Apr. 07 
14:00  15:30 RIMS 206 
Yuhei Suzuki (Tokyo)
Construction of minimal skew products of amenable minimal dynamical systems
We give a generalization of a result of Glasner and Weiss. This provides many new examples of amenable minimal dynamical systems of
exact groups. We also study the pure infiniteness of the crossed products of minimal dynamical systems arising from this result. For this purpose, we introduce and study a notion of the finite filling property for etale groupoids, which generalizes a result of Jolissaint and Robertson. As an application, we show that for any connected closed topological manifold M, every countable nonamenable exact group admits an amenable minimal free dynamical system on the product of M and the Cantor set whose crossed product is a Kirchberg algebra. This extends a result of Rørdam and Sierakowski.

May 12 
15:00  16:30 RIMS 206 
Igor Klep (Auckland)
Commuting Dilations and Linear Positivstellensätze
Given a tuple $A=(A_1,...,A_g)$ of real symmetric matrices of the same size, the affine linear matrix polynomial $L(x):=I\sum A_j x_j$ is a monic linear pencil. The solution set $S_L$ of the corresponding linear matrix inequality, consisting of those $x$ in ${\mathbb R}^g$ for which $L(x)$ is positive semidefinite (PsD), is called a spectrahedron. It is a convex semialgebraic subset of ${\mathbb R}^g$. We study the question whether inclusion holds between two spectrahedra. We identify a tractable relaxation of this problem by considering the inclusion problem for the corresponding free spectrahedra $D_L$. Here $D_L$ is the set of tuples $X=(X_1,...,X_g)$ of symmetric matrices (of the same size) for which $L(X):=I\sum A_j \otimes X_j$ is PsD.
We explain that any tuple $X$ of symmetric matrices in a bounded free spectrahedron $D_L$ dilates, up to a scale factor, to a tuple $T$ of commuting selfadjoint operators with joint spectrum in the corresponding spectrahedron $S_L$. The scale factor measures the extent that a positive map can fail to be completely positive. In the case when $S_L$ is the hypercube $[1,1]^g$, we derive an analytical formula for this scale factor, which as a byproduct gives new probabilistic results for the binomial and beta distributions.
The talk is based on joint work with Bill Helton, Scott McCullough and Markus Schweighofer.
