the existence of a universal space for the class of Banachspaces of some fixed larger density. The version which concerns isometric universality wasproved to be undecidable by results of Parovicenko and Shelah-Usvyatsov. Our main result isto prove the consistency of the nonexistence of a universal Banach space of density continuum.

On the approximate fixed point property in abstract spaces and applicationsCleon S. BarrosoUniversidade Federal do [email protected]

Coauthors: O. F. K. Kalenda and P.-K. Lin

In this talk we will present a research line known as approximate fixed point property inabstract spaces. With views to the structure of Banach spaces, some related problems andcontributions will be presented. We will also present some interesting applications related tothe fixed point theory, and also to the theory of ordinary differential equations in Banach spaces.

Lelek fan and projective Fraısse theoryDana BartosovaUniversidade de Sao [email protected]

Coauthors: Aleksandra Kwiatkowska

We construct the Lelek fan (the unique subfan of the Cantor fan with a dense set of end-points) as a natural quotient of a projective Fraısse family of finite directed trees. This allows usto show a homogeneity and projectivity property of the Lelek fan in the class of all smooth fans.We further explore the group of homeomorphisms of the Lelek fan, for instance its simplicity,dense conjugacy classes or connectivity properties. We also mention connections with the DualRamsey theory.

The theory of projective Fraısse classes and their limit was originated by Irwin and Soleckiin their model-theoretic and combinatorial treatment of the pseudoarc.

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As desigualdades de Bohnenblust–HilleDaniel PellegrinoUniversidade Federal da [email protected]

Resumo. A desigualdade multilinear de Bohnenblust–Hille, demonstrada em 1931 no Annalsof Mathematics, garante que para cada inteiro positivo m existe uma constante Cm ≥ 1 tal que N∑

i1,...,im=1

∣∣T (ei1 , ..., eim)∣∣ 2mm+1

m+12m

≤ Cm ‖T‖

para todos inteiros positivos N e todas formas m-lineares T definidas em `N∞ × · · · × `N∞. Adesigualdade polinomial de Bohnenblust–Hille e um resultado similar, que controla a normados coeficientes dos polinomios homogeneos. Embora tenham sido concebidas como ferramen-tas para o estudo de problemas relacionados a series de Dirichlet, atualmente as desigualdadesde Bohnenblust–Hille tem aplicacoes em diferentes reas da matemtica e ate mesmo em Teoriada Informacao Quantica. Curiosamente, em tais aplicacoes, o controle das constantes Cm tempapel central. Apresentaremos resultados recentes que mostram que, em forte contraste comas previsoes dos ultimos 80 anos, as constantes Cm tem um crescimento muito lento. Tambemmostramos que o crescimento das constantes da desigualdade polinomial de Bohnenblust–Hille,no caso de escalares complexos, e subexponencial. Aplicamos esse resultado para obter o com-portamento assintotico exato do raio de Bohr do polidisco n-dimensional.

References

1. N. Albuquerque, F. Bayart, D. Pellegrino and J. B. Seoane-Sepulveda, Sharp generaliza-tions of the multilinear Bohnenblust–Hille inequality, J. Funct. Anal.,DOI: 10.1016/j.jfa.2013.08.013 (to appear in print).

2. F. Bayart, D. Pellegrino and J. B. Seoane-Sepulveda, The Bohr Radius of the n-dimensionalpolydisk is equivalent to

√(log n)/n, arXiv:1310.2834v2 [math.FA], 15 Oct 2013.

3. D. Diniz, G.A. Munoz-Fernandez, D. Pellegrino, J.B. Seoane-Sepulveda, The asymptoticgrowth of the constants in the Bohnenblust-Hille inequality is optimal. J. Funct. Anal.263 (2012), no. 2, 415–428.

4. D. Nunez-Alarcon, D. Pellegrino, and J.B. Seoane-Sepulveda. On the Bohnenblust-Hilleinequality and a variant of Littlewood’s 4/3 inequality. J. Funct. Anal. 264 (2013), no.1, 326–336.

