$L_2$-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.

Accepted for IMRN