2017/08/18 by Fredy Vides, Vides, Fredy
Computer Science · Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Matrix Theory and Algorithms #math.FA #math.OA #msc:15A27 #msc:15A60 #msc:47A58
paper · pdf · doi:10.48550/arxiv.1708.05777
arxiv created 2017/08/18 · arxiv updated 2017/08/22
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.