  We systematize and generalize recent results of Gerlach and Gl\"uck on the
strong convergence and spectral theory of bounded (positive) operator
semigroups $(T_s)_{s\in S}$ on Banach spaces (lattices). (Here, $S$ can be an
arbitrary commutative semigroup, and no topological assumptions neither on $S$
nor on its representation are required.) To this aim, we introduce the
"semigroup at infinity" and give useful criteria ensuring that the well-known
Jacobs--de Leeuw--Glicksberg splitting theory can be applied to it.
  Next, we confine these abstract results to positive semigroups on Banach
lattices with a quasi-interior point. In that situation, the said criteria are
intimately linked to so-called AM-compact operators (which entail kernel
operators and compact operators); and they imply that the original semigroup
asymptotically embeds into a compact group of positive invertible operators on
an atomic Banach lattice. By means of a structure theorem for such group
representations (reminiscent of the Peter--Weyl theorem and its consequences
for Banach space representations of compact groups) we are able to establish
quite general conditions implying the strong convergence of the original
semigroup.
  Finally, we show how some classical results of Greiner (1982), Davies (2005),
Keicher (2006) and Arendt (2008) and more recent ones by Gerlach and Gl\"uck
(2017) are covered and extended through our approach.
