Popa, Sorin; Shlyakhtenko, Dimitri; Vaes, Stefaan Cohomology and \(L^2\)-Betti numbers for subfactors and quasi-regular inclusions. (English) Zbl 1415.46042 Int. Math. Res. Not. 2018, No. 8, 2241-2331 (2018). 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. Cited in 1 ReviewCited in 15 Documents MSC: 46L37 Subfactors and their classification 55N99 Homology and cohomology theories in algebraic topology Keywords:\(L^2\)-Betti numbers; homology; cohomology; standard invariant; subfactor PDFBibTeX XMLCite \textit{S. Popa} et al., Int. Math. Res. Not. 2018, No. 8, 2241--2331 (2018; Zbl 1415.46042) Full Text: DOI arXiv