On projections in the group algebra of unimodular groups
Mahmood Alaghmandan
University of Waterloo
Coauthors: M. Ghandehari, N. Spronk, and K. F. Taylor
Let G be a locally compact group, and L^1(G) is its group algebra. Self-adjoint
idempotent elements of L^1(G) are called projections. This talk concerns
with the study of projections in L^1(G) in the special case where G is unimodular.
We show that L^1-projections in the unimodular case have a rather special
form, namely they belong to the Fourier algebra. We then study minimal projections
in details, and investigate how the support of a projection relates to its
representing form in the Fourier algebra.
A Peak-Point Theorem for Uniform Algebras on Real-Analytic
Varieties
John T. Anderson
College of the Holy Cross
Coauthors: Alexander J. Izzo
It was once conjectured that if $A$ is a uniform algebra on its maximal
ideal space $X$, and if each point of $X$ is a peak point for $A$, then
$A=C(X)$. This peak-point conjecture was disproved by Brian Cole in 1968.
We establish a peak-point theorem for uniform algebras generated by real-analytic
functions on real-analytic varieties, generalizing previous results of
the authors and John Wermer. To appear in \emph{Mathematische Annalen}.
Generalizing C*-techniques to Banach and operator algebras
David Blecher
University of Houston
With Charles Read we have introduced and studied a new notion of (real)
positivity in operator algebras, with an eye to extending certain C*-algebraic
results and theories to more general algebras. As motivation note that
the ‘completely' real positive maps on C*-algebras or operator systems
are precisely the completely positive maps in the usual sense; however
with real positivity one may develop a useful order theory for more general
spaces and algebras. This is intimately connected to new relationships
between an operator algebra and the C*-algebra it generates. We have continued
this work together with Read, and also with Matthew Neal, and with Narutaka
Ozawa we have investigated the parts of the theory that generalize further
to Banach algebras. We describe some of this work, emphasizing the most
recent papers; and some refinements and advances to date, in the framework
of generalizing various C*-algebraic techniques and results.
Derivations on semigroup algebras and Fourier algebras
Yemon Choi
Lancaster University
The study of continuous derivations from commutative Banach algebras
into certain modules has a long history, yet there still remain natural
examples where --- in my view --- there is scope for further exploration.
One fundamental problem, which has been resolved only for a few particular
algebras, is to find a concrete description of the Banach-algebraic version
of the K\"ahler module.
Historically, semigroup algebras of various sorts have attracted more
attention than Fourier algebras, but recent work of various authors suggests
that the latter class may be more accessible than previously thought.
In this talk I will sketch of some of the known results for various examples,
some of which might not usually be thought of as semigroup algebras. I
will then give an outline of recent developments for the Fourier algebras
of connected Lie groups; some of these results are joint work with M.
Ghandehari (Waterloo). Time permitting, I will suggest some possible directions
for future research.
Characterization of commutative subalgebras of the algebra
of smooth operators
Tomasz Cias
Adam Mickiewicz University in Poznan
We show that closed commutative ${}^*$-subalgebras of the Fr\'echet ${}^*$-algebra
$\mathcal{L}(s',s)$ of smooth operators (also known as the algebra of
rapidly decreasing matrices) can be identified with the class of biprojective
K\"othe sequence algebras with the so-called property (DN). As a
consequence, every closed subspace of the Fr\'echet space $s$ of rapidly
decreasing sequences (and thus every closed subspace of the Schwartz space
$\mathcal{S}(\mathbb{R})$ and many other spaces of smooth functions) with
Schauder basis is isomorphic as a Fr\'echet space to some closed commutative
${}^*$-subalgebra of $\mathcal{L}(s',s)$.
Closed quantum subgroups of amenable quantum groups are
amenable
Jason Crann
Carleton University and University of Lille 1
It is well-known that closed subgroups of locally compact amenable groups
are amenable. In this talk, we will show that the corresponding relation
holds at the level of locally compact quantum groups using a new homological
characterization of amenability.
Banach function algebras and BSE norms
H. G. Dales
Lancaster University
I shall discuss Banach function algebras on a locally compact space.
I shall define the BSE norm on such an algebra and explain when the algebra
is a
BSE algebra. This will include an explanation of what `BSE' stands for.
There will be a variety of examples of algebras which do and do not have
a BSE norm/are BSE algebras. Classes of Banach function algebras considered
will include Banach sequence algebras, the Fourier algebra, the Figà-Talamanca-Herz
algebras, and certain Segal algebras based on a locally compact group.
The talk is based on the following two joint papers with Ali Ülger
of Istanbul:
1. H. G. Dales and A. Ülger, Approximate identities in Banach
function algebras, Studia Mathematica, 226 (2015), 155{187.
2. H. G. Dales and A. Ülger, Banach function algebras and BSE norms,
pp. 30, in preparation.
Biggest open ball in invertible elements of a Banach algebra
Sukumar Daniel
Indian Institute of Technology Hyderabad
Coauthors: Geethika Sebastian
Let $A$ be a unital Banach algebra and $G(A)$ be the set of invertible
elements of $A$. For an element $a$ in $G(A)$, we know that $B\left( a,\frac{1}{\|a^{-1}\|}\right)$
is contained in $G(A)$. An element $a$ in $G(A)$ is said to satisfy BOBP
(Biggest open ball property) if this is the biggest ball which is centered
at $a$ and is contained in $G(A)$. That is the boundary of this ball necessarily
intersects the set of non-invertible elements. A Banach algebra $A$ is
said to satisfy BOBP if every element $a$ in $G(A)$ satisfies BOBP. We
make an attempt on understanding Banach Algebras which satisfy BOBP. If
not then conditions on the elements to satisfy BOBP. The origin of this
problem is connected with condition spectra and almost multiplicative
functions. %\\Can we characterize all the elements of a Banach Algebra,
which do not satisfy BOBP? To actualise the above mentioned objective,
we discuss the observations made in some classical Banach Algebras.
