Topological radicals, IV. Frattini theory for Banach Lie algebras

The paper develops the theory of topological radicals of Banach Lie algebras and studies the structure of Banach Lie algebras with sufficiently many Lie subalgebras of finite codimensions – the intersection of all these subalgebras is zero. It is shown that the intersections of certain families of Lie subalgebras (closed Lie subalgebras of finite codimension, closed Lie ideals of finite codimension, closed maximal Lie subalgebras of finite codimension, closed maximal Lie ideals of finite codimension) correspond to different preradicals, and that these preradicals generate the same radical, the Frattini radical. The main attention is given to structural properties of Frattini-semisimple Banach Lie algebras and, in particular, to a new infinite-dimensional phenomenon associated with the strong Frattini preradical introduced in this paper. A constructive description of Frattini-free Banach Lie algebras is obtained.