Connecting Commuting Normal Matrices

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 ε > 0 and any fixed but arbitrary m-tuple X ∈ Mn(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 Mn(C) m and the δ-ball B ð (X, δ) centered at X for some δ > 0, with respect to some suitable metric ð in Mn(C) m induced by the operator norm. Two of the key features of these matrix paths is that δ can be chosen independent of n, and that the paths stay in the intersection of B ð (X, ε), 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 * -representations of some universal commutative C * -algebras. Some connections with the local connectivity properties of completely positive linear maps on matrix algebras are studied as well.