Caracterisation et proprietes des ensembles localement pics de A∞(D).

On dense ideals in spaces of analytic functions

“…The following remarkable characterization of locally peak sets was obtained by Chaumat and Chollet in [4], as a completion of some prior results due to Hakim and Sibony. (See also [3] A second part of the main result in [4] asserts that one can impose the condition T^(N) C T°z(90) at any point of TV, and then the condition on N to be totally real is automatically satisfied.…”

“…In both definitions we can choose equivalently a function g which is identically zero on E and with Re(^) > 0 elsewhere. (See [4] for details. )…”

“…[3], [4]). For that reason the proof proposed below cannot apparently be adapted to more general situations, as for instance a polydomain and an ideal with a singular zero locus.…”

Manifolds of holomorphic mappings from strongly pseudoconvex domains

“…, z 5 ) ∈ B : z 1 z 2 · · · z 5 = √ 5 −5 } is a real four dimensional submanifold of the boundary of B which is complex tangential to the sphere bB at each point. By Chaumat and Chollet ( [5], [6]) every compact subset of M is a peak interpolation set for A ∞ (B), the Fréchet algebra of functions holomorphic on the ball and smooth up to the boundary. Let H be a closed Hartogs figure in C 2 :…”

“…Since Φ( B) contains the Hartogs figure H × {0} 6 , any open Stein neighborhood of it will also contain the unit bidisc in C 2 × {0} 6 ; as this bidisc is not included in Φ( B), the latter set has no basis of Stein neighborhoods. The answer is easily seen to be affirmative if Y = C N and, by the embedding theorem, also if Y is a Stein manifold.…”