Invariant subspaces of subalgebras of $M_n(C)$ Given a subalgebra E of $M_n$ (nxn complex valued matrices), what can we say about the subspaces F of $M_n$ such that $EF \subset F$? Googling for an answer gives me the reference:
Israel Gohberg, Peter Lancaster, and Leiba Rodman (2006). Invariant Subspaces of Matrices with Applications. 
However, my library doesn't have this book. Is there a nice survey article available anywhere on this?
Thanks
 A: You can say (without fear of contradiction) that they form a modular lattice. I learnt this from "Algebra" (Second Edition) by MacLane and Birkoff (which is showing my age) in XIV 5.
There may be other things you can say. You would get a better response if you provided some context to your question.
A: This was originally tagged fa.functional analysis, I think.  So he's an Operator Algebraic answer.  I'm going to make the strong assumption that E is self-adjoint (i.e. closed under taking the hermitian transpose).  If not, then really this is an algebraic question, and it's probably irrelevant that you are working with the complex numbers...
Anyway, then E is a finite-dimensional von Neumann algebra.  The action of E on M_n is the same as identifying M_n with $\mathbb C^n \otimes \mathbb C^n = \ell^2_n \otimes \ell^2_n$ and letting E act as $E \otimes 1$.  Then invariant subspaces for $E$ correspond to orthongonal projections in the commutant of E, which by Tomita is $E' \otimes M_n$ where $E' = \{ A\in M_n : AB=BA (B\in E)\}$ the commutant of $E$ in $M_n$.  We identify $E'\otimes M_n$ with $M_n(E')$, and then it's just (ahem!) a case of working out the projections (self-adjoint idempotents) here.  In concrete cases, this is probably not too hard...
