Holomorphic Mappings of Domains in Operator Spaces

Our object is to give an overview of some basic results about holomorphic mappings of circular domains in various spaces of operators. We begin by considering C*-algebras and pass to J*-algebras and other spaces when this seems natural. Our first result is a simple extension of the maximum principle where the unitary operators play the role of the unit circle. We illustrate the power of this result by deducing some classical theorems of functional analysis in a straightforward way. Next we apply Cartan’s uniqueness theorem to determine biholomorphic mappings and to show that linear mappings between certain operator domains are Jordan isomorphisms. This motivates the discussion of homogeneous domains. Our first examples of homogeneous domains are the open unit balls of J*-algebras (which include all the classical domains.) Next we discuss some affinely homogeneous upper half-planes in spaces of operators called operator Siegel domains of genus 2. Although these are always holomorphically equivalent to a bounded domain, they are (as far as I know) not necessarily holomorphically equivalent to a ball. Our last examples are the domains of linear fractional transformations. These are symmetric affinely homogeneous domains which are never holomorphically equivalent to a bounded domain. Finally we show how certain extensions of the Riemann removable singularity theorem allow us to determine the automorphisms of domains where an indefinite operator-valued form is positive. This includes the operator analogue of the exterior of the open unit disc.