  We begin this note with a von Neumann algebraic version of the elementary but
extremely useful fact about being able to extend inner-product preserving maps
from a total set of the domain Hilbert space to an isometry defined on the
entire domain. This leads us to the notion of when `good' endomorphisms of a
factorial probability space $(M,\phi)$ (which we call equi-modular) admit a
natural extension to endomorphisms of $L^2(M,\phi)$. We exhibit examples of
such extendable endomorphisms.
  We then pass to $E_0$-semigroups $\alpha = {\alpha_t: t \geq 0}$ of factors,
and observe that extendability of this semigroup (i.e., extendability of each
$\alpha_t$) is a cocycle-conjugacy invariant of the semigroup. We identify a
necessary condition for extendability of such an $E_0$-semigroup, which we then
use to show that the Clifford flow on the hyperfinite $II_1$ factor is not
extendable.
