If a and b are trace-class operators, and if u is a partial isometry, then

We consider the Hilbert bimodule associated with a closed relation on the torus obtained by an infinite winding of an interval about the torus. The algebra of compact operators is identified and the associated Cuntz-Pimsner C^*-algebra formed.

Let A_q denote the rotation C^*-algebra generated by unitaries U and V satisfying VU = e^{2 pi q} UV where q is a fixed real number. Let r and k denote the canonical order 6 and 3 transforms on A_q respectively, and let 6_q = A_q \rtimes_r Z₆ and 3_q = A_q \rtimes_k Z₃ denote the associated C^*-crossed products by corresponding cyclic groups. In this talk we discuss how to obtain injections Z¹⁰\hookrightarrow K₀(6_q) and Z⁸ \hookrightarrowK₀(3_q) by computing the Connes-Chern characters of the projections and modules which form linearly independent classes in the K₀ groups. Along the way, we obtain the unbounded traces on the canonical dense smooth *-subalgebras, which for irrational q give Connes' cyclic cohomology groups of order zero HC⁰(6_q) @ C⁹, HC⁰(3_q) @ C⁷.

Jones' planar algebra formalism provides the most elegant and powerful description of the standard invariant of a finite index extremal II₁ subfactor, allowing the use of diagramatic techniques to prove results in the theory of subfactors. We will first review the theory of planar algebras and subfactors and then discuss extensions of the planar algebra results to general finite index subfactors and also to infinite index II₁ subfactors.

A free semigroup algebra is the unital weak operator topology closed algebra generated by n isometries with pairwise orthogonal ranges. We show that the unit ball of the norm closed algebra is weakly dense in the whole ball if and only if the weak-* closure agrees with the weak operator closure. This fails only when the weak closure is a von Neumann algebra but the weak-* closure is not-and no examples of this phenomenon are known to exist.

We shall discuss the problem of classifying various kinds of C^*-dynamical systems up to equivariant isomorphisms using invariants.

An attempt is made to evaluate the significance of recent results on the classification of amenable C^*-algebras, which suggest that new invariants may be needed.

I will discuss some of the tools used for computation of invariants of braid subfactors. I will also mention a generalisation to subfactors from tensor categories. Work in progress with H. Wenzl.

In his work on classification of type IIE₀-semigroups, R. T. Powers has introduced a standard form for spatial E₀-semigroups that appears to correspond to the standard form of von Neumann algebra theory. It was noticed that for some particular examples of E₀-semigroups in standard form (CAR/CCR flows, Tsirelson's type II₀-examples), the gauge groups act transitively on the set of normalized units. A result of A. Alevras implies that for these examples, conjugacy and cocycle-conjugacy are equivalent concepts. It was conjectured by A. Alevras and W. Arveson that the equivalence between conjugacy and cocycle-conjugacy holds for all E₀-semigroups in standard form.

The goal of my talk is to discuss my recent results on this problem.

Both in the measurable and in the Borel case, hyperfinite actions are well understood and classified up to orbit equivalence. In particular any free Borel action of Z² is (orbit equivalent to) hyperfinite. In the case of (minimal) topological actions of Z² on the Cantor set, the same question is still open. In this talk, I will present recent developments in the study of this problem. These developments come from a work in progress with I. Putnam (Victoria) and C. Skau (Trondheim).

A substantial part of Riemannian geometry has been concerned with geodesic dynamics. For example, significant relationships have been established between the entropy of the geodesic flow and topological and geometric invariants. In noncommutative geometry, however, this dynamical aspect has been largely absent. We initiate a study of growth in noncommutative geodesic flows in the context of Rieffel's work on group C^*-algebras as quantum metric spaces.

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

(Joint work with Peter C. Gibson, and Gary F. Margrave)

We consider a continuous version of Gabor multipliers: operators consisting of a short-time Fourier transform, followed by multiplication by a distribution on phase space (called the Gabor symbol), followed by an inverse short-time Fourier transform, allowing different localizing windows for the forward and inverse transforms. For a given pair of forward and inverse windows, which linear operators can be represented as a Gabor multiplier, and what is the relationship between the (non-classical) Kohn-Nirenberg symbol of such an operator and the corresponding Gabor symbol? These questions are answered completely for a special class of "compatible" window pairs. In addition, concrete examples are given of windows that, with respect to the representation of linear operators, are more general than standard Gaussian windows. The results in the paper help to justify techniques developed for seismic imaging that use Gabor multipliers to represent nonstationary filters and wavefield extrapolators.

