Date of this Version
In , J. Dyer, A. Pedersen and P. Porcelli announced that an affirmative answer to the invariant subspace problem would imply that every reductive operator is normal. Their argument, outlined in , provides a striking application of direct integral theory. Moreover, this method leads to a general decomposition theory for reductive algebras which in turn illuminates the close relationship between the transitive and reductive algebra problems.
The main purpose of the present note is to provide a short proof of the technical portion of  : that invariant subspaces for the direct integrands of a decomposable operator can be assembled "in a measurable fashion". The general decomposition theory alluded to above will be developed elsewhere in a joint work with C. K. Fong, though we do present a summary of some of its consequences below.