THE COMPLETE REDUCIBILITY OF LOCALLY COMPLETELY REDUCIBLE FINITARY LINEAR-GROUPS

Let V be a vector space over some division ring D, and G a finitary su bgroup of GL(V). If G is locally completely reducible, then the D-G mo dules V,[V, G] and V/C-V(G) need not be completely reducible, even if dim, V is finite. Moreover, if F is a field, then V and V/C-V(G) need not be completely reducible. We prove here that if D is a finite-dimen sional division algebra and G is locally completely reducible, then [V , G] is always a completely reducible D-G module.