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Zbl 0343.46042
Connes, Alain
Classification of injective factors. Cases $\mathrm{II}_1$, $\mathrm{II}_\infty$, $\mathrm{III}_\lambda$, $\lambda\neq 1$. (English)
[J] Ann. Math. (2) 104, 73-115 (1976). ISSN 0003-486X; ISSN 1939-0980/e

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$_1$.

Display scanned Zentralblatt-MATH page with this review.
[J. Tomiyama]

MSC 2000:: *46L10 General theory of von Neumann algebras
46M10 Projective / injective objects in categories of topol. linear spaces

Citations: Zbl 0175.14201; Zbl 0131.33201; Zbl 0060.26903

Cited in: Zbl 1237.46043 Zbl 1154.46035 Zbl 1040.43001 Zbl 0982.44005 Zbl 0954.46037 Zbl 1036.46048 Zbl 0721.46041 Zbl 0733.46033 Zbl 0638.28014 Zbl 0628.46061 Zbl 0624.46040 Zbl 0586.46047 Zbl 0562.46036 Zbl 0526.46061 Zbl 0534.46043 Zbl 0492.46049 Zbl 0408.46042

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