The Riemann sphere of a commutative Banach algebra

Abstract

Introduction. The Riemann sphere, or extended complex plane C«,, has long played an important role in classical function theory. Therefore in abstract function theory, where the range or both the domain and range of an analytic function lie in a Banach space or algebra A, it is natural to pose the problem : how should A be extended to a new structure Am which plays the role of the Riemann sphere? Here we give the solution to this problem, valid when A is a complex commutative Banach algebra with identity. Av is provided with quasi-algebraic, topological, and analytic structure. In §1, the quasi-algebraic structure of Ax is studied, with the exposition being given for the surprisingly abstract context of a commutative ring with identity. We say quasi-algebraic rather than algebraic because elements of A w may only sometimes be added and sometimes multiplied. AK may be regarded as the set obtained by adjoining to A all formal quotients afb of elements of A, where b is singular and both a and b lie in no proper ideal of A. The 2x2 matrices with coefficients in A and invertible determinant induce the fractional linear group of bijections of Am. Ring homomorphisms induce Riemann sphere homomorphisms and fractional linear group homomorphisms which are related ; this lends a functorial tinge to the subject. In §2, we equip Am (where A is now required to be a commutative complex £-algebra with unit) with the unique topology so that A is an open subspace and each fractional linear transformation is a homeomorphism. Locally compact Riemann spheres occur iff A is finite dimensional, and compact Riemann spheres are even rarer. The lifting of algebra homomorphisms to Riemann sphere homomorphisms enables the extension of the Gelfand representation of A to Aœ, as well as the definition of the spectrum of an element of Am. The Riemann sphere construction enables us to replace one disconnectedness phenomenon by another. The mapping a -*■ a~1 is usually regarded as being defined on the (perhaps not connected) set / of invertible elements of A. However, here we regard fl^-a^'asa mapping of all of A x onto itself, and clearly / is contained in the component of Ax which contains A. Unfortunately Example 2.6.5 shows that Am itself need not be connected. A related phenomenon is that the spectrum of an element of A M may be the whole extended plane ; whenever A x is disconnected there are such elements, but not conversely. In §3, we give A „ an analytic structure, in which each fractional transformation