Free Function Theory Through Matrix Invariants

Abstract

Abstract This paper concerns free function theory. Freemaps are free analogs of analytic functions in several complex variables and are defined in terms of freely noncommuting variables. A function of g noncommuting variables is a function on g-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions, and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps with involution, free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invarianttheoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involutionfree counterparts.