Noncommutative ball maps

TL;DR

This paper characterizes noncommutative analytic maps that preserve matrix balls and their boundaries, revealing that such maps are essentially direct sums of identity and ball-to-ball maps, with implications for noncommutative geometry and control theory.

Contribution

It provides a classification of NC analytic maps that send noncommutative matrix balls to themselves, showing they are structurally simple and extending understanding of noncommutative ball transformations.

Findings

NC ball maps are direct sums of identity and analytic maps.

Characterization of boundary-preserving NC maps is achieved.

Results have implications for noncommutative LMI and control theory.

Abstract

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call "NC ball maps". We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, "NC ball maps" are very simple, in contrast to the classical result of D'Angelo on such analytic maps over C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables.