Derivations of von Neumann algebras into the compact ideal space of a semifinite algebra

Introduction and statement of results.Let M be a semifinite von Neumann algebra and let ate(M) be the norm closed two-sided ideal generated by the finite projections of M. Let N _ M be a subalgebra of M. A derivation of N into aC'(M) is a linear application : N aC(M) satisfying 6(xy) 6(x)y + x(y) for x, y N.For instance, if K aC(M), then the derivation (x) (ad K)(x) Kx xK is of this type.Such derivations implemented by elements in o9"(M) are called inner.There are many examples of derivations of *-subalgebras N _M into the ideal at(M) which are not inner.A typical such example is as follows: Take M ,,(L2(]l", p,)), where/ is the Lebesgue measure on the torus qi, let N C01") act on L2(]l",/.t)by left multiplication, and define 8(x)= (ad PH 2)(x), where PH is the projection onto the Hardy subspace H2(]1,/)).Then it is easy to see that 8(x) )U(o') =aC('(gd)) for x C(ql) and that 8 is not implemented by a compact operator.We will, however, show in this paper that if N is self-adjoint and w-closed in M, then, except for certain situations, all derivations of N into o(M) are inner.Moreover, for the most typical excepted case we'll construct a counterexample.This derivation problem was initiated in the case M--'() and at(M) OU() by Johnson and Parrott in a paper of the early '70s ([3]).In that paper Johnson and Parrott wanted to characterize the commutant modulo the ideal of compact operators g(of) _ '(9') for a yon Neumann algebra N _ ().They noted that in order to identify it with the compact perturbations of the commutant of N in '(9'), it suffices to show that any derivation is inner.They proved that this is indeed the case if N has no certain type IIx factors as direct summands.To do this they first solved the case when N is abelian, the other cases being rather easy consequences of it.The general type II case was proved recently in [7] by different techniques and using more of the ergodic theory of the type II factors.In [4] this derivation problem is studied in the more general setting when ,,() is replaced by a semifinite yon Neumann algebra, (/') by the ideal ,,'(M), and the center of N is assumed to contain the center of M.Under this hypothesis it is proved that if N is either an abelian or a properly infinite yon Neumann algebra, then any derivation of N into at(M) is inner.

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