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

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].