On quasi-orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of quasi-orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no quasi-orthogonal decomposition of M_n©\otimes M_n©\equiv M_{n^2}© into a number of maximal abelian subalgebras and factors isomorphic to M_n© in which the number of factors would be 1 or 3. In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B \subset M_n© are to being quasi-orthogonal. It is shown that in the main cases of interest, c(A’,B’) - where A’ and B’ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding quasi-orthogonal systems.