Finitely generated algebras and algebras of solutions to partial differential equations

John Anderson

Vol. 133 (1988), No. 1, 1–12

DOI: 10.2140/pjm.1988.133.1

Abstract

We consider two types of uniform algebras A on the closure Ω of a domain Ω ⊂ Rⁿ: those generated by finitely many smooth functions and those consisting of solutions to Lu = 0 where L is a smooth complex vector field on Ω. Under certain conditions we prove the existence of one of two types of analytic structure in the maximal ideal space M_A of such an algebra: local foliations of Ω by complex manifolds on which the functions in the algebra are holomorphic, or foliations of a subset of M_A∖Ω by analytic disks. Some open questions suggested by this line of inquiry are discussed.

Mathematical Subject Classification 2000

Primary: 46J15

Secondary: 35A99

Milestones

Received: 1 June 1987

Published: 1 May 1988