Duality, convexity and peak interpolation in Drury-Arveson
space
Kenneth Davidson
University of Waterloo
The closed algebra $\mathcal{A}_d$ is generated by the polynomial multipliers
on the Drury-Arveson space. We identify $\mathcal{A}_d^*$ as the direct
sum of the preduals of the full multiplier algebra and of a commutative
von Neumann algebra. This provides a natural analogue of classical results
concerning the dual of the ball algebra. These results is applied to questions
about peak interpolation for multipliers. We also provide some applications
to multivariable operator theory.
This is joint work with Raphaël Clouâtre.
Numerical range of adjointable operators on a Hilebert
C*-module and application to problem of positivity preserving linear maps
of C*-algebras.
Rachid El Harti
University Hassan I Morocco
We define and investigate a new numerical range of adjointable operator
on some Helbert C$\sp*$-module over C$\sp*$ algebra. It is used to characterize
positive linear maps between the algebras of adjointable operators.
Cosine families close to scalar cosine families
Jean Esterle
University of Bordeaux
Let $G$ be an abelian group and let $A$ be a commutative unital Banach
algebra. A $G$-cosine family is a family $(C(g))_{g\in G}$ of elements
of $A$ satisfying the so-called d'Alembert equation $$C(0)=1_A, C(s+t)+C(s-t)=2C(s)C(t)
\ \ s,t \in G.$$ The purpose of the talk is to present recent results
proved by Chojnacki, Bobrowski, Gregoriewicz, Schwenninger, Zwart and
the author, which follow a recent paper by Chojnacki and Bobrowski, which
describe cosine families close to bounded scalar ones. We will present
in particular \begin{itemize} \smallskip \item A first zero-two law: if
a cosine function $(C(t))_{t\in \R}$ satisfies lim sup$_{t\to 0^+}\Vert
1_A-C(t)\Vert <2,$ then $C(t)=1_A$ for $t\in \R.$ \smallskip \item
A second zero-two law: if a cosine function $(C(t))_{t\in \R}$ satisfies
sup$_{t\in \R}\Vert C(t)-cos(at)1_A\Vert <2$ for fome $a\in \R,$ then
the closed algebra generated by $(C(t))_{t\in \R}$ is finite dimensional.
A similar result holds for cosine sequences $C(n)_{n\in \Z},$ but not
for cosine falimies over general groups. \smallskip \item A zero-${8\over
3\sqrt 3}$ law: if a cosine function $(C(t))_{t\in \R}$ satisfies sup$_{t\in
\R}\Vert C(t)-cos(at)1_A\Vert <{8\over 3\sqrt 3},$ then $C(t)=cos(at).1_A$
for $t\in \R.$ \smallskip \item A zero-${\sqrt 5\over 2}$ law: if a cosine
family $(C(g))_{g\in G}$ satisfies sup$_{g\in G}\Vert C(g)-c(g)\Vert <{\sqrt
5\over 2},$ for some bounded scalar cosine family $(c(g))_{g\in G},$ then
$C(g)=c(g)$ for $g \in G.$ \end{itemize} The constants above are optimal.
The author's methods involve the theory of commutative radical Banach
algebras, and elementary but nontrivial considerations about scalar bounded
cosine sequences of functions
Representations of C*-algebras and set theory
Ilijas Farah
York University
`Maximal quantum filters’ are analogues of ultrafilters on C*-algebras.
They correspond to their pure states and provide insight into their irreducible
representations
Regularity points and Jensen measures for R(X)
Joel Feinstein
The University of Nottingham, UK
Coauthors: Hongfei Yang
Most of the material in this talk is joint work with one of my research
students, Hongfei Yang. When studying the failure of regularity for Banach
function algebras, two types of `regularity point' were introduced in
a paper by Feinstein and Somerset, `Non-regularity for Banach function
algebras', Studia Mathematica 141 (2000), 53-68. These types of points
were called points of continuity and R-points. We show that, even for
the natural uniform algebras $R(X)$ (for compact plane sets $X$), these
two types of regularity point can be different. We then give a new method
for constructing Swiss cheese sets $X$ such that $R(X)$ is not regular,
but such that $R(X)$ has no non-trivial Jensen measures. (The original
construction appears in my earlier paper `Trivial Jensen measures without
regularity', Studia Mathematica 148 (2001), 67-74.) Our new approach to
constructing such sets is more general, and allows us to obtain additional
properties. In particular, we use our construction to give an example
of such a Swiss cheese set $X$ with the property that the set of points
of discontinuity for R(X) has positive area. Finally we show that there
are compact plane sets $X_1$ and $X_2$ such that $R(X_1)$ and $R(X_2)$
are both regular, but such that $R(X_1 \cup X_2)$ is not regular.
Ideals in the Fourier Algebra and Related Algebras
Brian Forrest
University of Waterloo
Coauthors: D.R.~Farenick, B.E.~Forrest, and L.W.~Marcoux
I have spent a good part of the last 30 years thinking about the nature
of ideals in the Fourier algebra and its related algebras. In this talk,
I will survey what we know, what we don't know, and what we should know.
I will focus much of my attention on two problems:
1)When does a closed ideal $I$ in $A(G)$ have a bounded approximate identity?
2)When is a closed ideal $I$ complemented in $A(G)$?
Of course, in the case of amenable groups the answer to problem 1) is
known. However, for non-amenable groups it remains open. I will try to
shed light on a possible solution to this problem for non-amenable groups.
State of the art in generalized notions of amenability
Fereidoun Ghahramani
University of Manitoba
In the year 2004, Richard J. Loy and I published a paper in which we
introduced several generalizations of the notion of amenability for Banach
algebras. This talk is about subsequent adavances made in the general
theory for these notions, and about the study of these notions for special
classes of Banach algebra. I shall put particular emphasis on more recent
results. The results obtained are the outcome of collaborative work with
several colleagues, whose names appear in the "References" below.
