2022/05/27 by Greg Friedman, Friedman, Greg, Efton Park +1
Mathematics · #15A18 #55S35 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Operator Algebras (math.OA) #Primary: 47B15 #Secondary: 55R40 #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2205.13729
openalex publication_date 2022/05/27 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28
This paper continues the authors' work on the question of unitary equivalence of matrices with entries in the complex-valued functions of a topological space (matrices over spaces). Specifically, we here consider the question of unitary equivalence for pairs of normal matrices over a space that share a common characteristic polynomial that can be globally factored into distinct linear factors. We show that such a matrix is diagonalizable if and only if the first Chern classes of its eigenbundles all vanish and derive as an application that all such matrices over ℂPm are diagonalizable for m > 1. Next, given a CW complex X and a polynomial μ in C(X)[λ] that globally splits into distinct linear factors, we prove that the number of unitary equivalence classes of matrices with μ as a characteristic polynomial depends only on the space X and the degree of μ, and we give some estimates on how many unitary equivalence classes there can be. In the case that X is a CW complex of dimension at most three, we demonstrate a bijection between the unitary equivalence classes of n × n normal matrices with characteristic polynomial μ and elements of the group (H2(X))n-1. Finally, when X is a smooth manifold and we restrict to matrices with smooth entries, we construct a de Rham cohomology class whose nonvanishing is an obstruction to unitary equivalence.