On the subequivalence relations induced by a Bernoulli action
Author: Adrian Ioana and joint work with Ionut Chifan
Let G be a countable group, (X,m) be a non-trivial probability space and denote by S the equivalence relation induced by the Bernoulli action of G on (X,m)^G. I will prove that for any subequivalence relation R of S, there exists a partition X_0,X_1,.. of X^G with R-invariant measurable sets such that the restriction R|X_0 is hyperfinite and R|X_i is strongly ergodic, for i=1,2,..
Taking Groupoid C*-algebras to the limit
Author: Aviv Censor and joint work with Daniel Markiewicz
Let T be a compact Hausdorff space. Raeburn and Taylor associated groupoid C*-algebras to open covers of T, and showed that any element in the Brauer group of T can be realized by such an algebra. We study the asymptotic behavior of the Raeburn-Taylor C*-algebras related to a sequence of open cover refinements. This is accomplished via a limit groupoid G which we construct, along with a groupoid 2-cocycle. Our main result presents the groupoid C*-algebra of G as a certain generalized direct limit of the Raeburn-Taylor algebras. As a special case, our construction produces all UHF C*-algebras as algebras of the form C*(G). This talk is based on joint work with Daniel Markiewicz.
Fourier transform on locally compact quantum groups
Author: Byung-Jay Kahng
The notion of Fourier transform is among the more important tools in analysis, which has been generalized in abstract harmonic analysis to the level of abelian locally compact groups. The aim of this paper is to further generalize the Fourier transform: Motivated by some recent works by Van Daele in the multiplier Hopf algebra framework, and by using the Haar weights, we define here the (generalized) Fourier transform and the inverse Fourier transform, at the level of locally compact quantum groups. We will then consider the analogues of the Fourier inversion theorem, Plancherel theorem, and the convolution product. Along the way, we also obtain an alternative description of the dual pairing map between a quantum group and its dual.
A Kurosh-Type Theorem for Type III Factors
Author: Jason Asher
We will present an extension of the Kurosh-Type Theorem of N. Ozawa to the case of the reduced free product of II_1 factors with non-tracial states. The argument will proceed via a generalization of S. Popa's intertwining-by-bimodules technique.
On K-groups of the Crossed Products associated with a Class of Transformations of Tori
Author: Kamran Reihani
In this talk, we first review how the Pimsner-Voiculescu exact sequence can be employed to compute the K-groups of the crossed products associated with a general transformation of an n-torus. An error in the literature is addressed regarding the torsion parts of the K-groups of crossed products of Furstenberg transformations. Then for a class of transformations of tori including Furstenberg ones, we explicitly compute the free ranks of the K-groups based on the combinatorial properties of representations of the simple Lie algebra sl_2(C). This leads to finding generating functions and the asymptotic behavior of the sequence of ranks in terms of the dimension of the transformed torus.
C*-extreme completely positivemappings
Author: Martha Gregg
The generalized state space of a commutative C*-algebra, denoted SH(C(X)), is the set of positive unital maps from C(X) to the algebra B(H) of bounded linear operators on a Hilbert space H . C*-convexity is one of several noncommutative analogs of convexity which have been discussed in this context. In this paper it is shown that a C*-extreme point of SH(C(X)) satisfies a certain spectral condition on the operators in the range of the associated positive operator-valued measure. If H is finite dimensional, D. Farenick and P. Morenz have shown that every C*- extreme point of SH(C(X)) is multiplicative. However, their method is fundamentally different from the one used here, which enables us to show that C*-extreme maps from C(X) into K+ , the algebra generated by the compact and scalar operators, are multiplicative. It is then possible determine the structure of these maps.
Subfactors obtained from Hadamard matrices
Author: Richard Burstein
A Hadamard matrix may be used to construct a symmetric commuting square, which gives a subfactor via iteration of the basic construction. Certain Hadamard subfactors arise from the group construction of Bisch and Haagerup. We will discuss methods for computing the principal graphs and standard invariants of these examples.
Group cocycles and the ring of affiliated operators
Author: Jesse Peterson and joint work with Andreas Thom
I will present some results (joint work with Andreas Thom) on cocycles from a group into it's left regular representation and also into the ring of affiliated operators of the group von Neumann algebra. Specifically I will be interested in when a group has positive first l2-Betti number and what does this mean about the structure of certain subgroups when this is the case. I will include a strong generalization of a result of Luck and Gaboriau which states that a finitely generated normal subgroup of a group with positive first l2-Betti number is either a finite subgroup or is of finite index. I will also generalize some classical results of Karrass, Solitar, Griffiths, and Baumslag.