Connecting Commuting Normal Matrices

TL;DR

This paper investigates the local path connectivity of sets of commuting normal matrices with geometric spectral constraints, showing paths exist within certain bounds independent of matrix size, and explores implications for matrix representations of C*-algebras.

Contribution

It establishes the existence of matrix paths within spectral and geometric constraints, independent of matrix dimension, and applies these results to matrix C*-algebra representations.

Findings

Paths exist between commuting normal matrix contractions within spectral bounds

Path connectivity is independent of matrix size n

Applications to local connectivity of matrix C*-algebra representations

Abstract

In this document we study the local path connectivity of sets of $m$-tuples of commuting normal matrices with some additional geometric constraints in their joint spectra. In particular, given $\varepsilon>0$ and any fixed but arbitrary $m$-tuple $\mathbf{X}\in {M_n(\mathbb{C})}^m$ in the set of $m$-tuples of pairwise commuting normal matrix contractions, we prove the existence of paths between arbitrary $m$-tuples in the intersection of the previously mentioned sets of $m$-tuples in ${M_n(\mathbb{C})}^m$ and the $\delta$-ball $B_\eth(\mathbf{X},\delta)$ centered at $\mathbf{X}$ for some $\delta>0$, with respect to some suitable metric $\eth$ in ${M_n(\mathbb{C})}^m$ induced by the operator norm. Two of the key features of these matrix paths is that $\delta$ can be chosen independent of $n$, and that the paths stay in the intersection of $B_\eth(\mathbf{X},\varepsilon)$, and the set pairwise commuting normal matrix contractions with some special geometric structure on their joint spectra. We apply these results to study the local connectivity properties of matrix $\ast$-representations of some universal commutative $C^\ast$-algebras. Some connections with the local connectivity properties of completely positive linear maps on matrix algebras are studied as well.