Intrinsic topologies on H-contraction groups with applications to semistability

Abstract

Semistable continuous convolution semigroups on Lie groups with non-trivial idempotent are characterized by semistable continuous convolution semigroups with trivial idempotent on a contractible, hence homogeneous Lie group. (Cf., e.g. [9], [10], III, theorem 3.5.4.) In fact, this homogeneous group is obtained by a retopologization of the contractible subgroup on which the original semistable laws are concentrated. In [26] E. Siebert investigated such intrinsic topologies for contractible subgroups of Polish groups, generalizing partially the before mentioned situation of Lie groups. Here we use these ideas to obtain intrinsic topologies for H-contractible subgroups of Polish groups, where H denotes a compact subgroup. This allows, under additional assumptions (which are satisfied in the Lie group case) to obtain similar characterization of semistable laws with non-trivial idempotents.