Gradient forms and strong solidity of free quantum groups

Abstract:

In their fundamental paper Ozawa and Popa introduced the notion of strongly solid von Neumann algebras: the von Neumann algebra generated by the normalizer of any amenable von Neumann subalgebra is amenable again. They proved that the free group factors are strongly solid, in particular implying that they do not have a Cartan subalgebra, that they are prime factors and that they cannot be written as crossed products. In this talk we show that semi-group theory can be used to obtain strong solidity results. In particular we show that arbitrary free quantum groups are strongly solid. A key result in the proof is the study of the `gradient bimodule' associated with a natural semi-group on the free quantum groups and to show that it is weakly contained in the coarse bimodule (or `biregularbimodule').