## 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, 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, $$\operatorname{Int}N$$, for a factor of type II$$_1$$.
Reviewer: J. Tomiyama

### MSC:

 46L10 General theory of von Neumann algebras 46M10 Projective and injective objects in functional analysis

### Citations:

Zbl 0175.14201; Zbl 0131.33201; Zbl 0060.26903
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