References:
1 . F. Ghahramani and C. J. Read, Approximate amenability is not bounded
approximate amenability. J. Math. Anal. Appl. 423 (2015), no. 1, 106
- 119.
2. F. Ghahramani and C. J. Read. Approximate amenability of K(X). J.
Funct. Anal. 267 (2014), no. 5, 1540-1565.
3. F. Ghahramani and C. J. Read, Approximate identities in approximate
amenability. J. Funct. Anal. 262 (2012), no. 9, 3929-3945.
4. Y. Choi and F. Ghahramani. Approximate amenability of Schatten classes,
Lipschitz algebras and second duals of Fourier algebras. Q. J. Math.
62 (2011), no. 1, 39-58.
4. F. Ghahramani, E. Samei, and Yong Zhang, Generalized amenability
properties of the Beurling algebras. J. Aust. Math. Soc. 89 (2010),
no. 3, 359-376.
5. Y. Choi, F. Ghahramani,and Y. Zhang, Approximate and pseudo-amenability
of various classes of Banach algebras. J. Funct. Anal. 256 (2009), no.
10, 3158-3191.
6. F. Ghahramani, R. J. Loy and Y. Zhang. Generalized notions of amenability.
II. J. Funct. Anal. 254 (2008), no. 7, 1776-1810.
7. Fereidoun Ghahramani, and Ross Stokke, Approximate and pseudo- amenability
of the Fourier algebra. Indiana Univ. Math. J. 56 (2007), no. 2, 909-930.
8. F. Ghahramani and Y. Zhang, Pseudo-amenable and pseudo-contractible
Banach algebras. Math. Proc. Cambridge Philos. Soc. 142 (2007), no.
1, 111-123.
9. F. Ghahramani, and R. J. Loy, Generalized notions of amenability.
J. Funct. Anal. 208 (2004), no. 1, 229-260.
Weierstrass polynomials over compact groups
Renat Gumerov
Kazan Federal University
This talk is concerned with the polynomials in one variable over the
Banach algebra of complex-valued continuous functions defined on a compact
connected group $G$ and the covering spaces of $G$. There is a close relation
between polynomials with functional coefficients and covering spaces.
For example, in the theory of analytic functions the covering spaces determined
by those polynomials arise naturally in connection with the Weierstrass
preparation theorem. Polynomials over function algebras are called \emph{the
Weierstrass polynomials}, and covering mappings defined by such polynomials
are called \emph{the polynomial coverings}. Different properties of Weierstrass
polynomials over Banach algebras of continuous functions were studied
by various authors( see, for example, [1] and the references therein).
The theory of polynomial coverings was developed in the works of V.~L.~Hansen.
He obtained algebraic and topological criteria for a covering of a connected
topological space to be equivalent to a polynomial covering. In particular,
it was proved that each finite-sheeted covering mapping of the unit circle
in the complex plane is equivalent to a polynomial covering [2, Theorem~8.3].
The following theorem is a generalization of this fact. {\bf Theorem.}
\emph{Every finite covering space of a compact connected abelian group
$G$ is equivalent to a polynomial covering.} We present results from [3]--[6]
concerning properties of Weierstrass polynomials and covering spaces over
compact connected groups. To this end, we discuss some topological tools,
in particular, an analog of the theorem on lifting a topological group
structure to the covering space of a topological group [7, Theorem~79].
Note that the results about the covering spaces of solenoids are used
for investigating Exel--Larsen crossed products in [8].
References:
[1] Hansen V.L. Braids and Coverings - Selected Topics. London Math.
Soc. Stud. Texts, Vol. 18, Cambridge Univ. Press, Cambridge, 1989.
[2] Hansen V.L. Coverings defined by Weierstrass polynomials// J. Reine
Angew. Math., 314 (1980), 29-39.
[3] Gumerov R.N., On finite-sheeted covering mappings onto solenoids//
Proc. Amer. Math. Soc. 133(2005) 2771-2778.
[4] Grigorian S.A., Gumerov R.N. On the structure of finite coverings
of compact connected groups// Topology Appl., 153(2006), 3598 -3614.
[5] Gumerov R.N. Weierstrass polynomials and coverings of compact groups
// Sib. Matem. Zhurn. 54(2), 320-324(2013)[Sib. Math. J., 54(2), 243-246
(2013)]
[6] Gumerov R.N., Characters and coverings of compact groups, Russian
Math., V.58, N 4, 2014, P. 7- 13.
[7]. Pontryagin L. S. Continuous groups, (in Russian), Moscow, Nauka,
1984.
[8]. Brownlowe, N. D., Raeburn, I. F. Two families of Exel-Larsen crossed
products // J. Math. Anal. Appl., 2013, 398 (1), 68-79.
The smoothness of orbital measures on classical simple
Lie algebras
Kathryn Hare
University of Waterloo
Coauthors: Sanjiv Gupta, Sultan Qaboos University
By orbital measures on a Lie algebra we mean the purely singular, uniform
measures supported on adjoint orbits. A convolution product of orbital
measures is either purely singular or purely absolutely continuous, depending
on whether or not the sum of the corresponding orbits is of measure zero.
In this talk we will characterize the orbital measures whose convolution
product is absolutely continuous in terms of roots of the Lie algebra,
and also in terms of eigenvalues and eigenspaces of associated operators.
Classification of multiplier algebras of Nevanlinna-Pick
spaces
Michael Hartz
University of Waterloo
Nevanlinna-Pick spaces are Hilbert function spaces for which an analogue
of the Nevanlinna-Pick interpolation theorem from complex analysis holds.
