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arXiv · 1905.11619

$L_2$-cohomology, derivations and quantum Markov semi-groups on $q$-Gaussian algebras

Martijn Caspers·Yusuke Isono·Mateusz Wasilewski
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Abstract

We study (quasi-)cohomological properties through an analysis of quantum Markov semi-groups. We construct higher order Hochschild cocycles using gradient forms associated with a quantum Markov semi-group. By using Schatten-$\mathcal{S}_p$ estimates we analyze when these cocycles take values in the coarse bimodule. For the 1-cocycles (the derivations) we show that under natural conditions they imply the Akemann-Ostrand property (using the Riesz transform). We apply this to $q$-Gaussian algebras $\Gamma_q(H)$. As a result $q$-Gaussians satisfy AO$^+$ for $| q | \leqslant \dim(H)^{-1/2}$. This includes a new range of $q$ in low dimensions compared to Shlyakhtenko.

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Martijn Caspers, Yusuke Isono, Mateusz Wasilewski. 2019-05-28. $L_2$-cohomology, derivations and quantum Markov semi-groups on $q$-Gaussian algebras. https://arxiv.org/abs/1905.11619

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