Sheaves and functional calculus

G. Deferrari, Angel Rafael Larotonda and Ignacio Zalduendo

Vol. 141 (1990), No. 2, 279–286

DOI: 10.2140/pjm.1990.141.279

Abstract

Let A be a commutative Banach algebra with identity over the complex field, ℂ. Let a₁,…,a_n be elements of A, and sp(a) their joint spectrum. In this paper we seek to characterize the functional calculus

as part of a cohomology sequence of certain sheaves, and the algebra A as the algebra of sections

of a sheaf 𝒜, which is related to the Putinar structural sheaf. This is obtained under certain conditions on a₁,…,a_n. The problem is related also to the unique extension property and to the local analytic spectrum σ(a,x) of x with respect to a.

Section 2 is devoted to attacking this problem. In §1, some preliminary results are obtained. We also prove that if σ(a,x) is empty, then x is nilpotent.

Mathematical Subject Classification 2000

Primary: 46M20

Secondary: 46H30, 46J05, 47A60

Milestones

Received: 20 October 1987

Revised: 26 September 1988

Published: 1 February 1990