Their multiplier algebras, which are commutative semisimple Banach algebras,
have attracted considerable attention in recent years. The investigation
of the classification problem for these algebras was initiated by Davidson,
Ramsey and Shalit.
I will report on the current state of this problem and talk about recent
work which uses a somewhat different perspective on these algebras. In
particular, I will indicate that a Nevanlinna-Pick space is completely
determined by the Banach algebra structure of its multiplier algebra.
Moreover, I will present a complete classification result for a certain
class of multiplier algebras.
Function spaces invariant under group actions
Alexander J. Izzo
Bowling Green State University
Motivated by his work on a conjecture of William Arveson in operator
theory, Ronald Douglas raised a question regarding function algebras on
the unit sphere in complex n-space invariant under the torus action. Surprisingly,
the answer to Douglas' question depends on the dimension. The speaker's
work on Douglas' question led him to formulate a conjecture regarding
function algebras that are invariant under a transitive group action.
This invariant function algebra conjecture, which can be regarded as a
replacement for the disproved peak point conjecture, has been proven in
many special cases. Among the results to be presented is a complete description
of all function algebras on the circle invariant under a transitive group
action. This description shows, in particular, that the invariant function
algebra conjecture holds for function algebras on the circle.
Isometric isomorphisms of Beurling algebras
Safoura Jafar-Zadeh
University of Manitoba
By a weighted locally compact group, we mean a pair $(G, \omega)$ where
$G$ is a locally compact group and $\omega$ is a continuous weight function
on $G$. In this talk, we first define what it means for two weighted locally
compact groups to be isomorphic. We then show that any weighted locally
compact group $(G, \omega)$ is completely determined by its Beurling measure
algebra $M(G,\omega)$, Beurling group algebra $L^1(G,\omega)$, $LUC(G,\omega^{-1})^*$,
and $L^1(G,\omega)^{**}$. Here, $LUC(G,\omega^{-1})$ is the weighted analogue
of $LUC(G)$, the space of left uniformly continuous functions on $G$,
for weighted locally compact groups. We will also provide a complete description
of the isometric isomorphisms on Beurling group and measure algebras in
terms of topological group isomorphisms and continuous characters.
Hamana boundary of $\ell^1$-algebra of discrete (quantum)
groups
Mehrdad Kalantar
IMPAN, Poland
We introduce the Hamana boundary of a pair $(\mathcal A, \epsilon)$ of
a Banach algebra $\mathcal A$ and a character $\epsilon$ on $\mathcal
A$. We consider the case $\mathcal A = \ell^1(G)$, where $G$ is a discrete
(quantum) group, and present applications of this notion in the study
of various analytic properties of $G$.
This talk is based on joint works with Emmanuel Breuillard, Matthew Kennedy,
and Narutaka Ozawa.
On Riesz Operators
Ur Koumba
University of Johannesburg
Coauthors: Heinrich Raubenheimer
The interest of this talk lies in the existence of a natural enlargement
of a complex Banach space X via the concept of filtration. We will provide
a characterization of Riesz operators in terms of Riesz operators defined
on ultrapowers and add to this discussion the non-existence of non-compact
positive Riesz operators defined on a Hilbert space.
Purely infinite C*-algebras associated to Fell bundles
over discrete groups
Bartosz Kwasniewski
Southern University of Denmark
Coauthors: Wojciech Szymanski (Southern University of Denmark)
In this talk we present conditions implying (strong) pure infiniteness
of the reduced cross-sectional $C^*$-algebra $C^*_r(\mathcal{B})$ of a
Fell bundle $\mathcal{B}$ over a discrete group $G$. We introduce notions
of aperiodicity, $\mathcal{B}$-paradoxicality and residual $\mathcal{B}$-infiniteness.
We discuss their relationship with similar conditions studied, in the
context of crossed products, by the following duos: Laca, Spielberg; Jolissaint,
Robertson; Sierakowski, R{\o}rdam; Giordano, Sierakowski and Kirchberg,
Sierakowski. The obtained results are shown to be optimal when applied
to graph $C^*$-algebras. They are also applied to a class of Exel-Larsen
crossed products.
Ideal structure of the algebra of bounded operators acting
on a Banach space
Niels Laustsen
Lancaster University, UK
Coauthors: Tomasz Kania
We construct a Banach space~$Z$ for which the Banach algebra~$\mathcal{B}(Z)$
of bounded operators on~$Z$ contains exactly four non-trivial closed ideals,
namely the compact operators, the inessential operators and two maximal
ideals. It appears to be the first example of a Banach space~$X$ for which~$\mathcal{B}(X)$
has finitely many closed ideals and they are not linearly ordered. We
determine which kinds of approximate identities (bounded/left/right),
if any, each of the four non-trivial closed ideals of~$\mathcal{B}(Z)$
contains, and we show that one of the two maximal ideals is generated
as a left ideal by two operators, but not by a single operator, thus answering
a question left open in our recent collaboration with Dales, Kania, Kochanek
and Koszmider (Studia Math.~2013). In contrast, the other maximal ideal
is not finitely generated as a left ideal. The Banach space~$Z$ is the
direct sum of Argyros and Haydon's Banach space~$X_{\text{AH}}$ which
has very few operators and a certain subspace~$Y$ of~$X_{\text{AH}}$.
The key property of~$Y$ is that every operator from~$Y$ into~$X_{\text{AH}}$
is the sum of a scalar multiple of the inclusion mapping and a compact
operator.
Spectra of weighted Fourier algebras on non-compact Lie
groups: the case of the Euclidean motion group
Hun Hee Lee
Seoul National University
Coauthors: Nico Spronk
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 Fourier algebras.
We will first introduce a model for a weighted version of Fourier algebras
on non-compact Lie groups and then 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 the Euclidean motion
group. The main difficulty here is that there is no abstract vs concrete
Lie theory correspondence available for us. The key ingredient to overcome
this difficulty is to use the underlying Euclidean structure on the group
and solve a Cauchy type functional equation for certain functionals. This
is a joint work with Nico Spronk.
