Minimal upper bounds of commuting operators

Let $(x_{i})$ be a finite collection of commuting self-adjoint elements of a von Neumann algebra $\mathcal {M}$. Then within the (abelian) C*-algebra they generate, these elements have a least upper bound $x$. We show that within $\mathcal {M}$, $x$ is a minimal upper bound in the sense that if $y$ is any self-adjoint element such that $x_{i} \leq y \leq x$ 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 $\mathcal {N} \subset \mathcal {M}$ are von Neumann algebras, $\Phi : \mathcal {M} \to \mathcal {N}$ is a faithful conditional expectation, and $x \in \mathcal {M}$ is positive, then $\Phi (x^{n})^{1/n}$ converges in the strong operator topology to the ``spectral order majorant'' of $x$ in $\mathcal {N}$.