×

Cohomology and \(L^2\)-Betti numbers for subfactors and quasi-regular inclusions. (English) Zbl 1415.46042

Summary: 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.

MSC:

46L37 Subfactors and their classification
55N99 Homology and cohomology theories in algebraic topology
PDFBibTeX XMLCite
Full Text: DOI arXiv