Spectral sets as Banach manifolds

Angel Rafael Larotonda and Ignacio Zalduendo

Vol. 120 (1985), No. 2, 401–416

DOI: 10.2140/pjm.1985.120.401

Abstract

Let A be a commutative Banach algebra, X its spectrum, and M a closed analytic submanifold of an open set in Cⁿ. We may consider the set of germs of holomorphic functions from X to M, 𝒪(X,M). Now let ν be the functional calculus homomorphism from 𝒪(X,Cⁿ) to Aⁿ, and A_M = ν(𝒪(X,M)).

It is proven that A_M is an analytic submanifold of Aⁿ, modeled on projective A-modules of rank = dimM.

Mathematical Subject Classification 2000

Primary: 58B12

Secondary: 32C25, 46H30, 46J05, 46M20

Milestones

Received: 8 March 1984

Published: 1 December 1985