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Classification of injective factors. Cases ${\mathrm{II}}_{1}$, ${\mathrm{II}}_{\infty }$, ${\mathrm{III}}_{\lambda }$, $\lambda \ne 1$. (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 $B\left(H\right)$ the algebra of all bounded linear operators in $H$. $N$ is said to be injective if there is a projection of norm one of $B\left(H\right)$ 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, Tĥoku math. J., II. Ser. 19, 315–323 (1967; Zbl 0175.14201); E. Effros and C. Lanee, Tensor products of operator algebras, to appear in Advances Math.]. 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 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 II${}_{1}$, a hyperfinite factor and the hyperfinite factor of type II${}_{\infty }$ is unique. It is also now clear that all subfactors of a hyperfinite factor $R$ of type 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, $IntN$, for a factor of type II${}_{1}$.
Reviewer: J. Tomiyama

##### MSC:
 46L10 General theory of von Neumann algebras 46M10 Projective / injective objects in categories of topol. linear spaces