Lemma 1

Let M be a semisimple left A-pseudomodule. Let B be the bicommutant of the $\widetilde{A}$-module M. Then every A-sub-pseudomodule of M is a B-submodule of M.

The $\widetilde{A}$-module M is semisimple. Let N be an A-sub-pseudomodule of M. Then N is an $\widetilde{A}$-submodule of M. By Corollary 2 of VIII, p. 56, there exists a projector $p$ of the A-module M with image N. Since we have the relation $pb=bp$ for every $b\in B$, we obtain that N is a B-submodule of M.