##
**Classification of injective factors. Cases \(\mathrm{II}_1\), \(\mathrm{II}_\infty\), \(\mathrm{III}_\lambda\), \(\lambda\neq 1\).**
*(English)*
Zbl 0343.46042

The paper contains definitive results an hyperfiniteness and injectivity of von Neumann algebras, which give the solutions of many important problems in the theory of operator algebras. Let \(N\) be a von Neumann algebra on a Hilbert space \(H\) and \(B(H)\) the algebra of all bounded linear operators in \(H\). \(N\) is said to be injective if there is a projection of norm one of \(B(H)\) to \(N\) or equivalently if, for a \(C^*\) algebra \(A\) and its \(C^*\)-subalgebra \(B\), any completely positive map of \(B\) into \(N\) has a completely positive extension to \(A\) [J. Hakeda and the reviewer, Tôhoku Math. J. (2) 19, 315–323 (1967; Zbl 0175.14201); E. G. Effros and E. C. Lance, Tensor products of operator algebras, Adv. Math. 25, 1–34 (1977; Zbl 0372.46064)]. The algebra \(N\) is also said to be semidiscrete if the identity map \(N\to N\) is approximated in \(\sigma\)-weak topology by a net of completely positive maps of finite rank. The author’s main result asserts that for a factor \(N\) of type \(\mathrm{II}_1\) in a separable Hilbert space the notions of injectivity and semidiscreteness are equivalent to the hyperfiniteness of \(N\), the weak closure of an ascending sequence of matrix algebras (results are stated in separated theorems). He also proved further equivalence of these properties to those of the property \(P\) by J. T. Schwartz [Commun. Pure Appl. Math. 16, 19–26 (1963; Zbl 0131.33201)] and the property \(\Gamma\) [F. J. Murray and J. von Neumann, Ann. Math. (2) 44, 716–808 (1943; Zbl 0060.26903)]. Thus, as natural consequences of these results one knows that up to isomorphisms there is only one injective factor of type \(\mathrm{II}_1\), a hyperfinite factor and the hyperfinite factor of type \(\mathrm{II}_\infty\) is unique. It is also now clear that all subfactors of a hyperfinite factor \(R\) of type \(\mathrm{III}_1\) are isomorphic to \(R\) or finite dimensional. The equivalences of those properties are further shown to be valid for any factor in a separable Hilbert spare. Besides these remarkable consequences, the result implies the following answer to the conjecture by Kadison and Singer; any representation of a solvable separable locally compact group or a connected locally compact separable group in a Hilbert space generates a hyperfinite von Neumann algebra. The paper also contains characterizations of an automorphism which lies in the closure of the inner automorphism group, \(\operatorname{Int}N\), for a factor of type \(\mathrm{II}_1\).

Reviewer: J. Tomiyama