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

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 $|q|⩽\dim (H{)}^{-1/2}$. This includes a new range of $q$ in low dimensions compared to Shlyakhtenko.

arXiv commentAccepted for IMRN

References in corpus (6)

• Some prime factorization results for type II_1 factors
• The Haagerup property for locally compact quantum groups
• Non injectivity of the q-deformed von Neumann algebra
• Feynman Diagrams and Wick Products Associated with $q$-Fock spaces
• The Haagerup approximation property for von Neumann algebras via quantum Markov semigroups and Dirichlet forms
• Strong Solidity of the q-Gaussian Algebras for all -1 < q < 1

Cited by in corpus (1)

• Riesz transforms on compact quantum groups and strong solidity