On unbounded, non-trivial Hochschild cohomology in finite von Neumann algebras and higher order Berezin's quantization

We introduce a class of densely defined, unbounded, 2-Hochschild cocycles ([PT]) on finite von Neumann algebras $M$. Our cocycles admit a coboundary, determined by an unbounded operator on the standard Hilbert space associated to the von Neumann algebra $M$. For the cocycles associated to the $\mathrm{\Gamma }$-equivariant deformation ([Ra]) of the upper halfplane $(\mathrm{\Gamma }=PS{L}_2(\mathbb{Z}))$, the “imaginary” part of the coboundary operator is a cohomological obstruction - in the sense that it can not be removed by a “large class” of closable derivations, with non-trivial real part, that have a joint core domain, with the given coboundary.