COMMUTATOR ESTIMATES IN W*-FACTORS

Let M be a W*-factor and let S (M) be the space of all measurable operators affiliated with M. It is shown that for any self-acljoint element alpha is an element of S(M) there exists a scalar lambda(0) is an element of R, such that for all epsilon > 0, there exists a unitary element u(epsilon) from M, satisfying vertical bar[alpha, u(epsilon)]vertical bar >= (1 - epsilon)vertical bar alpha - lambda(0)1 vertical bar. A corollary of this result is that for any derivation delta on M with the range in an ideal I subset of M, the derivation delta is inner, that is, delta(center dot) = delta(alpha)(center dot) = [alpha, center dot], and a is an element of I. Similar results are also obtained for inner derivations on S(M).