Measurable Choice and the Invariant Subspace Problem

In [1], 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 [1], 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 [1] : 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. All Hubert spaces discussed in this paper will be separable and all operators will be bounded. We use the term 'algebra' to refer to an identity—containing algebra of operators which is closed in the weak operator topology. A transitive algebra is an algebra having no nontrivial invariant subspaces; more generally, an algebra is called reductive if it is reduced by each of its invariant subspaces. The reader is referred to [2] or [3] for the details of direct integral theory; the primary purpose of the following summary is to fix notation. Let JU be the completion of a finite positive regular Borel measure supported on a or-compact subset of a separable metric space A and let {en}9 l^n^co, be a collection of disjoint Borel subsets of A with union A. Let hx ç h2 s • • • ç/joo be a sequence of Hubert spaces with hn having dimension n and h^ spanned by the remaining hn s. We write h = $A 0 h{X)fi (dX) for the Hubert space of (equivalence classes of) weakly measurable functions ƒ from A into / ^ such that for X e en9 f(X) e h(X)=hn9 and JA \\f(X)\\ fx (dX)<oo. The element in h represented by the function X-^f(X) is denoted by J A 0 / ( % ( ^ ) .