On quasi-orthogonal systems of matrix algebras

TL;DR

This paper investigates the existence of certain quasi-orthogonal systems of matrix subalgebras, proving non-existence results for specific configurations and introducing new tools to analyze their structure.

Contribution

It establishes new non-existence results for quasi-orthogonal decompositions of matrix algebras and introduces a novel measure c(A,B) to analyze subalgebra relationships.

Findings

No quasi-orthogonal decomposition of M_{n^2}(C) into 1 or 3 maximal abelian subalgebras and factors isomorphic to M_n(C).

Introduces the measure c(A,B) to quantify quasi-orthogonality between subalgebras.

Provides formulas relating c(A',B') to c(A,B) and subalgebra dimensions, leading to new obstructions.

Abstract

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(C)\otimes M_n(C)\equiv M_{n^2}(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(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 \subset M_n(C) 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.