Topological Radicals and Frattini Theory of Banach Lie Algebras

In this paper we develop the theory of topological radicals of Banach Lie algebras and apply it to the study of the structure of Banach Lie algebras with sufficiently many closed Lie subalgebras of finite codimensions, that is, the intersection of all these subalgebras is zero. The first part is devoted to the radical theory of Banach Lie algebras; the second develops some technique of construction of preradicals via subspace-multifunctions and analyses the corresponding radicals, and the third part contains the Frattini theory of infinite-dimensional Banach Lie algebras. It is shown that the multifunctions of 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) produce 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 novel infinite-dimensional phenomenon associated with the strong Frattini preradical introduced in this paper. A new constructive description of Frattini-free Banach Lie algebras is obtained.