  We investigate uniform, strong, weak and almost weak stability of
multiplication semigroups on Banach space valued $L^p$-spaces. We show that,
under certain conditions, these properties can be characterized by analogous
ones of the pointwise semigroups. Using techniques from selector theory, we
prove a spectral mapping theorem for the point spectra of the pointwise and
global semigroups and apply this as a major tool for determining almost weak
stability.
