CMS-MITACS Joint Conference 2007

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.

Let M_n be the algebra of n×n complex matrices, and for x Î M_n denote by s(x) and r(x) the spectrum and spectral radius of x respectively. Let D be a domain in M_n containing 0, and let F : D ®M_n be a holomorphic map. We prove:

(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.

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).

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.

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.

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 DF-spaces.

Let G be a locally compact group and let A_p(G) be the Figá-Talamanca-Herz algebra of G. In this talk, we will present some results on the approximation properties of A_p(G) and A_p^r(G) = A_p(G) ÇL^r(G) in their multiplier algebras.

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 (ir)regularity.

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.

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.

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?

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.

Let L¹_w(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 L¹_w(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 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 algebras.

This is joint work with Mahya Ghandehari and Hamed Hatami.

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.

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 s_p (Tf+Tg) = s_p(f+g) for all f,g Î A, where s_p(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.

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 space E.

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.