# On quasi-orthogonal systems of matrix algebras - Mathematical Physics

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Abstract: In this work it is shown that certain interesting types of quasi-orthogonalsystem of subalgebras whose existence cannot be ruled out by the trivialnecessary conditions cannot exist. In particular, it is proved that there isno quasi-orthogonal decomposition of M nC\otimes M nC\equiv M {n^2}C intoa number of maximal abelian subalgebras and factors isomorphic to M nC inwhich the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantitycA,B, which measures -how close- the subalgebras A,B \subset M nC are tobeing quasi-orthogonal. It is shown that in the main cases of interest,cA-,B- where A- and B- are the commutants of A and B, respectively - can bedetermined by cA,B and the dimensions of A and B. The corresponding formulais used to find some further obstructions regarding quasi-orthogonal systems.

Author: ** Mihály Weiner**

Source: https://arxiv.org/