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

We introduce a class of densely de ned, unbounded, 2-Hochschild cocycles [14] on nite 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 -equivariant deformation [17] of the upper half-plane ( = PSL2(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. As a byproduct, we prove a strengthening of the non-triviality of the Euler cocycle in the bounded cohomology H2 bound(Gamma; Z)