Riesz transforms and strong solidity

Martijn Caspers (TU Delft)

Thursday 14th January, 2021 16:00-17:00 Contact the organizers for Zoom coordinates

Abstract

The Riesz transform is one of the most important and classical examples of a Fourier multiplier on the real numbers. It may be described as the operator "∇_j Δ^-1/2" where ∇_j = d/dx_j is the derivative and Δ is the Laplace operator. In a more general context the Riesz transform may always be defined for any diffusion semigroup on the reals. In case the generator of this semi-group is the Laplace operator the classical Riesz transform is retrieved. In quantum probability the quantum Markov semi-groups play the role of the diffusion semi-groups and again a suitable notion of Riesz transform can be described.

In this paper we show that the Riesz transform may be used to prove rigidity properties of von Neumann algebras. We focus in particular on examples from compact quantum groups and q-Gaussian algebras. The rigidity properties include the Akemann-Ostrand property and strong solidity. Part of this talk is based on joint work with Mateusz Wasilewski and Yusuke Isono. 

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