
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
JanuaryJune
2014
Thematic Program on Abstract Harmonic Analysis,
Banach and Operator Algebras
May
2014
Theme Period on
Operator Spaces, Locally Compact Quantum Groups
and Amenability
Organizer:
Volker Runde



May 1223, 2014
MiniCourses: Operator Spaces, Locally Compact Quantum Groups and Amenability
Video archive
of talks

Monday
May 12
Room 230

Tuesday
May 13
Room 230

Wednesday
May 14
Room 230

Thursday
May 15
Room 230

Friday
May 16
Room 230

10:00 a.m. 
Volker Runde
Amenability of Banach algebras,1

Volker Runde
Amenability of Banach algebras,3


Michael White
Cohomology of Banach and topological algebras,1
(lecture notes)

Michael White
Cohomology of Banach and topological algebras,3
(lecture notes)

11:00 a.m. 

David Blecher,
Operator spaces,1
(lecture notes) 
David Blecher,
Operator spaces,3
(lecture notes) 
Piotr Soltan
On amenability and injectivity for locally compact quantum groups



Lunch Break 
2:00 p.m.

Volker Runde
Amenability of Banach algebras,2

Volker Runde
Amenability of Banach algebras,4 

Michael White
Cohomology of Banach and topological algebras.2

Michael White
Cohomology of Banach and topological algebras,4

3:30 p.m. 

David Blecher
Operator spaces,2

David Blecher
Operator spaces,4



Piotr Soltan
On amenability and injectivity for locally compact quantum groups
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 2023

Monday
May 19

Tuesday
May 20
Room 230

Wednesday
May 21
Room 230

Thursday
May 22
Stewart Library

Friday
May 23
Stewart Library

10:00 a.m. 
Victoria Day Holiday





11:00 a.m. 
Zinaida Lykova
Higherdimensional amenability,1
(lecture notes) 
Zinaida Lykova
Higherdimensional amenability
(lecture notes) 
Ebrahim Samei
Quantization of topological homology 
Ebrahim Samei
Quantization of topological homology 

Lunch Break 
2:00 p.m. 
Matt Daws
Locally Compact Quantum Groups,2
(lecture notes)




3:30 p.m. 
Zinaida Lykova
Higherdimensional amenability,2

Zinaida Lykova
Higherdimensional amenability

Ebrahim Samei
Quantization of topological homology

Ebrahim Samei
Quantization of topological homology

May 2630, 2014 Workshop
on Operator Spaces, Locally Compact Quantum Groups and Amenability

Monday

Tuesday

Wednesday

Thursday

Friday


10:0010:50




Rui
Okayasu
Haagerup approximation property and positive cones associated with
a von Neumann algebra


10:0010:50


Coffee Break

11:1012:00 
Reiji
Tomatsu
Product type actions of compact quantum groups 
Hun
Hee Lee
Weighted Fourier algebras on noncompact Lie groups and their spectrumy

Yemon
Choi
Unitarizable group representations and amenable operator algebras


Jason Crann
An uncertainty principle for unimodular quantum groups

11:1011:35


Yuhei
Suzuki
Amenable minimal Cantor systems of free groups arising from
diagonal actions

11:4012:05

12:1013:00



Yong
Zhang
Amenability properties of Banach algebra valued continuous functions


David
Kyed
Dimensionflatness and Lück's amenability conjecture.

12:1013:00


Lunch
Break 
15:3016:20


Coxeter
Lecture
Sorin Popa
On II1 factors arising from free groups acting on spaces

Coxeter
Lecture
Sorin Popa
On II1 factors arising from free groups acting on spaces

Coxeter
Lecture
Sorin Popa
On II1 factors arising from free groups acting on spaces


15:3016:20



Coxeter
Lecture Reception 

Sutanu
Roy
The Haagerup property of the generalised Drinfel'd double. 
Gilles Pisier
A continuum of $\mathrm{C}^*$norms on ${{\mathbb B}}(H)\otimes {{\mathbb
B}}(H)$ and related tensor products

16:30 
Mahmood Alaghmandan
Hypergroups and their amenability notions ( slides)
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.
Michael Brannan
$L_p$representations of discrete quantum groups.
Given a unimodular discrete quantum group $\mathbb G$, we
define and study unitary representations of $\mathbb G$ associated to the
noncommutative $L_p$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.
Martijn Caspers
The Haagerup property for arbitrary von Neumann algebras ( slides)
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, semifinite, 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
semifinite 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 groupHaagerup 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.
Yemon Choi
Unitarizable group representations and amenable operator algebras
( slides)
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 nonunitarizable 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.
Jason Crann
An uncertainty principle for unimodular quantum groups
( slides)
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 nonunimodular setting. Time permitting, we will
present a potential application to noncommutative random walks
on arbitrary discrete quantum groups. This is joint work with
Mehrdad Kalantar.
Pierre Fima
Graphs of quantum groups and Kamenability
We construct the fundamental group of a graph of quantum groups and
we show that it is Kamenable whenever the initial quantum groups are
amenable. This is a joint work with A. Freslon.
Mateusz Jurczynski
Quantum Wiener chaos expansion ( slides)
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 LieTrotter product formula for quantum stochastic evolutions
and the construction of Levy processes on compact quantum groups.
Mehrdad Kalantar
Quantum groups actions and noncommutative boundaries
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.
David Kyed
Dimensionflatness and Lück's amenability conjecture.
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.
Hun Hee Lee
Weighted Fourier algebras on noncompact Lie groups and their spectrum
( slides)
In this talk we will discuss a model for a weighted version
of Fourier algebras on noncompact 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.
ChiKeung Ng
Property T for locally compact quantum groups ( slides)
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]
Rui Okayasu
Haagerup approximation property and positive cones associated with a
von Neumann algebra ( slides)
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 oneparameter family of positive cones
due to H. Araki. We also discuss the Haagerup approximation property for
noncommutative Lpspaces. This is based on a joint work with Reiji Tomatsu.
Gilles Pisier
A continuum of $\mathrm{C}^*$norms on ${{\mathbb B}}(H)\otimes {{\mathbb
B}}(H)$\\ and related tensor products ( slides)
( abstract)
This is an account of joint work with N. Ozawa.
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 nonnuclear
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 nonseparability
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.
Alexei Pirkovskii
Taylor’s functional calculus and derived categories ( abstract)
( slides)
J. L. Taylor’s functional calculus theorem (1970)
asserts that every commuting ntuple 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 finitedimensional Stein space. The idea of TaylorPutinar’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 TaylorPutinar’s theorem to the setting of
derived categories. We believe that this is exactly the environment in
which TaylorPutinar’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 TaylorPutinar’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.
Krzysztof Piszczek
Amenability properties of the noncommutative Schwartz space
( slides)
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.
Sutanu Roy
The Haagerup property of the generalised Drinfel'd double .
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.
Yuhei Suzuki
Amenable minimal Cantor systems of free groups arising from diagonal
actions ( slides)
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 Kgroups for the diagonal actions of the boundary
action and the odometer transformations. These computations with Matui's
theorem classify their topological full groups.
Reiji Tomatsu
Product type actions of compact quantum groups (slides)
Abstract: A faithful product type action of the qdeformation
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.
Leonid Vainerman
Coamenability and quantum groupoids (slides)
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.
NgaiChing Wong
Isometries of real Hilbert C*modules ( slides)
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 2isometry;
(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 MingHsiu Hsu.
Yong Zhang
Amenability properties of Banach algebra valued continuous functions
( slides)
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 Avalued continuous functions is a Banach algebra. We investigate
amenability, weakly amenability and generalized amenability of this
algebra.
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