Remark 3

Endow $S_{\lambda}$, for $\lambda \in \mathscr{S}_M$, with the C-module structure given by Proposition 7, a). If we replace A with B and B with C in Proposition 5 (VIII, p. 85), then we see that for every $\lambda \in \mathscr{S}_M$, the left C-module $S_{\lambda}$ is simple, with commutant $D_{\lambda}$, that the isotypical component of type $S_{\lambda}$ of the C-module M is equal to $M_{\lambda}$, and that the mapping $\lambda \mapsto$ cl$_C(S_{\lambda})$ is a bijection from the support of the A-module M to the support of the C-module M. Finally, observe that the A-linear mappings and C-linear mappings from $S_{\lambda}$ to M are identical, as are the A-submodules and C-submodules of M, and that the rings End$_A(M)$ and End$_C(M)$ are equal.

Let M be a semisimple A-module. Denote by Z the center of the bicommutant C of the A-module M; it is also the center of the commutant B of M. Endow M and the $S_{\lambda}$, for $\lambda \in \mathscr{S}_M$, with Z-module structures deduced by restriction of scalars from the C-module structures. For every $\lambda \in \mathscr{S}_M$, denote the center of the field $D_{\lambda}$ by $Z_{\lambda}$.