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 [14] 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 Gamma-equivariant deformation [17] of the upper half-plane (Gamma= 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 H-bound(2) (Gamma, Z) [2].