Bilinear Schur products and second order perturbation of
functional calculus
Christian Le Merdy
Universite de Franche-Comte
Let $A$ be a possibly unbounded self-adjoint operator on a Hilbert space
$\mathcal H$, let $K\in S^2(\mathcal H)$ be a self-adjoint Hilbert-Schmidt
operator and consider a $C^2$-function $f\colon\mathbb R\to \mathbb R$
with a bounded second derivative. The main result of this talk is that
the perturbation operator $$f(A+K) -f(A) -\frac{d}{dt}\bigl(f(A+tK)\bigr)_{\vert
t=0}$$ does not necessarily belong to the trace class $\mathcal S^1(\mathcal
H)$.
This result relies on a characterization of bounded bilinear Schur products
$\mathcal S^2 \times \mathcal S^2 \to \mathcal S^1$. This is a joint work
with C. Coine, D. Potapov, F. Sukochev and A. Tomskova.
Schur idempotents and hyperreflexivity
Rupert Levene
University College Dublin
Coauthors: Ivan Todorov, Georgios Eleftherakis
A subspace X of B(H) is hyperreflexive if the distance from an operator
T to X is equivalent to the Arveson distance from T to X. We will discuss
hyperreflexivity properties of the subspaces of the form X=range(Phi)
where Phi is an idempotent Schur multiplier; informally, this means that
for an operator T in B(H), the matrix of the operator Phi(T) is the matrix
of T with certain entries replaced by zeros.
Compactness of weighted composition operators on Lipschitz
spaces
Hakimeh Mahyar
Kharazmi University
Coauthors: Azin Golbaharan (Kharazmi University)
Let $(X,d)$ be a compact metric space and $0< \alpha \leq 1$. The Lipschitz
space $Lip(X,d^\alpha)$ is the space of all complex-valued Lipschitz functions
of order $\alpha$ on $X$. Let $u$ be a complex-valued function on $X$ and
$\varphi$ a self-map& of $X$ . We& give a necessary and sufficient
condition on functions $u$ and& $\varphi$ for which a weighted composition
operator $uC_\varphi$& on $Lip(X,d^\alpha)$ to be bounded (well-defined)
and compact. We also obtain a lower bound for the essential norms of weighted
composition operators on $Lip(X,d^\alpha)$ when $0<\alpha < 1$
Inductive limits of operator systems
Linda Mawhinney
Queen's University Belfast
Coauthors: Prof. Ivan Todorov
Inductive limits are an important tool in the study of $C^*$-algebras.
In this talk we will explore the inductive limit adapted to the category
of operator systems with unital completely positive maps. We will discuss
several results on the interplay between the inductive limit and other well-studied
operator system structures including the minimal and maximal operator systems
and operator system tensor products.
On character amenability of semigroup algebras
Oluwatosin T. Mewomo
School of Mathematics, Statistics and Computer Science, University of Kwazulu-Natal,
Durban, South Africa
Coauthors: S.M. Maepa, Department of Mathematics and Applied Mathematics,
University of Pretoria, South Africa.
We study the character amenability of semigroup algebras. We work on general
semigroups and certain semigroups such as inverse semigroups with a finite
number of idempotents, inverse semigroups with uniformly locally finite
idempotent set, Brandt and Rees semigroup and study the character amenability
of the semigroup algebra $l^{1}(S)$ in relation to the structures of the
semigroup $S.$ In particular, we show that for any semigroup $S,$ if ${\ell}^{\,1}(S)$
is character amenable, then $S$ is amenable and regular. We also show that
the left character amenability of the semigroup algebra ${\ell}^{\,1}(S)$
on a Brandt semigroup $S$ over a group $G$ with index set $J$ is equivalent
to the amenability of $G$ and $J$ being finite. Finally, we show that for
a Rees semigroup $S$ with a zero over the group $G,$ the left character
amenability of ${\ell}^{\,1}(S)$ is equivalent to its amenability, this
is in turn equivalent to $G$ being amenable.
Homomorphisms of Bland-Feinstein Algebras
Sam Morley
University of Nottingham
Let $X$ be a perfect compact subset of the complex plane, and let $D^{1}(X)$
denote the algebra of complex-differentiable functions on $X$. Then $D^{1}(X)$
is a normed algebra of functions but, in many cases, fails to be a Banach
function algebra. So we can ask whether the completion is necessarily
a Banach function algebra. However, there are examples of compact sets
$X$ in the complex plane such that the completion of $D^{1}(X)$ fails
to be semisimple. Bland and Feinstein investigated the completions of
the algebra $D^{1}(X)$, for certain sets $X$, by considering $\mathcal{F}$-differentiable
functions on $X$. Given a suitable collection of paths $\mathcal{F}$,
we say a continuous function $g:X\to\mathbb{C}$ is $\mathcal{F}$-differentiable
on $X$ if there is a continuous function $h:X\to\mathbb{C}$ so that integrating
$h$ along each path in $\mathcal{F}$ yields the difference of the values
of $g$ evaluated at the endpoints of that path. In this talk, we discuss
the properties of algebras of $\mathcal{F}$-differentiable functions.
These algebras are Banach function algebras and are called Bland-Feinstein
algebras. We also discuss homomorphisms of algebras of functions which
have $\mathcal{F}$-derivatives of all orders, analogous to the algebras
investigated in a paper by Dales and Davie.
