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The first part of this article is a brief survey of the properties ofso-called almost interior points in ordered Banach spaces. Those vectors can beseen as a generalization of ``functions which are strictly positive almosteverywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banachlattices.  In the second part we study the long--term behaviour of strongly positiveoperator semigroups on ordered Banach spaces; these are semigroups which, in asense, map every non-zero positive vector to an almost interior point. Usingthe Jacobs--de Leeuw--Glicksberg decomposition together with the theorypresented in the first part of the paper we deduce sufficiency criteria forsuch semigroups to converge (strongly or in operator norm) as time tends toinfinity. This generalises known results for semigroups on Banach lattices aswell as on normally ordered Banach spaces with unit.

Almost Interior Points in Ordered Banach Spaces and the Long--Term Behaviour of Strongly Positive Operator Semigroups