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



Invariant subspaces of subalgebras of $M_n(C)$
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...


