Separating Projection Of Subalgebras In Matrix Algebras

In this paper, the author discusses the existence of the separating projection of subalgebras in matrix algebras.The existence of the separating projections means that assuming that M=Mn(C)(?)Mk(C), N=Mn(C)(?)Ik(N(?)M), is there a projection P∈M with rank(P)=r(1≤r≤nk-1), such that if T∈N satisfies (I-P)TP=0, then T∈CI?In this paper, we researched the case n≤k and completely characterized the relationship among r,n and k. As for the case n> k, we obtained some partial results. Furthermore, we discussed the existence of the separating projection for the subalgebras of a finite-dimensional C*-algebra and obtained some results relative to an infinite-dimensional separable Hilbert space.