Operator algebra of foliations with projectively invariant transverse measure

TL;DR

This paper investigates the operator algebras linked to foliations with projectively invariant measures, revealing conditions under which holonomy invariant measures are absent, using cyclic cohomology and modular automorphisms.

Contribution

It introduces a framework connecting ergodicity, cyclic cohomology, and modular automorphisms to analyze operator algebras of foliations with projectively invariant measures.

Findings

Holonomy invariant transverse measure can be absent under certain ergodicity conditions.

Cyclic cohomology class associated with the transverse fundamental cocycle characterizes measure properties.

Modular automorphism group plays a key role in the structure of these operator algebras.

Abstract

We study the structure of operator algebras associated with the foliations which have projectively invariant measures. When a certain ergodicity condition on the measure preserving holonomies holds, the lack of holonomy invariant transverse measure can be established in terms of a cyclic cohomology class associated with the transverse fundamental cocycle and the modular automorphism group.