The algebra of decomposable operators in direct integrals of not necessarily separable Hilbert spaces

Considering direct integrals of not necessarily separable Hilbert spaces we examine the question whether the algebra of decomposable operators is the commutant of the algebra of diagonalizable operators. Using the continuum-hypothesis we prove this relation, if the set of square integrable vector fields is generated by a subset ${\Gamma _0}$ such that $\vert{\Gamma _0}\vert \leq \vert{\mathbf{R}}\vert$. For the general case, a counterexample is given.