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-Hoch- schild 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 -equivariant deformation ((Ra)) of the upper halfplane ( = PSL 2(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 re al 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 ((Br))H 2 bound ( ,Z).