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\(H^ 4(BK,{\mathbb Z})\) and operator algebras. (English) Zbl 1087.46041

Let \(K\) be a compact Lie group and let \(BK\) be a classifying space for \(K\). Let \(U(K,{\mathbb T})\) denote the unitary group of the abelian von Neumann algebra \(L^\infty K\). Left translation induces an action of \(K\) on \(U(K,{\mathbb T})\). There is a natural isomorphism between the cohomology groups \(H^4(BK,{\mathbb Z})\) and \(H^2_K(U(K,{\mathbb T})/{\mathbb T})\), giving an identification between elements \(l\in H^4(BK,{\mathbb Z})\) and isomorphism classes of abelian noncentral extensions \(E_l\): \[ 0\rightarrow {\mathbb T}\rightarrow U(K,{\mathbb T})\rightarrow E_l\rightarrow K\rightarrow 0. \] Building on the seminal work [J. Feldman and C. C. Moore, Trans. Am. Math. Soc. 234, 325–359 (1977; Zbl 0369.22010)], the author constructs, when \(K\) is connected and semisimple, a faithful representation of the above exact sequence of the form \[ 0\rightarrow {\mathbb T}\rightarrow N_{U({\mathcal M})} ({\mathcal A})\rightarrow \text{ Aut} ({\mathcal M},{\mathcal A})\rightarrow \text{ Out} ({\mathcal M},{\mathcal A})\rightarrow 0, \] where \({\mathcal M}={\mathcal M}(R,\sigma)\) is a type \(II_1\) factor defined by an ergodic countable equivalence relation \(R\) on \(K\) and a cocycle \(\sigma \in H^2(R,{\mathbb T})\), and \({\mathcal A}=L^\infty K\) is a Cartan subalgebra in \({\mathcal M}\). He also poses, for every level \(l\), the question of existence of a factor \({\mathcal M}_l\), a group \({\mathcal E}\subset \text{ Aut}({\mathcal M}_l)\), and of an extension equivalent to the extension of \(E_l\), of the form \[ 0\rightarrow {\mathbb T}\rightarrow U({\mathcal M}_l)\rightarrow {\mathcal E}\rightarrow K\rightarrow 0. \] Such a realization of \(H^4(BK,{\mathbb Z})\) is thought to play a role in elliptic cohomology. The paper concludes with a discussion of the case \(K=\text{ Spin}(n)\).

MSC:

46L10 General theory of von Neumann algebras
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
22D40 Ergodic theory on groups
22E67 Loop groups and related constructions, group-theoretic treatment
46L55 Noncommutative dynamical systems

Citations:

Zbl 0369.22010
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