Connecting Commuting Normal Matrices

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.