MINIMAL UPPER-BOUNDS OF COMMUTING OPERATORS

Let (x(i)) be a finite collection of commuting self-adjoint elements o f a von Neumann algebra M. Then within the (abelian) C-algebra they g enerate, these elements have a least upper bound Ic: We show that with in M, x is a minimal upper bound in the sense that if y is any self-ad joint element such that x(i) less than or equal to y less than or equa l to x for all i, then y = x. The corresponding assertion for infinite collections (x(i)) is shown to be false in general, although it does hold in any finite von Neumann algebra. We use this sort of result to show that if N subset of M are von Neumann algebras, Phi : M --> N is a faithful conditional expectation, and x epsilon M is positive, then Phi(x(n))(1/n) converges in the strong operator topology to the ''spec tral order majorant'' of x in N.