Speaker
Malte Leimbach, Max-Planck-Institut für Mathematik, Bonn
Abstract
We use operator systems to relate the following two problems for discrete groups:
(1) Does every positive semi-definite function on a given symmetric unital subset admit an extension to a positive semi-definite function on the whole group?
(2) Does every positive element of the group \(C^*\)-algebra which is a finite linear combination of some fixed unitary generators admit a decomposition into sums of squares?
As applications, we obtain a statement about (non-)nuclearity of a certain pairs of operator systems, and we exemplify how one can use the \(C^*\)-envelope to disprove existence of positive semi-definite extensions and sums-of-squares decompositions. Based on joint work with Evgenios Kakariadis, Ivan Todorov, and Walter van Suijlekom.