Proposition 7

a) For every element $c$ of the bicommutant C of M and every $\lambda \in \mathscr{S}_M$, there exists a unique element $c_{\lambda}$ of End$_{D_{\lambda}}(S_{\lambda})$ such that for every $\varphi \in$ Hom$_A(S_{\lambda},M)$ and every $s\in S_{\lambda}$, we have $c\varphi (s) =\varphi (c_{\lambda}s)$.

b) Endow the $S_{\lambda}$, for $\lambda$ running through $\mathscr{S}_M$, with the C-module structure defined by a). Then the canonical mapping $\alpha_M$ from $\bigoplus_{\lambda\in\mathscr{S}_M}S_{\lambda}\otimes_{D_{\lambda}}V_{\lambda}$ to M is an isomorphism of $(C,B^o)$-bimodules.

c) The mapping $c\mapsto (c_{\lambda})_{\lambda\in\mathscr{S}_M}$ is an isomorphism from C to $\prod_{\lambda\in\mathscr{S}_M}$ End$_{D_{\lambda}}(S_{\lambda})$.

Assertions a) and c) follow from VIII, p. 71, Proposition 9 because the canonical mapping $\alpha_M$ from $\bigoplus_{\lambda\in\mathscr{S}}(S_{\lambda}\otimes_{D_{\lambda}}W_{\lambda})$ to M provides a description of the B-module M (Remark 2). Moreover, $\alpha_M$ is $(C,B^o$)-linear, which proves b).