Corollary 1

Suppose that A = K or that A = K(i). Let B be a subalgebra of $\mathscr{L}_A(E)$, stable under the mapping u → u*. Then E is a semisimple B-module, and is the direct sum of simple submodules pairwise orthogonal. The algebra B is semisimple.

Indeed, since every B-submodule V of E has a supplement, for example V⁰, the B-module V is semisimple (chap. VIII, § 3, No. 3, prop. 7). Since every B-submodule ≠ {0} and minimal in E is simple, E is the direct sum of simple submodules pairwise orthogonal. Finally B is a semisimple algebra, since it admits a semisimple and faithful module E whose contramodule is of finite type (chap. VIII, § 5, No. 1, prop. 3).