MR3299506 Reviewed Rădulescu, Florin(I-ROME2) On unbounded, non-trivial Hochschild cohomology in finite von Neumann algebras and higher order Berezin's quantization. (English summary) Rev. Roumaine Math. Pures Appl. 59 (2014), no. 2, 265–292.

Abstract

If (At)t>1 is a family of finite von Neumann algebras with a Chapman-Kolmogorov set of linear maps (symbol system) (Φs,t), and if αt:A→A are isomorphisms in a finite family of von Neumann algebras, the corresponding Hochschild cocycles are related to an obstruction to the deformation of the set of linear maps (Φs,t) in the corresponding Chapman-Kolmogorov system (Φs,t)˜ of completely positive maps. In this set-up, the author introduces an invariant (c,Z) for a finite von Neumann algebra M, consisting of a 2-Hochschild cohomology cocycle c and a coboundary unbounded operator Z for c. With some assumptions on c and Z=α+X+iY (α>0, Y is antisymmetric), the existence of an unbounded derivation δ is considered, such that the "perturbed'' operator Z+δ has imaginary part of cocycle c with antisymmetric coboundary operator defined on identity operator 1. In this sense, it turns out that the imaginary part of Z is a "cohomological obstruction'', in the sense that Z+δ loses some properties of Z for a large class of δ. The results are considered in Γ-equivariant Berezin quantum deformation of the upper half-plane with Γ=PSL2(Z), and in the case of von Neumann algebra L(F∞) with free group F∞.