Banach Algebras and Abstract Harmonic Analysis
Org: Yong Zhang (Manitoba)
- YEMON CHOI, University of Manitoba, Winnipeg, R3T 2N2
Biflatness for Clifford semigroup algebras
We show how one may use classical ideas from the representation theory
of Clifford semigroups (in particular certain Möbius inversion
formulae) to characterize the biflatness of their l1-convolution
algebras. A key tool is the old result of Duncan and Namioka that the
l1-convolution algebra of a semilattice is amenable
precisely when the semilattice is finite.
- CONSTANTIN COSTARA, Univ. Laval, Dep. de Math. et Stat., Quebec, Canada,
On local irreducibility of the spectrum
Let Mn be the algebra of n×n complex matrices,
and for x Î Mn denote by s(x) and r(x) the
spectrum and spectral radius of x respectively. Let D be a domain
in Mn containing 0, and let F : D ®Mn be a holomorphic map. We prove:
Both results are special cases of theorems expressing the
irreducibility of the spectrum s(x) near x=0.
(1) if s( F(x) ) Çs(x) ¹ Æ
for x Î D, then s( F(x) ) = s(x) for x Î D;
(2) if r( F(x) ) = r(x) for x Î D, there
exists l of modulus one such that s( F(x) ) = ls(x) for x Î D.
Joint work with T. J. Ransford.
- BRIAN FORREST, University of Waterloo
Projective Operator Spaces, Almost Periodicity and
The concept of an operator space or quantized Banach space has proved
to be extremely useful in addressing problems in Abstract Harmonic
Analysis. In this talk we will focus on the operator space analog of
Grothendieck's notion of projectivity for Banach spaces. We will show
how projective operator spaces arise naturally as spaces of almost
periodic functions. In particular, we will show that a locally
compact group is compact if and only if its Fourier-Stieltjes algebra
(or equivalently its Fourier algebra A(G)) is projective as an
operator space. From this we see that if K is a compact subgroup of
G, then the ideal I(K) of functions in A(G) vanishing on K is
completely complemented in A(G).
- NIELS GRONBAEK, Department of Mathematics, University of Copenhagen,
Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
The L1(H)-module L1(G)H
Let G be a locally compact group, and let H be a closed subgroup.
Identifying Haar measure on H with an H-supported Borel measure on
G, the Banach space L1(G) is naturally a right module over the
convolution algebra L1(H), denoted L1 (G)H. We prove that L1(G)H is a strictly flat generator of the category of essential right
L1(H)-modules. This result is the key to understand the
representations of G in terms of the representations of its closed
subgroups. As an illustration we give a characterization of the
strongly continuous 1-parameter group actions which are induced from
some doubly power bounded operator.
- ZHIGUO HU, University of Windsor, Windsor, Ontario
Multipliers on Banach algebras and applications to the
second dual Banach algebras
We prove results on multiplier algebras for a large class of Banach
algebras A. They are used to characterize the predual of a locally
compact quantum group under a representation of its multipliers.
Applications are obtained on the second dual Banach algebras A**
of A. Some results on the group algebra L1(G) and the Fourier
algebra A(G) of a locally compact group G are extended and unified
through an abstract Banach algebraic approach.
The talk is based on joint work with M. Neufang and Z-J. Ruan.
- ZINAIDA LYKOVA, School of Mathematics and Statistics, Newcastle University,
Newcastle upon Tyne, NE1 7RU, UK
The Hochschild and cyclic cohomology of simplicially
trivial topological algebras
We give explicit formulae for the continuous Hochschild and cyclic
homology and cohomology of simplicially trivial
[^(Ä)]-algebras. We show that, for a continuous morphism
j: X* ® Y* of complexes
of complete nuclear DF-spaces, the isomorphism of cohomology groups
Hn(j) : Hn(X*) ®Hn(Y*) is automatically topological. The continuous
cyclic-type homology and cohomology are described up to topological
isomorphism for the following classes of biprojective
[^(Ä)]-algebras: the algebra of smooth functions
E(G) on a compact Lie group G, the algebra of
distributions E*(G) on a compact Lie group G; the
tensor algebra E [^(Ä)] F generated by the duality (E,F,á·, ·ñ) for nuclear Fréchet spaces E and
F or for nuclear DF-spaces E and F; nuclear biprojective
Köthe algebras l(P) which are Fréchet spaces or
- TIANXUAN MIAO, Lakehead University
Approximation Properties of Ap(G) and Apr(G)
Let G be a locally compact group and let Ap(G) be the
Figá-Talamanca-Herz algebra of G. In this talk, we will present
some results on the approximation properties of Ap(G) and Apr(G) = Ap(G) ÇLr(G) in their multiplier algebras.
