L2-Cohomology, Derivations, and Quantum Markov Semi-Groups onq-Gaussian Algebras

Abstract

AbstractWe 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 we obtain the Akemann–Ostrand property. We apply this to $q$-Gaussian algebras $\Gamma _q(H_{{\mathbb{R}}})$. As a result $q$-Gaussians satisfy AO$^+$ for $\vert q \vert \leqslant \dim (H_{{\mathbb{R}}})^{-1/2}$. This includes a new range of $q$ in low dimensions compared to Shlyakhtenko [ 34].