5. D. Nunez-Alarcon, D. Pellegrino, J.B. Seoane-Sepulveda, D.M. Serrano-Rodrıguez. Thereexist multilinear Bohnenblust-Hille constants (Cn)∞n=1 with limn→∞(Cn+1 − Cn) = 0. J.Funct. Anal. 264 (2013), no. 2, 429–463.

6. D. Pellegrino, J.B. Seoane-Sepulveda, New upper bounds for the constants in the Bohnenblust-Hille inequality. J. Math. Anal. Appl. 386 (2012), 300–307.

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Quasi-lattice ordered groups and C∗-algebrasEduardo ScarparoUniversidade Federal de Santa [email protected]

Quasi-lattice ordered groups were introduced by A. Nica, who associated C∗-algebras tothese structures and defined amenability in a natural way. We will present some examples andresults due to Nica, Laca and Raeburn about these C∗-algebras and also discuss a possiblegeneralization of quasi-lattice ordered groups.

Partial actions and partial representations of Hopf algebrasEliezer [email protected]

Partial actions is a subject of interest for operator algebras, dynamical systems and purealgebra as well. The intent of this talk is to present a generalization of the concept of partialactions and partial representations to the context of Hopf algebras. Partial representations ofHopf algebras are directly related to Hopf algebroids and this important connection allows oneto describe the monoidal category of partial modules. We present some nontrivial examples, asthe category of partially G graded modules and partially G graded algebras.

Descricao das algebras de Cuntz-Li como Produtos Cruzados ParciaisGiuliano BoavaUFSC, Florianopolis, [email protected]

Ha poucos anos, Cuntz e Li introduziram a C∗-algebra A[R] associada a um domınio deintegridade R com quocientes finitos por ideais principais. Nesta apresentacao, mostramos queA[R] e uma algebra parcial de grupo com um conjunto apropriado de relacoes e que o espectro detais relacoes e homeomorfo ao completamento profinito do anel R. A partir desta representacao,exibimos uma caracterizacao de A[R] como produto cruzado parcial. Por fim, mostramos quetais caracterizacoes podem ser usadas para recuperar resultados provados por Cuntz e Li como,por exemplo, que A[R] e simples (concluindo que a acao do sistema e topologicamente livre eminimal).

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Marginally outer trapped surfaces and geodesic incompleteness of spacetimesIvan Pontual Costa e SilvaUniversidade Federal de Santa [email protected]

In the context of Lorentzian geometry and mathematical Relativity, there has been a greatsurge of interest in (marginally) outer trapped surfaces, whose importance stems from thefact that they are both quasi-local versions of black hole horizons and natural generalizations ofminimal surfaces in initial data sets. In this lecture we review recent work relating the existenceof such objects in a spacetime with causal geodesic incompleteness therein.

Outerness of partial actionsJohan OinertLund University, [email protected]

Coauthors: Patrik Nystedt (University West, Sweden)

In this talk, I will explain how one can extend the classicial notion of an outer action α of agroup G on a unital ring A, to the case when α is a partial action on ideals, all of which havelocal units. Using a very general result from the theory of graded rings, we shall see that if αis an outer partial action of an abelian group G, then its associated partial skew group ringA ?α G is simple if and only if A is G-simple. Finally, I will explain how this result can beapplied in order to obtain generalizations of recent results by D. Goncalves (UFSC) on partialskew group algebras associated with certain types of partial dynamical systems.

Isometries of a generalized numerical radiusMaria Inez Cardoso [email protected]

Coauthors: A. R. Sourour, V. Pestov

For 0 < q < 1, the q-numerical radius is defined on the algebra Mn of all n × n complexmatrices by

rq(A) = max{|x∗Ay| : x, y ∈ Cn, ‖x‖ = ‖y‖ = 1, 〈y, x〉 = q}.

This definition can easily be extended to infinite dimension.During this talk I’ll characterize all the isometries of the q-numerical radius both on Mn an

on the space K(H) of compact operators on an infinite-dimensional Hilbert space H.While the strategy of the proof is broadly similar to the finite dimension case, the infinite

dimensional case poses numerous difficulties and requires new techniques. The main challengewas the usage of extreme points, which was the key component of the proofs in the finitedimension case.

