With a closed relation on a locally compact Hausdorff space X arising from a continuous positive map of X to the space of Radon measures on X we associate a C^*-algebra, namely the Cuntz-Pimsner algebra of a particular Hilbert bimodule constructed from the relation. The relations may have branch points and there are no local homeomorphism requirements. This family of C^*-algebras contains the C^*-algebras of directed graphs, and the crossed product C^*-algebras of topological dynamical systems. Some examples are considered.

It is often a complicated matter to estimate the C^*-norm (the usual Hilbert-space operator-norm) of a complex matrix. Nevertheless, an ultimate answer (without hard computation) can be sought for the best bound of the norm of T=A + iB where A and B are (non-commuting) hermitian operators with known eigenvalues. Moreover, the main result can be extended to cover the case of the sum of two normal matrices.

(This is a joint work with Chi-Kwong Li.)

If T is a bounded operator on a separable Hilbert space H which is not of the form scalar plus compact, then every bounded linear operator on H can be written as a linear combination of 14 or fewer operators unitarily equivalent to T, as a linear combination of 6 or fewer operators similar to T, and as a sum of 8 or fewer operators similar to T. When T is not polynomially compact, the set of all sums of 2 operators similar to T is dense in B(H), while if T is polynomially compact, but not of the form scalar plus compact, then the set of sums of 3 operators similar to T is dense in B(H).

A summary is given of recent progress in the classification of separable amenable C^*-algebras.

The study of E₀-semigroups of a type I_¥ factor was initated by R. Powers and W. Arveson, and many interesting classification results were obtained in the last years. Our purpose, in this talk, is to investigate the structure of E₀-semigroups that act on arbitrary von Neumann algebras. We show that such a semigroup can be canonically decomposed as the direct sum of an inner E₀-semigroup and a properly outer E₀-semigroup. This decomposition is stable under conjugacy and cocycle conjugacy. We also show that the class of inner E₀-semigroups can be completely characterized in terms of product systems.

The index theorem of Connes and Moscovici provides a formula for Chern character in cyclic cohomology involving residues of zeta functions associated to elliptic operators. I shall give a streamlined account of the `residue cocycle' discovered by Connes and Moscovici and the associated Fredholm index formula.

I will present joint work with B. Rangipour on invariant cyclic homology. This theory extends cyclic homology of Hopf algebras defined by Connes and Moscovici and its dual theory defined by present authors. I will also give several computations and conjectures regarding this new theory.

A q-white noise is the von Neumann algebra generated by q-Brownian motion on q-Fock space. In the case -1 < q < 1 we characterize bounded L^p-martingales (1 < p < ¥) w.r.t a canonical filtration as non-commutative Hardy spaces. This result generalizes work of Pisier and Xu on Itô-Clifford martingales which correspond to the case q=-1.

Absorbing extensions are C^*-algebra extensions having the property that their sum with a trivial extension are unitarily equivalent to the given extension. It can be shown that an extension that is absorbing in this sense must necessarily be full. In this talk, we attempt to modify the absorption property so that fullness is replaced, in a natural way, by a weaker condition we call local fullness.

I will discuss the structure of the equilibrium state space of quasi-free dynamics on the C^*-algebras associated by M. Pimsner to a Hilbert bimodule. (Current joint work with S. Neshveyev.)

(joint work with A. Nica)

Let (W, P) be a probability space and X be self-adjoint Gaussian random matrix, i.e. X: W®M_n(C)_s.a. is a matrix valued random variable with independent and normally distributed entires. If we write X = (f_i,j)_{i,j = 1}ⁿ and X is normalized so that E(f_i,j) = 0 for all i and j, and E(Â(f_i,j)²) = E(Á(f_i,j)²) = 1/(2n) for i ¹ j and E(f_i,i²) = 1/n, then E. Wigner showed that

We shall show that

It is a characteristic feature of completely bounded operators on B(H) to admit an amplification to the level of B(H Ä₂ K), where H and K are Hilbert spaces. Using Wittstock's Hahn-Banach principle and Tomiyama's slice map theorem, one deduces that, more generally, any completely bounded map on M can be amplified to a map on the von Neumann tensor product M [`(Ä)] N, whenever M and N are either von Neumann algebras or dual operator spaces with at least one of them sharing the w<sup>*</sup> operator approximation property. Our aim is to show that there is a simple and explicit formula of an amplification of completely bounded operators for all such pairs (M,N), thus providing a constructive approach to the amplification problem. The key idea is to combine two fundamental concepts in the theory of operator algebras, one being classical, the other one fairly modern: Tomiyama's slice maps on the one hand, and the description of the predual of M [`(Ä)] N given by Effros-Ruan in terms of the projective operator space tensor product, on the other hand.

We will further discuss the question of uniqueness of such an amplification, but mainly focus on various applications of our construction, such as:

• a generalization of the so-called Ge-Kadison Lemma;
• the amplification of completely bounded module homomorphisms;
• an algebraic characterization of normality for completely bounded maps in terms of a commutation relation for the associated amplification.

A topological group G has the fixed point on compacta (f.p.c.) property if there is a fixed point in each compact space upon which G acts continuously. This is a very strong version of amenability, which is why such groups are also called extremely amenable. The property is closely linked to Ramsey theory and to geometry of high-dimensional structures. Among a number of known groups with the f.p.c. property, many are linked to operator algebras and ergodic theory, and we will dwell on some of the recent developments in this direction.

We begin with a unital C^*-algebra A and a unital C^*-subalgebra, Z of the centre of A. We assume that we have a faithful, unital Z-trace t and a continuous action a:R®Aut(A) leaving t and hence Z invariant. We let d be the infinitesimal generator of a on A.

We have in this setting a largest (in the sense of quasi-containment) *-representation of A on a Hilbert space which carries a faithful, unital u.w.-continuous Z^-u.w.-trace [`(t)]: A<sup>-u.w.</sup> ® Z<sup>-u.w.</sup> extending t. We assume that A is concretely represented on this Hilbert space. We denote by A and Z respectively, the ultraweak closures of A and Z. One shows that there is an u.w.-continuous action [`(a)]:R ®Aut(A) extending a and leaving [`(t)] and Z invariant.

At this point we construct a representation, Ind = [(p)\tilde]×l of A\rtimesR on a certain self-dual Hilbert-Z module H_A constructed from a certain ``Z-Hilbert Algebra,'' A. We let M=Ind(A\rtimesR)^¢¢ which contains Z in its centre and has a faithful, normal semifinite Z-trace [^(t)]. This construction is half the battle. We let H denote the image of the Hilbert Transform in M and let P = [1/2](H+1) in M. We then consider the semifinite von Neumann algebra,

We prove the following theorem.

Theorem 1 Let A be a unital C^*-algebra and let Z Í Z(A) be a unital C^*-subalgebra of the centre of A. Let t: A ® Z be a faithful, unital Z-trace which is invariant under a continuous action a of R. Then for any a Î A^-1Çdom(d), the Toeplitz operator T_a is Fredholm relative to the trace [^(t)] on N=P(Ind(A\rtimesR)^¢¢)P,and

I will describe recent work with T. Giordano (Ottawa) and C. Skau (Trondheim) on topological orbit equivalence for actions of higher rank free abelian groups on the Cantor set.

I will present some recent results from free probability theory.