- MATTHIAS NEUFANG, Carleton University, School of Mathematics and Statistics,
1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6
Jordan Arens irregularity
Given a Banach algebra A, its product admits two canonical
extensions to the bidual, yielding the left and right Arens products.
If these coincide, A is called Arens regular; this is the
case, e.g. for all C*-algebras A. However, most
Banach algebras arising in abstract harmonic analysis are preduals of
Hopf-von Neumann algebras, and multiplication in the second dual is
typically highly irregular. The degree of irregularity can be
measured through the so-called left and right topological centres,
i.e., the sets of elements in A** for which the left,
resp. right, Arens product is separately w*-w*-continuous.
Arens regularity means precisely that both topological centres
coincide with A**; if the left, resp. right,
topological centre equals A, following Dales-Lau, we call
A left, resp. right, strongly Arens irregular (SAI).
There are natural examples of Banach algebras (due to Dales-Lau and
myself) which are left but not right SAI-such as the algebra (T (L2 (G)),* ), for non-compact
G, equipped with a certain convolution type product I
introduced and studied; its right topological centre can also be
described explicitly for all discrete groups G.
A canonical way to symmetrize a given Banach algebra is to consider
the corresponding Jordan product. This prompts the question to what
extent this symmetry is carried over to the second dual-in other
words: what is the (algebraic) centre of the second dual of a Jordan
algebra? We are thus led to the notion of Jordan Arens
We shall prove that for any discrete ICC group G, the
centre of the bidual of l1 (G) endowed with the Jordan
product, is exactly l1 (G). The proof relies on our
simultaneous left/right factorization theorem for bounded sequences in
l¥ (G) through elements in l1(G)** and a single function in l¥(G). As a consequence, we shall derive that for the same
class of groups, ( T (l2 (G)),* )
is Jordan SAI as well.
This is joint work with my Master's student, Chris Auger.
- CHI-KEUNG NG, Nankai University, Tianjin 300071
Property (T) and strong property (T) for C*-algebras
We study two (T)-type properties for unital C*-algebras, namely,
the property (T) as introduced by Bekka and a slightly stronger
version of it, called the strong property (T). Similar to a result
of Bekka, if G is a discrete group, then C*(G) have
strong property (T) if and only if G have property (T). We
will give some interesting equivalent formulations as well as some
permanence properties for both property (T) and strong
property (T). We will also relate them to some (T)-type
properties of the unitary groups of the C*-algebras.
- THOMAS RANSFORD, Université Laval, Québec (QC), G1K 7P4
Pseudospectra and power growth
The pseudospectrum of a matrix is the set of level curves of the norm
of the resolvent. It has become a very useful tool in the study of
the evolution of the powers of the matrix. In this talk I shall
discuss the following question: to what extent are the norms of the
powers actually determined by the pseudospectrum?
- ZHONG-JIN RUAN, University of Illinois at Urbana-Champaign
Completely Bounded Multipliers on Locally Compact Quantum
I will discuss some of recent progress on completely bounded
multipliers on locally compact quantum groups. I will also discuss
the related representation theorems, something old and something new.
