Veranstaltungen am Mathematischen Institut

In operator algebra theory central sequences have long played a significant role in addressing problems in and around amenability, having been used both as a mechanism for producing various examples beyond the amenable horizon and as a point of leverage for teasing out the finer structure of amenable operator algebras themselves. One of the key themes on the von Neumann algebra side has been the McDuff property for II_1 factors, which asks for the existence of noncommuting central sequences and is equivalent, by a theorem of McDuff, to tensorial absorption of the unique hyperfinite II_1 factor. We will show that, for a topologically free minimal action of a countable amenable group on the Cantor set, the von Neumann algebra of the associated dynamical alternating group is McDuff. This yields the first examples of simple finitely generated nonamenable groups for which the von Neumann algebra is McDuff. This is joint work with Spyros Petrakos.

Given some large positive integer N, what is the largest possible size of a subset of {1,...,N} which does not contain a three-term arithmetic progression (i.e. without three distinct elements x,y,z satisfying x+z=2y)? Similarly, given a prime p and a large positive integer n, what is the largest possible size of a subset of the vector space F_p^n which does not contain a three-term arithmetic progression (i..e without three distinct vectors x,y,z satisfying x+z=2y)? These are long-standing problems in additive combinatorics. This talk will explain the known bounds for these problems, give an overview of some of the proof techniques, and discuss additional applications of these techniques to other additive combinatorics problems.

Abstract: For a finite group G, an object with a G-action in some semi-additive category, and any additive functor, there always exists a factorization square involving assembly, coassembly and norm maps. One can use this to give a completely formal proof that the assembly map in K- and L-theory for finite groups is rationally and K(n)-locally split-injective. I have some open conjectures for how one might deal with infinite groups as well.

Based on previous results on ideals in reduced groupoid C*-algebras, Christian Bönicke and I proved in 2018 that all ideals in a reduced groupoid C*-algebra C_r*(G) are dynamical if and only if the underlying étale groupoid G is inner exact and has the residual intersection property. Recall that an ideal I in the reduced groupoid C*-algebra C_r*(G) is dynamical if I is uniquely determined by an open invariant subset U of G^0, the unit space of the groupoid G. Equivalently, I=I(U) the ideal generated by C_0(U) in C_r*(G). In the ongoing project with Jiawen Zhang, we are investigating ideals in reduced C*-algebras of general étale groupoids. In this general setting, each ideal I in C_r*(G) is associated with two open invariant subsets of G^0, namely the inner support U_I and the outer support V_I. Let I be an ideal in C_r*(G), we call I is of type I if I(U_I ) ? I ? I(V_I ), and I is of type II if U_I = V_I. It is known that an ideal I in C_r^*(G) is dynamical if and only if I is of both type I and type II; and a non-dynamical ideal I can be either type I or type II. It turns out that every ideal in C_r*(G) can be uniquely reconstructed from type I and type II ideals. Most importantly, we also provide characterizations for type I and type II ideals by means of underlying groupoids. In my talk I will present several examples and applications.