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Isospectral Alexandrov spacesMartin WeilandtUniversidade Federal de Santa [email protected]

Coauthors: Alexander Engel (Universitt Augsburg)

The quotient of a compact Riemannian manifold by a compact group of isometries can beseen as an Alexandrov space. Generalizing Schth’s version of the so-called torus method to thissetting, we construct the first non-trivial examples of compact non-isometric Alexandrov spaceswhich are isospectral with respect to the Laplacian (but do not fall into the simpler categoryof Riemannian orbifolds).

Generic Dynamics of the Cantor SpaceNilson C. Bernardes Jr.Instituto de Matemtica - Universidade Federal do Rio de [email protected]

Coauthors: Udayan B. Darji

We develop unifying graph theoretic techniques to study the dynamics and the structureof spaces H(C) and C(C), the space of homeomorphisms and the space of self-maps of theCantor space, respectively. Using our methods, we give characterizations which determinewhen two homeomorphisms of the Cantor space are conjugate to each other. We also give anew characterization of the comeager conjugacy class of the space H(C). The existence of thisclass was established by Kechris and Rosendal and a specific element of this class was describedconcretely by Akin, Glasner and Weiss. Our characterization readily implies many old andnew dynamical properties of elements of this class. For example, we show that no element ofthis class has a Li-Yorke pair, implying the well known Glasner-Weiss result that there is acomeager subset of H(C) each element of which has topological entropy zero. Our analogousinvestigation in C(C) yields a surprising result: there is a comeager subset of C(C) such thatany two elements of this set are conjugate to each other by an element of H(C). Our descriptionof this class also yields many old and new results concerning dynamics of a comeager subset ofC(C). This is a joint work with Udayan B. Darji.

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K-theory of Cuntz-Pimsner algebrasRuy [email protected]

Given a C∗-algebra A and a corespondence M over A, one may construct an associatedC∗-algebra, usually denoted by OM , called the Cuntz-Pimsner algebra of M . Choosing theapropriate correspondence, one gets lots of interesting outcomes such as all Graph C∗-algebras,all Kirchberg algebras, and in fact many C∗-algebras that have been studied in the literaturein the last 35 years.

In the original 1997 paper where Pimsner introduced this construction a method was descri-bed for the computation of the K-theory groups of OM . However the application of the methodin concrete examples presents some daunting difficulties, sometimes preventing a satisfactoryfinal answer.

In this talk I will briefly describe the construction of the Cuntz-Pimsner algebras and amore pedestrian way to compute K-theory groups under certain (quite often verifiable) specialhypothesis. Time permiting I will show how these ideas may be used to study certain C∗-algebras associated to integer dilation matrices.

This is joint work with Astrid an Huef and Iain Raeburn, from the University of Otago,New Zealand.

Annihilators in C∗-AlgebrasTristan [email protected]

I outline my recent research regarding a quite basic C∗-algebraic structure which has, uptill now, received little attention from the operator algebra community. Specifically, I lookat C∗-algebra annihilators, the hereditary C∗-subalgebras coming from left/right annihilatorideals. It turns out that these are, in many ways, the best C∗-algebra analogs of projectionsin von Neumann algebras and, indeed, every annihilator in a von Neumann algebra A is of theform pAp for some p in A. Unlike projections, however, annihilators still exist in abundanceand form a complete lattice in an arbitrary C∗-algebra. They also admit a simple notion ofequivalence which is completely consistent with the classical notion of Murray-von Neumannequivalence. Moreover, we can use annihilators to decompose an arbitrary C∗-algebra in a waythat is again completely consistent with the classical von Neumann algebra type decomposition.

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CR Singularities in Codimension 2: Normal FormsValentin [email protected]

I will survey some results about the real submanifolds in the complex space and the normalforms constructions near a CR singularity. In particularly, I shall focus on the case when thecodimension of M is 2. Depending on the available time, I will present also some of my veryrecent work in this direction.

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