Speaker: Gabor Szabo (KU Leuven)
Title: The equivariant Kirchberg-Phillips theorem
Time/Date: 4:45-6:15pm, December 10 (Thu.), 2020
Room: This will be an online seminar on Zoom. Please ask Kawahigashi for the Zoom link.
Abstract: After the enormous breakthrough in the Elliott program over the last few years, it becomes an even more pressing question if one can push the successful classification machinery into the realm of C*-dynamics. While there exist some very satisfactory partial results, most of which were pioneered by the Japanese community, they concern actions of rather special groups and are technically underpinned by the Rokhlin property. A particularly natural but ambitious goal is to classify outer actions of amenable groups on Kirchberg algebras, which has been achieved for poly-Z groups in recent work of Izumi-Matui. In this talk I would like to announce that this can indeed be done in general. In particular, our main result asserts that outer actions of amenable groups on Kirchberg algebras are classified, up to cocycle conjugacy, by equivariant Kasparov theory. The same result turns out to be true if one considers amenable actions of arbitrary discrete groups. (In fact we can even do better, but I leave the details of that to my talk.) Our methodology builds in a crucial way on my recent work about a new categorical perspective on C*-dynamics, and involves two somewhat independent conceptual components. The first component involves new insights related to the equivariant KK-theory of arbitrary C*-dynamics in the spirit of the Lin-Dadarlat-Eilers stable uniqueness theorem. The second component is to establish existence and uniqueness theorems related to certain cocycle embeddings between the C*-dynamics mentioned in the main result. The talk will aim to give a broad conceptual overview of our approach towards the classification theorem. All of this is joint work with James Gabe.