Riesz transforms on compact quantum groups and strong solidity

Martijn Caspers*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call 'approximate linearity with almost commuting intertwiners'. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient-S2 condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann-Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

Original languageEnglish
Pages (from-to)2135-2171
Number of pages37
JournalJournal of the Institute of Mathematics of Jussieu
Issue number6
Publication statusPublished - 2021


  • compact quantum groups
  • quantum Markov semigroups
  • Riesz transforms
  • strong solidity


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