# A generalization of Schur’s lemma

(Dixmier) Let $A$ be a $\Bbb{C}$-algebra (in fact, any uncountable algebraically closed field will do) and $V$ be a simple $A$-module having a countable basis as a $\Bbb{C}$-vector space. Then $D=\mathrm{End}_A(V)\simeq \Bbb{C}$.

Brief outline:

1. By Schur’s lemma, $D$ is a division algebra. Moreover, it has a countable basis.
2. Suppose $\varphi\in D$ does not act as a scalar. Then $\varphi$ is trascendental over $\Bbb{C}$.
3. This is absurd since it would imply $\Bbb{C}(\varphi)\subseteq D$ has uncountable dimension.