# 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

-
I think you are just asking for left E-submodules of V^n where V is the natural left module of E, V=C^n. If modules over finite dimensional algebras are not familiar, then I can recommend some textbooks. If modules are familiar, it might help to explain how this description is not helpful enough. – Jack Schmidt Jun 23 '10 at 1:15
The book Phil mentions does indeed seem very close to Jack's comment. But I'm not actually sure that this is what Phil wants. I suppose you could split up a matrix in F into it's columns, and then look at invariant subspaces for each column, but that's not really the same. My guess is that this is quite a specialised question (as opposed to looking at E acting on C^n). Phil: do you have a more specific question?? – Matthew Daws Jun 23 '10 at 6:52
I agree with Jack, Matthew, and Bruce: you should explain the context. For example, there is a sharp description if the subalgebra $E$ is semisimple. – Victor Protsak Jun 23 '10 at 7:45
Matthew: The subspaces F in his problem are exactly the E-submodules of Mn(C). However, the E-module Mn(C) is isomorphic to the E-module V^n. Submodules of V^n can be quite a big more complicated than just "coordinate submodules" that work entry-by-entry in V or column-by-column in Mn(C). Submodules of the natural module and its direct powers are often studied, so this might help. In particular, the collection of all such F is a modular lattice since it is just a submodule lattice. However, submodules of E^n are very complicated over some fields, so maybe he needs something else. – Jack Schmidt Jun 23 '10 at 13:10
Sorry about the vague question. Matt's correct that I was interested in the case that E was a sub $C^*$ algebra of $M_n$ and that the book I mentioned wasn't what I was after at all! – Phil Ellison Jun 23 '10 at 16:22

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...