A non-commutative analogue of (almost) band preserving
operators
Timur Oikhberg
University of Illinois at Urbana-Champaign
Suppose $A$ is a von Neumann algebra, and $E$ is a Banach $A$-bimodule
(we are concerned with the cases of $E=A$ itself, or $E$ being an operator
function space). Taking a cue from the theory of Banach lattices, we say
that a linear map $T : E \to E$ is band-preserving (BP) if $p[Tx]p = Tx$
whenever $x = pxp$. Further, $T$ is $\epsilon$-band preserving ($\epsilon$-BP)
if $\|p[Tx]p - Tx\| \leq \epsilon \|x\|$ whenever $x = pxp$. We prove
that, under certain conditions, linear BP maps $T$ are automatically continuous,
and moreover, are of the form $Tx = ax$, where $a$ belongs to the center
of $A$. Furthermore, under certain conditions, we prove that, if $T$ is
continuous and $\epsilon$-BP, then there exists and BP maps $S$ so that
$\|T-S\| \leq K\epsilon$ ($K$ is an absolute constant).
Multipliers and Perfectness in Topological Algebras
Lourdes Palacios
Universidad Autónoma Metropolitana- Iztapalapa
Coauthors: Marina Haralampidou (University of Athens) Carlos Signoret (Universidad
Autónoma Metropolitana- Iztapalapa, Mexico)
Given an algebra A, a linear mapping T : A→A is called a left
(right) multiplier on A if T(xy) = T(x)y (resp. T(xy) = xT(y)) for all
x, y in A; it is called a two-sided multiplier on A if it is both a left
and a right multiplier. The notion of a perfect algebra was de ned by
M. Haralampidou in 2003 in terms of the description of the algebra as
a projective limit of algebras of a simpler type. This representation
is the classical Arens-Michael decomposi- tion in the case of locally
m-convex algebras and the generalized Arens-Michael decompostion of locally
m-pseudoconvex algebras. In this talk we consider a complete locally m-convex*-algebra
with con- tinuous involution, which is also a perfect projective limit,
and we describe its multiplier algebra M(A), under a weaker topology,
making it a locally C- algebra. This is applied to certain locally convex
H*-algebras. We consider two more cases: when A is a perfect complete
locally m-convex (resp. locally m- pseudoconvex) algebra with an approximate
identity and with complete Arens- Michael (resp. generalized Arens-Michael)
normed (resp. k -normed ) factors. In each case we describe the multiplier
algebra M(A) via the multiplier algebras of the corresponding factors.
Suitable examples will be given.
When a subalgebra of the Fourier algebra is the whole algebra?
Hung Pham
Victoria University of Wellington
Coauthors: Anthony To-Ming Lau
This talk is based on a recent joint work with Anthony To-Ming Lau. In
this talk, I will discuss conditions for a subalgebra of the Fourier algebra
$A(G)$ of a locally compact group $G$ to be the whole of $A(G)$. Our main
result is that if $A$ is a closed translation-invariant Tauberian subalgebra
of $A(G)$ with spectrum $\sigma(A)=G$ and if $A$ approximately contains
a nontrivial real function, then $A=A(G)$.
An invitation to operator algebras on L^p spaces
N. Christopher Phillips
University of Oregon
Somewhat surprisingly, there appears to be a rich theory of "C*
like" operator algebras on L^p spaces (despite the absence of an
adjoint), with significant similarities to and significant differences
from the theory of C*-algebras. Our main evidence for this is a collection
of classes of examples about which there are interesting theorems; we
still lack an abstract general theory.
In this talk, we will illustrate the similarities and differences in
the context of several classes of examples. We will describe both the
C* versions and the L^p versions, so that one does not need much knowledge
of C*-algebras to follow the talk; similarly, the talk should be accessible
to C*-algebraists with little familiarity with more general Banach algebras.
Grothendieck Inequality in the noncommutative Schwartz
space
Krzysztof Piszczek
Adam Mickiewicz University in Poznan
The famous Grothendieck Inequality says that there is a constant $K>0$
such that for any bounded bilinear form $\phi\colon C(X)\times C(Y)\to\mathbb{K}\,(X,Y\text{\,--\,compact},\,\mathbb{K}=\mathbb{R}\,\text{or}\,\mathbb{C})$
there are probabilities $\mu$ and $\nu$ on $X$ and $Y$, respectively such
that
\[\forall\,(x,y)\in C(X)\times C(Y)\colon\,\,\,\,|\phi(x,y)|\leqslant
K\|\phi\|\Bigl(\int_X|x|^2d\mu\Bigr)^{\frac12}\Bigl(\int_Y|y|^2d\nu\Bigr)^{\frac12}.\]
The best value $K_G$ of the above constant is the Grothendieck Constant
and is still unknown. This inequality was further generalized -- by Pisier
and Haagerup -- onto bilinear maps on arbitrary C*-algebras and later
on -- by Pisier/Shlyakhtenko and Haagerup/Musat -- onto bilinear maps
on operator spaces. In the first part of the talk we will briefly outline
the history of the Grothendieck Inequality. The second part will be devoted
to the Grothendieck Inequality in the framework of the noncommutative
Schwartz space $\mathcal{S}$. Recall that by $\mathcal{S}$ we mean the
Fr\'echet *-algebra of operators acting from the space of tempered distributions
into the Schwartz space of rapidly decreasing functions.
A Gleason-Kahane-Zelazko theorem for modules and applications
to holomorphic function spaces
Thomas Ransford
Université Laval
Coauthors: Javad Mashreghi
We generalize the Gleason-Kahane-Zelazko theorem to modules. As an application,
we show that every linear functional on a Hardy space that is non-zero
on non-vanishing functions is a multiple of a point evaluation. A further
consequence is that every linear endomorphism of a Hardy space that maps
non-vanishing functions to non-vanishing functions is a weighted composition
operator. In neither case is continuity assumed. We also consider extensions
to other function spaces.
The index for Fredholm elements in a Banach algebra via
a trace II
Heinrich Raubenheimear
University of Johannesburg
Coauthors: JJ Grobler and AM Swartz
We show that the index defined via a trace for Fredholm elements in a
Banach algebra has the property that an index zero Fredholm element can
be decomposed as the sum of an invertible and an element in the socle.
