
Free function theory through matrix invariants
This paper concerns free function theory. Free maps 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 \emph{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 finitedimensional slices is presented and then used to
establish power series expansions for analytic free maps about
scalar and nonscalar 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.
Keywords:free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomials Categories:16R30, 32A05, 46L52, 15A24, 47A56, 15A24, 46G20 