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

${}_{1}$.