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