Translation invariant Dirichlet forms in the context of locally compact quantum groups

Adam Skalski
Impan (Warsaw)
Monday, April 23, 2018 - 14:15 to 15:15

Since the work of Cipriani on one hand and Goldstein and Lindsay on the other in the 1990s it is known that certain natural class of symmetric Markov semigroups on a von Neumann algebra $M$ equipped with a faithful normal state admits extensions to associated Haagerup $L^p$-spaces and is characterised via a Dirichlet property of the generating quadratic form on the $L^2$-space. Recently Cipriani, Franz and Kula studied a special class of such semigroups associated to compact quantum groups. In this talk I will discuss how their results extend to the framework of locally compact quantum groups, where two new important technical features appear: there is no natural "algebraic" domain for the generator and one needs to work with weights, as opposed to finite states (using the extension of the appropriate Dirichlet form result provided by Goldstein and Lindsay). I will also present some applications of Dirichlet forms to the study of geometric properties of quantum groups. This is based on joint work with A.Viselter.

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