Let X be a Banach space on which a discrete group G acts by isometries. For certain natural choices of X, every element of the group algebra, when regarded as an operator on X, has empty residual spectrum. This turns out to be the case when X is l²(G) or the group von Neumann algebra VN(G), regardless of the choice of group. On the other hand, when X=l¹(G), an example of Willis shows that some condition on G is necessary.

In this talk I will discuss some of these results, using the notion of a surjunctive pair to try and develop a systematic picture. If time permits I will mention some partial results that can be obtained for X = l^p(G) when G is amenable; these rely on a majorization result of Herz.

In this talk we will focus on some results concerning the multipliers and the completely bounded multipliers of A(G). In particular, we will show amongst other things that there exits ideals in A(F₂) which are complemented, but not even weakly completely complemented. This answers a question raised previously by Peter Wood. Furthermore, when H is a closed subgroup of G, we can use this to shed light on the problem of extending functions in A_M(H).

This is joint work with Michael Brannan and Cameron Zwarich.

In this talk I will present a criterion that can be used to show that neither of the Schatten operator ideals, Lipschitz algebras, or certain Segal algebras on locally compact groups is approximately amenable. This answers two open questions which Rick Loy and I raised in around 2003.

This is joint work with Yemon Choi.

In 1966 Kahane gave a condition that implied a set is I₀. That condition is also (the new half of the characterization) implied by "I₀-ness". The proof of the new assertion uses a lemma (which might be of independent interest): there are constants 0 < c_N such that the Haar measure of Ç_n=1^N {g: |ág,x_n ñ- 1| < e} is at least c_N e^N; the c_N are independent of the x_n and the LCA group involved.

This comes from joint work with Kathryn Hare.

Orbital measures are the uniform measures supported on conjugacy classes in the Lie group G. They are always singular measures and are continuous provided the conjugacy class is non-trivial. These measures satisfy a striking L²-singular dichotomy: Either m^k Î L² or m^k is singular to Haar measure on G. In this talk we will discuss how the Weyl character formula and other ideas from Lie theory are used to determine the index where the change occurs.

Given a locally compact group G let J(G) denote the set of closed left ideals in L¹(G), of the form J_m = [L¹(G)*(d_e -m)] [` ] where m is a probability measure on G. Given a closed subgroup H of G let L¹₀(G,H) denote the kernel of the canonical mapping from L¹(G) to L¹(G/H). When G is totally disconnected and has polynomial growth, we prove that the following conditions are equivalent:

(i) J(G) = {L¹₀(G,H); H is a closed subgroup of G};
(ii) the group of inner automorphisms of G acts distally on G ;
(iii) every inner automorphism of G is distal;
(iv) the contraction subgroup of every inner automorphism of G is trivial;
(v) G is a SIN group.

This is a joint work with C. R. E. Raja.

Given a finite-dimensional representation p of a Banach algebra A, we discuss the role of p-invariant elements in A" in determination of finite-dimensional left ideals in A".

In this talk, we discuss to what extent, the orthogonality structure of a Hilbert C^*-module determines its C^*-valued inner product.

Let 1 < p < ¥. We discuss various approximation properties of the pseudofunction algebras PF_p(G) and the pseudomeasure algebras PM_p(G) in the category of p-operator spaces. More precisely, we show that a discrete group G is p-weakly amenable if and only if PF_p(G) has the p-completely bounded approximation property (respectively, PM_p(G) has weak^* continuous p-completely bounded approximation property). We also show that a discrete group G has the p-AP if and only if PF_p(G) has the p-OAP (respectively, PM_p(G) has the weak^* p-OAP). These results generalize the work of Haagerup and Kraus to the general case of 1 < p < ¥.

This is a joint work with Jung Jin Lee.

Let G be a locally compact group, let A(G) be the Fourier algebra of G, and let VN(G) be the von Neumann algebra generated by the left regular representation of G. We study the similarity problem for A(G), i.e., whether every bounded representation of A(G) on a Hilbert space H is similar to a *-representation. We show that the answer is affirmative if G is a SIN-group and we consider completely bounded representations.

We apply these results to classify the corepresentations of VN(G) for large classes of groups including SIN-groups, maximally almost periodic groups, and totally disconnected groups. This partially answers a question of Effros and Ruan.

This is a joint work with Michael Brannan (Queen's University).