May 12-23, 2014
Mini-Courses: Operator Spaces, Locally Compact Quantum Groups and Amenability
Video archive of talks

I will recall the notion of amenability of a locally compact quantum group and prove a theorem providing a characterization of this property in terms of injectivity of the von Neumann algebra associated to the dual (quantum) group.

Minicourses Week of May 20-23

May 26-30, 2014 Workshop on Operator Spaces, Locally Compact Quantum Groups and Amenability

Abstract: In this talk, after defining hypergroups and introducing algebras constructed on them, we discuss their different amenability notions. We specifically consider these notions for some classes of hypergroup structures related to locally compact groups. Subsequently, we demonstrate some applications to locally compact groups and their Banach algebras.

Given a unimodular discrete quantum group $\mathbb G$, we define and study unitary representations of $\mathbb G$ associated to the non-commutative $L_p$-spaces $L_p(\mathbb G)$. After discussing some general aspects of $L_p$-representations, we will show how this theory can be applied to construct new examples of exotic quantum group norms on the algebras of polynomial functions on duals of unimodular orthogonal free quantum groups. This is joint work with Z.-J. Ruan.

Abstract: We introduce a natural generalization of the Haagerup property of a finite von Neumann algebra to an arbitrary von Neumann algebra equipped with a normal, semi-finite, faithful weight and prove that this property does not depend on the choice of the weight. In particular this defines the Haagerup property as an intrinsic invariant of the von Neumann algebra. Our initial definition/approach is in terms of cp maps preserving a given weight and therefore stays close to the semi-finite case by M. Choda/P. Jolissaint. However, our techniques rely on crossed product duality.
We shall discuss stability properties of the Haagerup property regarding crossed products and free products. We also show how to define a noncommutative counterpart of the group-Haagerup property in terms of the existence of a proper, continuous, conditionally negative de?nite function.
Our results are motivated by recent examples from the theory of discrete quantum groups, where the Haagerup property appears a priori only with
respect to the Haar state. We will review some of these examples.
This is joint work with Adam Skalski.

All bounded continuous representations of a locally compact amenable group on a Hilbert space H are unitarizable inside B(H): this is an old result of Dixmier and Day. Is the same true for representations inside the Calkin algebra? It turns out that the answer is sometimes yes and sometimes no, with a crucial role played by the size of the group. In this talk I will explain more precisely what this means, and show how the non-unitarizable representations may be combined with an ingenious idea of Ozawa to produce the first known examples of amenable operator algebras that are not isomorphic to C*-algebras. If time permits I will discuss where the study of amenable operator algebras might go from here. This is based on joint work with I. Farah and N. Ozawa.

Heisenberg's celebrated uncertainty principle, concerning measurements of position and momentum, roughly states that a function and its Fourier transform cannot both be highly concentrated. In 1957, Hirschman extended this uncertainty relation to locally compact Abelian groups by using the relative entropy with respect to the Haar measure to quantify the degree of concentration. In this talk, we will extend Hirschman's result to unimodular quantum groups and discuss various consequences along with partial results in the non-unimodular setting. Time permitting, we will present a potential application to non-commutative random walks on arbitrary discrete quantum groups. This is joint work with Mehrdad Kalantar.

We construct the fundamental group of a graph of quantum groups and we show that it is K-amenable whenever the initial quantum groups are amenable. This is a joint work with A. Freslon.

Operator spaces provide a natural framework for investigating quantum stochastic processes. In this talk we will present some important classes of such processes, all of which have the special property of being representable as ``quantum Wiener integrals". Direct applications include Lie--Trotter product formula for quantum stochastic evolutions and the construction of Levy processes on compact quantum groups.

We study various properties of noncommutative Poisson boundaries of convolution maps arising from locally compact quantum group actions. We also introduce a quantum version of Jaworski's notion of SAT actions, and investigate basic properties of such actions. The is based on joint work with M. Amini and M. S. M. Moakhar, and M. Kennedy.

I will discuss the notion of dimension flatness, as well as its connection with amenability, and present a positive answer to a diffuse analogue of Lück's amenability conjecture. The talk is based on joint works with V. Alekseev and H.D. Petersen.

In this talk we will discuss a model for a weighted version of Fourier algebras on non-compact Lie groups. If we recall that the spectrum of the Fourier algebra is nothing but the underlying group itself (as a topological space), then it is natural to be interested in determining the spectrum of weighted algebras. We will demonstrate that the spectrum of the resulting commutative Banach algebra is realized inside the complexification of the underlying Lie group by focusing on the case of Heisenberg group and determine them in some concrete cases.

We generalized several equivalences of property T to the case of general locally compact quantum groups. When G is a locally com- pact quantum group of Kac type, we also show that G has property T if and only if every finite dimensional irreducible ¤-representation of Cu 0 (b G) is an isolated point in the spectrum of Cu 0 (b G). On the other hand, as a first step in an attempt to generalize the Delmorme- Guichardet theorem, we introduce, in the case of a second countable locally compact group G, a cohomology theory for the convolution ¤-algebra Cc(G), equipped with an inductive limit topology and the canonical character "G : Cc(G) ! C, and show that the vanishing of all such first cohomologies is equivalent to the property T of G.
[Based on joint works with Chen Xiao]

We discuss various definitions of the Haagerup approximation property for an arbitrary von Neumann algebra. As a consequence, we give a simple and direct proof that the definition given by M. Caspers and A. Skalski is equivalent to our original one defined by using the standard form. Our strategy is to use the one-parameter family of positive cones due to H. Araki. We also discuss the Haagerup approximation property for non-commutative Lp-spaces. This is based on a joint work with Reiji Tomatsu.

