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.
A generalization of Schur’s lemma