Theorem 1 (Jacobson’s density theorem)

Let M be a semisimple left A-pseudomodule. Let $b$ be an element of the bicommutant of the $\widetilde{A}$-module M. Let $F =\{x_1, . . . , x_n\}$ be a finite subset of M. Then there exists an element $a\in A$ such that $bx_i=ax_i$ for every $i\in [1, n]$.

Let B be the bicommutant of the $\widetilde{A}$-module M. The A-pseudomodule $M^n$ is semisimple. Let $\boldsymbol{x}= (x_1, . . . , x_n)\in M^n$. It follows from Lemma 2 that $\boldsymbol{x}\in$ $A\boldsymbol{x}$. The bicommutant of the $\widetilde{A}$-module $M^n$ coincides with the homotheties of the B-module $M^n$ (VIII, p. 79, Proposition 2). By Lemma 1, the A-sub-pseudomodule $A\boldsymbol{x}$ of $M^n$ is therefore a B-submodule of $M^n$. We therefore have the inclusion $B\boldsymbol{x}\subset A\boldsymbol{x}$. So there exists an $a\in A$ such that $b\boldsymbol{x}=a\boldsymbol{x}$. The result follows.