Cohomology and $L^2$-Betti numbers for subfactors and quasi-regular inclusions

TL;DR

This paper develops a unified (co)homology theory for subfactors and quasi-regular inclusions of von Neumann algebras, introducing $L^2$-Betti numbers and applying them to various algebraic structures.

Contribution

It introduces a general (co)homology framework for quasi-regular inclusions, recovering known invariants and computing new $L^2$-Betti numbers for specific subfactors.

Findings

$L^2$-Betti numbers vanish for amenable inclusions

Cohomological characterizations of property (T), Haagerup property, and amenability

Explicit calculations for Temperley-Lieb-Jones, Fuss-Catalan subfactors, free products, and tensor products

Abstract

We introduce $L^2$-Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II_1 factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups $\Gamma$, we recover the ordinary (co)homology of $\Gamma$. For Cartan subalgebras, we recover Gaboriau's $L^2$-Betti numbers for the associated equivalence relation. In this common framework, we prove that the $L^2$-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property and amenability. We compute the $L^2$-Betti numbers for the standard invariants of the Temperley-Lieb-Jones subfactors and of the Fuss-Catalan subfactors, as well as for free products and tensor products.