On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of 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 orthogonal decomposition of Mn(C)⊗Mn(C)≡Mn2(C) into a number of maximal abelian subalgebras and factors isomorphic to Mn(C) 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⊂Mn(C) are to being 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 orthogonal systems.