Completely reducible infinite-dimensional skew linear groups

Let V be a left vector space over a division ring D and GL(V) the group of all D-automorphisms of V. A subgroup G of GL(V) is completely reducible of V is completely reducible as D-G bimodule. Our aim in this brief note is to point out that in a sense the very useful notion of a local marker extends from V finite-dimensional to V infinite-dimensional. (A local marker of a subgroup G of GL(n, D) is any finitely generated subgroup X of G such that row n space D-(n) has least composition length as D-X bimodule. A local marker of G controls to a considerable extent the local behaviour of G.) Our main result is the following. Let G be a completely reducible subgroup of GL(V) and let W be any finite-dimensional D-subspace of V. Then G has a finitely generated subgroup X such that for every finitely generated subgroup Y of G containing X the D-Y submodule WY has a D-Y submodule M with W boolean AND M = {0} and WY/M completely reducible. We also give some examples and state without proof some stronger conclusions valid for various special subgroup G.