- EBRAHIM SAMEI, University of Waterloo
2-weak amenability of Beurling algebras
Let L1w(G) be a Beurling algebra on a locally compact abelian
group G. We look for general conditions on the weight which allows
the vanishing of continuous derivations of L1w(G) into its
iterated duals. This leads us to introducing vector-valued Beurling
algebras and considering the translation of operators on them. This
is then used to connect the augmentation ideals to the behavior of
derivations space. We apply these results to give examples of various
classes of 2-weakly amenable and none 2-weakly amenable Beurling
- NICO SPRONK, University of Waterloo
Amenability constants for semilattice algebras
For any finite unital commutative idempotent semigroup S, a unital
semilattice, we show how to compute the amenability constant
of its semigroup algebra l1(S), which is always of the form
4n+1. We then show that these give lower bounds to amenability
constants of certain Banach algebras graded over semilattices. Our
theory applies to certain natural subalgebras of Fourier-Stieltjes
This is joint work with Mahya Ghandehari and Hamed Hatami.
- ROSS STOKKE, University of Winnipeg, Department of Mathematics and
Statistics, 515 Portage Ave., Winnipeg, MB, R3B 2E9
Approximate and pseudo-amenability of the Fourier Algebra
The amenability of a Banach algebra can be defined in terms of the
existence of certain bounded nets. By dropping the requirement that
these nets are bounded, Ghahramani, Loy, and Zhang have introduced
several generalized notions of amenability, including approximate and
pseudo-amenability. Among many other things, these authors have shown
that for group algebras, L1(G), approximate amenability,
pseudo-amenability, and amenability are all equivalent. In this talk
I will discuss several results showing that for Fourier algebras,
A(G), the situation is very different.
This talk is based on joint work with Fereidoun Ghahramani.
- THOMAS V. TONEV, The University of Montana, Missoula, MT 59812
Peripheral additivity and isomorphisms between semisimple
We give sufficient conditions for mappings between semisimple
commutative Banach algebras, not necessarily linear, to be algebra
isomorphisms. Namely, if A and B are semisimple commutative
Banach algebras, then a mapping T: A® B is
peripherally-additive if sp (Tf+Tg) = sp(f+g) for all f,g Î A, where sp(f) is the peripheral
spectrum of f.
It is shown that under natural conditions every such mapping T is an
isometric algebra isomorphism from A onto B that preserves the
spectral radii, and therefore is linear and multiplicative. It is
shown that similar result holds also for symmetric semisimple
commutative Banach algebras.
- FARUK UYGUL, University of Alberta, 632 CAB, Edmonton, AB, T6G 2G1
A Representation Theorem for Completely Contractive Dual
In this talk, we prove that every completely contractive dual Banach
algebra A is completely isometric to a w*-closed
subalgebra of CB(E), for some reflexive operator
- MICHAEL C. WHITE, Newcastle University, England
The Bicyclic semigroup algebra
The bicyclic semigroup is generated by two elements, q and p,
subject to the relation qp = 1. For a model of this semigroup you
may think of the C*-algebra generated by the right shift and its
adjoint on Hilbert space. This C*-algebra is amenable and so most
of its cohomology is trivial. One can also consider the 1-normed
algebra generated by this semigroup. The algebra is not amenable, in
particular it has (non-inner) derivations into its dual, so is not
even weakly amenable. We will see how the semigroup's structure can
be used to calculate the cohomology of this algebra.
- YONG ZHANG, Department of Mathematics, University of Manitoba, Winnipeg
MB, R3T 2N2
Fixed point properties characterized by existence of left
invariant means on semigroups
Let S be a semitopological semigroup. Denote by AP(S), WAP(S)
and LUC(S) the spaces of almost periodic functions on S, weakly
almost periodic functions on S and left uniformly continuous
functions on S respectively. Existence of left invariant means (LIM
for short) on these spaces can characterize various fixed point
properties (FPP for short) of S acting on subsets of locally convex
spaces (and vice versa). We consider FPP of S acting as
non-expansive quasi equicontinuous mappings on a weakly compact convex
set. When S is separable we show, among other things, that this
type of FPP is equivalent to the existence of a LIM on WAP(S) or a
LIM on WAP(S) ÇLUC(S). Some FPP characterized by the existence
of LIM on AP(S) will also be discussed.
This is joint work with A. T.-M. Lau.