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

TL;DR

This paper explores unbounded Hochschild cocycles in finite von Neumann algebras, revealing cohomological obstructions in the context of higher order Berezin's quantization and $ ext{PSL}_2( ext{Z})$-equivariant deformations.

Contribution

It introduces a new class of unbounded Hochschild cocycles on finite von Neumann algebras and analyzes their coboundaries and obstructions in deformation settings.

Findings

Existence of unbounded Hochschild cocycles with non-trivial cohomological obstructions.

The imaginary part of the coboundary operator cannot be removed by certain derivations.

Identification of cohomological obstructions in $ ext{PSL}_2( ext{Z})$-equivariant deformation.

Abstract

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 $\Gamma$-equivariant deformation ([Ra]) of the upper halfplane $(\Gamma=PSL_2(\mathbb Z))$, 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.