Introduction to Classical and Quantum Markov Semigroups

Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups

IEEE Transactions on Information Theory ◽

10.1109/tit.2021.3065452 ◽

2021 ◽

Vol 67 (5) ◽

pp. 2878-2909

Author(s):

Ivan Bardet ◽

Marius Junge ◽

Nicholas Laracuente ◽

Cambyse Rouze ◽

Daniel Stilck Franca

Keyword(s):

Markov Semigroups

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RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000165 ◽

2021 ◽

pp. 1-37

Author(s):

Martijn Caspers

Keyword(s):

Quantum Group ◽

Quantum Groups ◽

Riesz Transform ◽

Wreath Products ◽

Compact Quantum Group ◽

Riesz Transforms ◽

Markov Semigroup ◽

Markov Semigroups ◽

Free Products ◽

Compact Quantum Groups

Abstract 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- ${\mathcal {S}}_2$ 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]).

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