Abstract
AbstractOne 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]).
Highlights
We 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
The proof of Ozawa and Popa [35] essentially splits into two parts. They show that weak amenablity of a group can be used to prove a so-called weak compactness property
We introduce a new property for a quantum Markov semigroups (QMSs) of central multipliers on a compact quantum group which we call ‘approximate linearity with almost commuting intertwiners’
Summary
By δ(x ∈ X) we denote the function that is 1 if x ∈ X and 0 otherwise. Inner products are linear in the left leg. We say that a von Neumann algebra M has the W∗CBAP if there exists a net (Φi)i of normal completely bounded finite-rank maps M → M such that:. We say that a finite von Neumann algebra with faithful normal state (M,τ ) has the Haagerup property if there exists a net (Φi)i of normal unital completely positive maps M → M such that τ ◦ Φi = τ , such that Φi is compact as a map L2(M,τ ) → L2(M,τ ) and such that for every x ∈ M we have Φi(x) → x strongly. A finite von Neumann algebra M is called strongly solid if for every diffuse amenable von Neumann subalgebra P ⊆ M , NorM (P )′′ is amenable, where the normaliser is defined as.
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