If X_n is a n ×n self-adjoint Gaussian random matrix then a theorem of Johansson (1998) shows that the random variables {Tr(T_k(X_n))}_k converge, as n ® ¥, to independent Gaussian random variables, where {T_k} are the Chebyshev polynomials of the second kind (suitably centred). Cabanal-Duvillard (2001) extended this result to the case of a pair of independent self-adjoint Gaussian random matrices X_n and Y_n, in that he showed that the random variables {Tr (T_j(X_n)),Tr(T_k(Y_n)),Tr(S_l,m(X_n,Y_n))}_j,k,l,m converge to independent Gaussian random variables, where S_m,n is a family of Chebyshev polynomials of the first kind in two non-commuting variables.

I shall give a diagrammatic interpretation of the result of Cabanal-Duvillard motivated by a theorem of Andu Nica and me which shows that the correlation of certain ensembles of random matrices is given by non-crossing annular diagrams. Some extensions to the case of Wishart ensembles will be given.

This is joint work with Tim Kusalik (Queen's) and Roland Speicher (Queen's).

The circular element of Voiculescu is a remarkable generator for the von Neumann algebra of the free group on two generators. The q-circular element is a deformation of the circular one, which appears in the context of the q-commutation relations of Bozejko and Speicher. The talk will survey a few results and (mostly) problems related to q-circular elements and to some of their siblings called "z-circular elements".

Let S be Riemann surface of genus n ³ 1. Denote by SpecS the length spectrum of S, i.e. the (countable) set of length of all simple periodic geodesics on S. Selberg proved that SpecS is a conformal invariant of S. Later on Wolpert showed that in "generic" case SpecS is in fact a complete conformal invariant, i.e. "module" of S.

In present talk we discuss how one can relate to SpecS a simple dimension group of rank 2n. In this setting, classification of dimension groups "translates" into the language of moduli of Riemann surfaces. In the simplest case n=1, one gets a correspondence between complex and noncommutative tori.

The distortion property, proved by Odell and Schlumprecht in 1994, has been one of the most important and stunning discoveries ever made about the geometry of Hilbert space. To date, there is no direct proof of the distortion property for Hilbert space, rather it is deduced indirectly using distortion in some special Banach spaces. Vitali Milman has suggested that one way to understand the distortion is to try and formulate its analogues for infinite-dimensional groups, such as the unitary group of the separable Hilbert space. In this talk, we will report on some preliminary progress in this direction. It turns out that the distortion property admits a very natural reformulation in the language of topological transformation groups, and when applied to concrete examples other than the unitary group, it allows to incorporate some known results of infinite combinatorics into the same scheme. Besides, there is an interesting open question which the speaker is unable to answer (as of September 15): does every topological group have the distortion property?

We generalise the local index formula of Connes and Moscovici to the case of unbounded spectral triples (A,N,D) for a *-subalgebra A of a general semifinite von Neumann algebra, N. In this setting it gives a formula for (type II) spectral flow along a path joining D, an unbounded self adjoint Breuer-Fredholm operator, affiliated to N, to uDu^* for a unitary u Î A.

We start from the spectral flow formula for finitely summable triples developed by Carey-Phillips and show how the the seemingly innocuous normalising constant in that formula actually gives a new approach to the Connes-Moscovici results. This is joint work with Alan Carey, Adam Rennie and Fyodor Sukochev.

The crossed product of every irrational rotation algebra by the noncommutative Fourier transform is AF. The crossed products by the standard actions of Z/3Z and Z/6Z always have tracial rank zero, and we are close to proving that these crossed products are also always AF. (Joint work with Wolfgang Lück and Sam Walters.)

We show that various cyclic and cocyclic modules attached to Hopf algebras and Hopf modules are related to each other via Connes' duality isomorphism for the cyclic category. This is a joint work with M. Khalkhali.

We give a construction ,applicable to any countable group G, of a continuous action of G on the Cantor set, which is free and minimal. Furthermore, the construction in question yields an invariant probability measure for the action, which is remarkable since the group do not have to be amenable. We will apply this to prove some new results.

Let A be a nonsingular n by n real matrix and let d = |detA|. For w Î L²(Rⁿ), k Î Z and x Î Rⁿ, let w_k,x(y) = d^-k/2w(A^-ky-x), for all y Î Rⁿ. We call w a tight frame generator (TFG) if {w_k,x:k Î Z,x Î Rⁿ} forms a normalized tight frame in L²(Rⁿ). We think of TFGs as semi-discrete wavelets. The existence of a TFG depends on the nature of A and smooth TFGs can exist only if A, or its inverse, is expansive. In this case, we show how smooth TFGs are intimately related to projections in L¹(G), where G is the semidirect product group formed by the action of the integers on Rⁿ through A.