Minimal and Maximal Semigroups Related to Causal Symmetric Spaces

G/H

be an irreducible globally hyperbolic semisimple symmetric space, and let S⊆G be a subsemigroup containing H not isolated in S. We show that if S ^o≠ 0 then there are H-invariant minimal and maximal cones C _min⊆C _max in the tangent space at the origin such that H exp C _min⊆S⊆HZ _K (a)expC _max. A double coset decomposition of the group G in terms of Cartan subspaces and the group H is proved. We also discuss the case where G/H is of Cayley type.