A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space (Q1071271)

The uniqueness of the hyperfinite \(II_{\infty}\)-factor and the classification of the hyperfinite \(III_{\lambda}\)-factors is deduced by \textit{A. Connes} [Ann. Math., II. Ser. 104, 73-115 (1976; Zbl 0343.46042)] from the equivalence of injectivity and hyperfiniteness. This equivalence was originally obtained by a detailed study of the automorphism groups of factors. In the present paper the proof is considerably simplified, avoiding the ''automorphism group machinery'' at all. As a by-product the uniqueness of the hyperfinite \(II_{\infty}\)-factor is proved without retreating on injectivity. The most difficult part is to prove that injectivity implies hyperfiniteness. Building on previous work of S. Wassermann the author reduces to prove that hyperfiniteness is implied by semi-discreteness as introduced by Effros and Lance. This takes the main part of the paper.