  Let $\mathbf{F}$ be a Banach space of continuous functions over a connected
locally compact space $X$. We present several sufficient conditions on
$\mathbf{F}$ guaranteeing that the only multiplication operators on
$\mathbf{F}$ that are surjective isometries are scalar multiples of the
identity. The conditions are given via the properties of the inclusion operator
from $\mathbf{F}$ into $\mathcal{C}\left(X\right)$, as well as in terms of
geometry of $\mathbf{F}$. An important tool in our investigation is the notion
of Birkhoff Orthogonality.
