L2-cohomology, derivations and quantum Markov semi-groups on q-Gaussian algebras

Journal article (2020)

Authors

M.P.T. Caspers Analysis - EEMCS
Yusuke Isono Kyoto University
Mateusz Wasilewski Katholieke Universiteit Leuven
Research Group
Analysis (EEMCS) (TU Delft)
Copyright: © 2020 M.P.T. Caspers, Yusuke Isono, Mateusz Wasilewski
DOI: https://doi.org/10.1093/imrn/rnaa044
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Published Date

2020

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

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-Sp estimates we analyze when these cocycles take values in the coarse bimodule. For the 1-cocycles (the derivations) we show that under natural conditions we obtain the Akemann–Ostrand property. We apply this to q-Gaussian algebras Γq(HR)⁠. As a result q-Gaussians satisfy AO+ for |q|⩽dim(HR)−1/2⁠. This includes a new range of q in low dimensions compared to Shlyakhtenko [ 34].

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