For any pair $M,N$ of von Neumann algebras such that the algebraic tensor product $M\otimes N$ admits more than one $\mathrm{C}^*$-norm, the cardinal of the set of $\mathrm{C}^*$-norms is at least $ {2^{\aleph_0}}$. Moreover there is a family with cardinality $ {2^{\aleph_0}}$ of injective tensor product functors for $\mathrm{C}^*$-algebras in Kirchberg's sense. Let $\mathbb B=\prod_n M_{n}$. We also show that, for any non-nuclear von Neumann algebra $M\subset \mathbb B(\ell_2)$, the set of $\mathrm{C}^*$-norms on $\mathbb B\otimes M$ has cardinality {\it equal to} $2^{2^{\aleph_0}}$. The talk will also recall the connection of such questions with the non-separability of the set of finite dimensional (actually $3$-dimensional) operator spaces which goes back to a 1995 paper with Marius Junge, and several recent ``quantitative" refinements obtained using quantum expanders.

J. L. Taylor’s functional calculus theorem (1970) asserts that every commuting n-tuple T = (T1, . . . , Tn) of bounded linear operators on a Banach space E admits a holomorphic functional calculus on any neighborhood U of the joint spectrum ¾(T). This means that there exists a continuous homomorphism ° : O(U) ! B(E) (where O(U) is the algebra of holomorphic functions on U and B(E) is the algebra of bounded linear operators on E) that takes the coordinates z1, . . . , zn to T1, . . . , Tn, respectively. The original Taylor’s proof was quite involved. In 1972, Taylor developed a completely different and considerably shorter proof based on methods of Topological Homology. Later it was simplified and generalized by M. Putinar (1980) to the case of Fr´echet O(X)-modules, where X is a finite-dimensional Stein space. The idea of Taylor-Putinar’s construction is to establish an isomorphism between a Fr´echet O(X)-module M satisfying ¾(M) ½ U and the 0th cohomology of a certain double complex C of Fr´echet O(U)-modules. Unfortunately, C depends on the choice of a special cover of X by Stein open sets, and there seems to be no canonical way of associating C to M.

Our goal is to extend Taylor-Putinar’s theorem to the setting of derived categories. We believe that this is exactly the environment in which Taylor-Putinar’s
theorem is most naturally formulated and proved. Given an object M of the derived category D-(O(X)-mod) of Fr´echet O(X)-modules, we define the spectrum ¾(M) ½ X, and we show that for every open set U ½ X containing ¾(M) there is an isomorphism M »= R¡(U,OX) b­LO(X)M in D-(O(X)-mod). In the special case where M is a Fr´echet O(X)-module, this yields Taylor-Putinar’s result. Moreover, we have C = R¡(U,OX) b­LO(X)M, so C is natural in M when viewed as an object of the derived category.

The talk will be devoted to S, the so called noncommutative Schwartz space. This LMC Fr´echet ¤-algebra is a noncommutative analogue of the very important
Schwartz space, appearing naturally in the structure theory of Fr´echet spaces. The Schwartz space has also several natural representations as a space of functions. We will look at S from the viewpoint of the automatic continuity theory and we will examine how ‘amenable’ this algebra is. By the result of Pirkovskii we know S is not amenable. The main result of the talk will tell us S is not boundedly approximately amenable however it is approximately amenable.

In this talk, we first discuss the generslised Drinfel'd double construction within the scope of modular or manageable multiplicative unitaries.
Our construction uses the following data: two C*-quantum groups and a bicharacter between them. Then we show that the generalised Drinfeld double has the Haagerup property whenever the underlying two quantum groups have the same. This shows that the Drinfel'd double of quantum ax+b, az+b and E(2)
groups has the Haagerup property.

We study amenable minimal Cantor systems of free groups. We show for every free group, (explicitly given) continuum many Kirchberg algebras are realized as the crossed product of an amenable minimal Cantor system of it. In particular this shows there are continuum many Kirchberg algebras such that each of which is decomposed tothe crossed products of amenable minimal Cantor systems of any virtually free group. We also give computations of K-groups for the diagonal actions of the boundary action and the odometer transformations. These computations with Matui's theorem classify their topological full groups.

Abstract: A faithful product type action of the q-deformation of a connected semisimple compact Lie group is discussed.
Our main theorem states that such an action is induced from a minimal action
of the maximal torus. I will sketch out its proof.

We are discussing in which way the definitions and results on coamenable compact quantum groups can be extended to the framework of compact quantum groupoids. Some motivating examples are presented.

Let T be a surjective real linear isometry between full real Hilbert C*-modules V and W, over real C*-algebras A and B, respectively. We show that the following conditions are equivalent.

(a) T is a 2-isometry;
(b) T is a complete isometry;
(c) T preserves ternary products;
(d) T preserves inner products;
(e) T is a module map.

When A and B are commutative, all these five conditions hold automatically.

This is a joint work with Ming-Hsiu Hsu.

Let X be a compact Hausdorff space and A a Banach algebra. Then, with pointwise operations and the uniform norm, the space C(X,A) of all A-valued continuous functions is a Banach algebra. We investigate amenability, weakly amenability and generalized amenability of this algebra.

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