We identify the set of index zero Fredholm elements as an upper semiregularity
with the Jacobson property. The Weyl spectrum is then characterized in
terms of the index.
The Algebra of Germs and the Invariant Subspace Problem
Charles Read
University of Leeds
Counterexamples to the Invariant Subspace Problem are by now well known
(Enflo, Acta Math., 1987; Read, BLMS, 1985), but several of the best
problems in the area remain unsolved. One lesser problem in the area
that has recently been solved is the following: is there a bounded operator
$T$ on a general complex Banach space $X$ such that, for each nonconstant
polynomial $p$, the operator $p(T)$ has no invariant subspace other
than $\{0\}$, $X$?
The question seems well known; we believe it was well known to Radjavi
and Rosenthal long before Read stated it as a ``fall-back problem''
in the paper '{\it A short proof concerning the invariant subspace problem}'
(JLMS 1986), which gave an accessible proof of the existence of an operator
without invariant subspaces on $X=l^1$.
In my talk I will describe - using the pictorial approach to combinatorial
complexity which is so much a part of the invariant subspace problem
- how to find an operator $T$ such that not only is $p(T)$ lacking in
invariant subspaces for every nonconstant polynomial $p$, but the same
is true if $p$ is an arbitrary nonconstant analytic function defined
on a complex neighbourhood of zero. This is joint work with Eva Gallardo.
So, one continuously embeds the entire algebra of germs into $B(X)$,
in such a way that the only germs with any invariant subspaces are the
constants. Of course, for $p(T)$ to make sense for an arbitrary analytic
function defined in a neighbourhood of zero, the operator $T$ must be
quasinilpotent. Now quasinilpotent solutions to the invariant subspace
problem are not new {\it per se} - they have been known to exist since
(Read, JLMS, 1997). But in order to find operators such that every nonconstant
``germ'' of that operator has no nontrivial invariant subspaces, one
has to - well - change the {\it picture}. The pictorial approach is,
as we have mentioned, a very useful tool when getting to grips with
the somewhat formidable combinatorics in the area.
Faithful actions of locally compact quantum groups on classical
spaces
Sutanu Roy
University of Ottawa
Coauthors: Debashish Goswami
A rigidity conjecture by Goswami states that existence of a smooth and
faithful action $\alpha$ of a compact quantum group $\mathbb{G}$ on a
compact connected Riemannian manifold $M$ forces $\mathbb{G}$ to be compact
group. In particular, whenever $\alpha$ is isometric, or $\mathbb{G}$
is finite dimensional, Goswami and Joardar have proved that the conjecture
is true. The first step in the investigation of a non-compact version
of this rigidity conjecture demands correct notion of faithful actions
of locally compact quantum groups on classical spaces. In this talk we
show that bicrossed product construction for locally compact groups provides
a large class of examples of non-Kac locally compact quantum groups acting
faithfully on connected manifolds.
Approximate diagonals for $C^\ast$-algebras
Volker Runde
University of Alberta
Let $A$ be a $C^\ast$-algebra. If $A$ has a \emph{bounded} approximate
diagonal, it is amenable and thus nuclear. We shall explore what the existence
of an \emph{unbounded} approximate diagonal entails.
Weak amenability of the Fourier algebra of a Lie group
Ebrahim Samei
University of Saskatchewan, Canada
Coauthors: Hun Hee Lee, Jean Ludwig, Nico Spronk
It has been long conjectured that the Fourier algebras $A(G)$ is weakly
amenable if and only if its connected component of the identity $G_e$
is abelian. In this talk, we show that, for a Lie group $G$, this conjecture
holds. Our main idea is to show that when $G$ is connected, weak amenability
of $A(G)$ implies that the anti-diagonal, $\{(g,g^{-1}):g\in G\}$, is
a set of local synthesis for $A(G\times G)$. We then show that this cannot
happen if $G$ is non-abelian. We also conclude for a locally compact group
$G$, that $A(G)$ can be weakly amenable only if it contains no closed
connected non-abelian Lie subgroups; in particular, for a Lie group $G$,
$A(G)$ is weakly amenable if and only if its connected component of the
identity $G_e$ is abelian. This is a joint work with Hun Hee Lee, Jean
Ludwig and Nico Spronk.
On generalized notions of operator amenability for Fourier
algebras
Miad Makareh Shireh
In this talk we define the notion of operator boundedly approximately
contractibility for a completely contractive Banach algebra and we will
show that for a locally compact group $G$, operator boundedly approximately
contractibility is equivalent to operator amenability for $A(G)$, and
hence by a result odue to Z.J-Ruan, if and only if $G$ is amenable. \\
At the end we will show that under certain conditions on the product,
operator amenability will be preserved for neighboring completely contractive
Banach algebras.
CPH-Semigroups
Michael Skeide
Università degli Studi del Molise
Coauthors: (partly) joint with Sumesh K. (ISI Bangalore)
It has shown to be a fruitful idea to analyze maps between Hilbert modules
by how ``nicely'' they can be extended to maps acting blockwise between
the linking algebras of the modules. For instance the maps between full
Hilbert modules that can be extended to blockwise homomorphisms between
the linking algebras, are precisely the ternary homomorphisms, that is,
the linear maps $T\colon E\rightarrow F$ fulfilling $T(x\langle y,z\rangle)=T(x)\langle
T(y),T(z)\rangle)$. For more general maps it turns out that one better
considers maps that allow for strict extensions to the multiplier algebras
of the linking algebras. But, then, a (strict) CP-map between Hilbert
modules woud be a map that allows for a strict blockwise CP-extension
to the multiplier algebras of the linking algebras. CPH-maps (a special
subclass of which has been introduced by Asadi in 2009) sit strictly in
between ternary homomorphisms and strict CP-maps and, therefore, should
not be called CP-maps. (``H'' stands for ``homomorphic'' extension and
refers to that one corner of the CP-extension is a homormorphism.) They
are charcterized as those fulfilling the quaternary condition $$ \langle
T(x'\langle x,y\rangle),T(y')\rangle = \langle T(y),T(x\langle x',y'\rangle)\rangle.
$$ We discuss semigroups of such maps and illustrate the connections with
the product systems both of a CP-semigroup and of a strict $E_0$--semigroup.
