I just posted this on overflow.... just can't figure it out.
Is there a direct proof of the following without going through composition series or Artin-Wedderburn theorem?
Let V be a finite-dimensional complex Hilbert space. Let A⊂End(V) be a self-adjoint subalgebra. Then A is semisimple.
I am using the following definition of semisimple algebra: semisimple algebra is a direct sum of simple algebras, and a simple algebra is one with no two-sided ideals other than 0 and itself. thanks!
On overflow it was stated that the hypotheses imply that V breaks up as a direct sum of simple modules... don't see it.