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 $A$ [{\it J. Hakeda} and the reviewer, Tĥoku math. J., II. Ser. 19, 315--323 (1967;

Zbl 0175.14201); {\it E. Effros} and {\it 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 {\it J. T. Schwartz} [Commun. Pure Appl. Math. 16, 19--26 (1963;

Zbl 0131.33201)] and the property $\Gamma$ [{\it F. J. Murray} and {\it 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$_1$.