Local Deformation of Matrix Words

TL;DR

This paper investigates local deformation properties of matrix representations of certain universal C*-algebras called Semi-Soft Cubes and Spheres, with implications for matrix equations and connections to physics and numerical analysis.

Contribution

It introduces new local deformation results for matrix words in Semi-Soft Cubes and Spheres using C*-algebraic methods, linking algebraic and geometric perspectives.

Findings

Characterization of local deformation properties of matrix words

Analysis of geometric aspects of matrix deformations

Connections to matrix numerical analysis and physics

Abstract

In this document we study some local deformation properties of matrix representations of the universal C$^*$-algebras denoted by $\mathbb{I}^{m}_\varepsilon[p_1,\ldots,p_m]$ and $\mathbb{S}^{m-1}_\varepsilon[p_1,\ldots,p_m]$, and that we call {\bf Semi-Soft Cubes} and {\bf Semi-Soft Spheres} respectively. We will use some C$^*$-algebraic technology to study the local deformation properties of matrix words in particular representations of Semi-Soft Cubes and Spheres, we will then use these results to study the local deformation properties of generic matrix equations on words. Some geometrical aspects of the local deformation of matrix words will be addressed, and some connections with matrix numerical analysis and computational physics will be outlined as well.