Introduction. In [8] we gave a proof of Connes' fundamental theorem that injectivity implies hyperfiniteness for finite von Neumann algebras. It consisted in first proving a certain FØlner type condition for injective algebras, and then in combining this FØlner condition with a local Rohlin lemma to get a local approximation property. Using a maximality argument, the latter gives the proof. We show here that by using a slightly stronger local Rohlin lemma, one can derive a similar proof of injectivity implies hyperfiniteness from Connes' FØlner type condition ([l],[2]). Although we will use the same ideas as in [8], by reordering some of the steps in [8] the proof will become here more conceptual and shorter. In fact the proof in [8] can also be shortened using- this reordering of the arguments.

Notation. Throughout this paper, M will denote a finite von Neumann algebra with a fixed normal finite faithful trace τ (τ(1) = 1), ||x||2 = τ(x*x)½, x∈M will be the Hilbert norm given by τ, and L2(M,τ), the completion of M in this norm. We let u(M) denote the unitaries in M.

Injectivity. The algebra M is called injective if there exists an M-invariant state Ψo on B(L2(M,τ)), i.e., Ψ0 satisfies Ψo(χT)=Ψo(Tχ) for all χ∈ M, T∈B(L2(M,τ)). Such a state is called a hypertrace on M and may be regarded as the operator algebra analogue of an invariant mean on an amenable group (see [2]).