Orthogonal Maximal Abelian *-Subalgebras of the N×n Matrices and Cyclic N-Roots

It is proved that for n = 5, there is up to isomorphism only one pair of orthogonal maximal abelian-subalgebras (MASA's) in the n n-matrices. The same result holds trivially for n = 2 and n = 3, but de la Harpe, Jones, Munemasa and Watatani have shown that, for every prime number n 7, there are at least two non-isomorphic pairs of MASA's in the n n matrices. We draw connections to the research of Backelin, Bjj orck and Frr oberg on cyclic n-roots, and use their classiication of cyclic 7-roots to construct ve non-isomorphic pairs of MASA's in the 7 7 matrices.