Real Algebraic and Analytic Geometry |

Noncommutative ball maps.

e-mail: , , ,

Submission: 2008, October 10.

*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 in $\CC$.
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.*

Mathematics Subject Classification (2000): 47A56, 46L07, 32H99, 32A99, 46L89.

Keywords and Phrases: noncommutative analytic function, complete isometry, ball map, linear matrix inequality.

**Full text**, 41p.:
dvi 263k,
pdf 402k.

Server Home Page