Abstract view
Higher Dimensional Spaces of Functions on the Spectrum of a Uniform Algebra


Published:20070301
Printed: Mar 2007
Abstract
In this paper we introduce a nested family of spaces of continuous functions defined
on the spectrum of a uniform algebra. The smallest space in the family is the
uniform algebra itself. In the ``finite dimensional'' case, from some point on the
spaces will be the space of all continuous complexvalued functions on the
spectrum. These spaces are defined in terms of solutions to the nonlinear
CauchyRiemann equations as introduced by the author in 1976, so they are not
generally linear spaces of functions. However, these spaces do shed light on the
higher dimensional properties of a uniform algebra. In particular, these spaces are
directly related to the generalized Shilov boundary of the uniform algebra (as
defined by the author and, independently, by Sibony in the early 1970s).