A geometric approach to the Bohr compactification of cones (Q1093747)

Mislove and Hazewinkel have raised the following problem: Let C be a proper cone of finite dimension, \(C^ b\) its Bohr compactification. By Friedberg's results we know that the semilattice of idempotents in \(C^ b\) is isomorphic with the semilattice of faces of the dual cone \(C^ *\) of C. (a) Which idempotents correspond to exposed faces of \(C^ *\) under this isomorphism ? (b) Is it true that these idempotents are exactly those which are contained in the \(C^ b\)-closure of a ray subsemigroup of C ? In the present paper we show that the answer to (b) is ``yes'', and we offer a characterization of the idempotents in (a) in terms of the geometry of the cone C. More generally, we present a ``geometrical'' approach to the theory of Bohr compactifications of cones and show how this approach yields a proof and a ``geometric'' understanding of Friedberg's results. For technical reasons we consider the slightly more general class of ``conoid semigroups'' (direct products of cones with topological groups) instead of cones.