Cocycles and positive functionals in higher cohomology

TL;DR

This paper generalizes the GNS construction to higher cohomology, linking positive functionals and cocycles, and characterizes cohomological vanishing via spectral gap conditions.

Contribution

It extends classical cocycle theory to higher degrees and relates spectral gap properties to cohomology vanishing in a novel way.

Findings

Higher degree GNS construction established

Cohomology vanishing characterized by spectral gap

Extension property for positive functionals proven

Abstract

We establish and explore the correspondence between positive functionals and cocycles in higher unitary cohomology. We generalize the classical cocycle version of the Gelfand-Naimark-Segal construction to higher degrees and apply it to characterize vanishing of higher unitary cohomology as an extension property for positive functionals. We also prove that under mild conditions the algebraic spectral gap for the one sided Laplacian characterizes cohomological vanishing instead of reducedness of unitary cohomology