Splittings of Extensions of Banach Algebras
Richard Skillicorn
Lancaster University, UK
Coauthors: N.J.Laustsen (Lancaster University, UK)
An extension of a Banach algebra $B$ is a short exact sequence $0 \to
I \to A \to B \to 0$ of Banach algebras and continuous algebra homomorphisms.
The extension splits algebraically if there is a subalgebra $C$ of $A$
such that $C\oplus I= A$ as a vector space, and splits strongly if there
is a closed subalgebra $D$ of $A$ so that $D\oplus I=A$. We answer some
questions of Bade, Dales and Lykova about extensions of the Banach algebra
of bounded operators on a Banach space. More specifically, given a Banach
space $X$, must an extension of $B(X)$ split algebraically? If it does
split algebraically, must it also split strongly? We show that the answer
to both these questions is no for certain Banach spaces, and demonstrate
a connection with homological bidimension; for some Banach spaces $X$
we show that the homological bidimension of $B(X)$ is strictly greater
than one.
Closed Convex Hulls of Unitary Orbits in C$^*$-Algebras
of Real Rank Zero
Paul Skoufranis
Texas A&M University
In matrix algebras, the notion of majorization completely characterizes
the closed convex hull of the unitary orbit of a self-adjoint operator
in terms of spectral data. This characterization is useful in many applications,
such as the Schur-Horn Theorem and generalized numerical ranges. Furthermore,
a similar characterization holds in II$_1$ factors. In this talk, we will
demonstrate that it is possible to describe the closed convex hull of
the unitary orbit of a self-adjoint operator in terms of spectral data
for many C$^*$-algebras with real rank zero.
Commuting contractive idempotents in measure algebras
Nico Spronk
University of Waterloo
Greenleaf characterized the contractive idempotents in measure algebras
of locally compact groups more than 50 years ago. I will discuss situations
under which products of such idempotents are thmeselves idempotents. I
will also consider certain intrinsic groups of measures at contractive
idempotents.
Amenable Cores for a Banach algebra
Arezou Valadkhani
Vancouver, B.C
In this work I studied, When a chain of amenable subalgebras of a Banach
algebra has an upper bound? As a consequence I introduced a new notion
called "Amenable Core" which is a maximal amenable Banach subalgebra
of a Banach algebra. I showed the existence of Amenable cores for finite
dimensional Banach algebras. Also, I proved that the tensor product of
Amenable cores in Banach algebras A and B, is an amenable core in the
tensor product of A and B.
Spatial $L^p$ AF algebra
Grazia Viola
Lakehead University
The talk will be an introduction to $L^p$ operator algebras and spatial
$L^p$ AF algebra. The main theorem we will discuss is a classification
theorem for spatial L^p AF algebras. We show that two spatial $L^p$ AF
algebras are isomorphic if and only if their scaled ordered $K_0$ groups
are isomorphic. Moreover, we prove that any countable Riesz group can
be realized as the scaled ordered $K_0$ group of a spatial $L^p$ AF algebra.
Therefore, the classification given by G. Elliott for AF algebras also
holds for spatial $L^p$ AF algebras. Lastly, we will talk about compressibility
and p-incompressibility for Banach algebras, and discuss the results we
have for a spatial $L^p$ AF algebra. This is joint work with Chris Phillips.
Weak mixing for locally compact quantum groups
Ami Viselter
University of Haifa
We present a generalization of weak mixing of unitary representations
to the framework of locally compact quantum groups. The standard characterizations
known for groups are extended and their equivalence is proved. This is
applied to complement the noncommutative Jacobs-de Leeuw-Glicksberg splitting
theorem obtained recently. In addition, we establish a relation between
(weak) mixing of state-preserving actions of discrete quantum groups and
(weak) mixing of ambient inclusions of von Neumann algebras that is known
to hold for discrete groups.
Weak* tensor products for von Neumann algebras
Matthew Wiersma
University of Waterloo
The category of C*-algebras is blessed with many different tensor products.
In contrast, virtually the only tensor product ever used in the category
of von Neumann algebras is the normal spatial tensor product. We propose
a definition of what a generic tensor product in this category should
be and study properties of von Neumann algebras in relation to these tensor
products.
Classicalisation of Swiss cheese sets
Hongfei Yang
University of Nottingham
Coauthors: J. Feinstein and S. Morley
Most of the material in this talk is joint work with J. Feinstein and
S. Morley. In this talk we discuss Swiss cheese sets,their applications
and some methods for improving the topological properties of such sets.
We use the term Swiss cheese set to describe compact subsets of the plane
obtained by deleting an appropriate sequence of open disks from a closed
disk. Without some additional conditions, every compact subset of the
plane would be a Swiss cheese set, so we usually require that the sum
of the radii of the deleted disks is finite and that the Swiss cheese
set has positive area. However, even under these conditions, Swiss cheese
sets need not have good topological properties, such as being locally
connected. Such sets serve as useful examples in the theory of rational
approximation and uniform algebras.
Weak amenability of central Beurling algebras
Yong Zhang
University of Manitoba
Coauthors: Varvara Shepelska
I will present some recent investigation on the weak amenability of the
center algebras of weighted group algebras. In particular, I will focus
on [FC]$^-$ and [FD] groups. This is joint work with Varvara Shepelska.
