Geometry and Analytic Boundaries of Marcinkiewicz Sequence Spaces

Abstract

We investigate the geometric structure of the unit ball of the Marcinkiewicz sequence space m 0 Ψ , giving characterisations of its real and complex extreme points and of the exposed points in terms of the symbol Ψ. Using our knowledge of the geometry of B m 0 Ψ we then give necessary and sufficient conditions for a subset of B m 0 Ψ to be a boundary for Au(B m 0 Ψ), the algebra of functions which are uniformly continuous on B m 0 Ψ and holomorphic on the interior of B m 0 Ψ. We show that it is possible for the set of peak points of Au(B m 0 Ψ) to be a boundary for Au(B m 0 Ψ) yet for Au(B m 0 Ψ) not to have aŠilov boundary in the sense of Globevnik.

Weighted Vector-Valued Holomorphic Functions on Banach Spaces

“…In case V ( ) is an algebra the constant, can be always taken 1 and, according to Globevnik, the sampling sets are called boundaries [20][21][22]. If ⊂ and = {V( ) : ∈ } ⊆ V ( ) * , it follows from the definitions that ⊆ * V , is sampling if and only if is norming, and is a set of uniqueness if and only if is total.…”

Boundaries for spaces of holomorphic functions on M-ideals in their biduals

“…Various types of boundaries have already been considered in [6] and [12]. For instance, the set T of complex extreme points of B ℓ∞ is a closed boundary in the sense of Definition 1.1 for the larger algebra A ∞ (B ℓ∞ ), but there are also closed boundaries Γ ⊂ B ℓ∞ such that Γ ∩ T = ∅.…”

“…On the other hand, S ℓp is the intersection of all closed boundaries for A ∞ (B ℓp ), 1 ≤ p < ∞. In [12], the authors describe boundaries for A u (B X ) for certain spaces X whose bidual is a Marcinkiewicz space. They show that the set of complex extreme points of B X is a non-empty closed set but that no minimal boundary for A u (B X ) exists.…”

“…Globevnik studied the boundaries for two important algebras of holomorphic functions that are closed subalgebras of C b (Ω) in case Ω is the closed unit ball of the complex Banach space c 0 (see [20,21]). The pioneering work of Globevnik was followed by many authors (see, e.g., [6], [27], [28], [13], [1], [2], [14], [4], [15] and [12]), who studied the existence and characterization of such generalized boundaries. These authors considered algebras of holomorphic mappings defined on the closed unit ball of concrete complex Banach spaces which were, by and large, sequence spaces.…”

On boundaries for spaces of holomorphic functions on the unit ball of a Banach space

“…The pioneering work of Globevnik was followed by many authors (see, e.g., [6], [27], [28], [13], [1], [2], [14], [4], [15] and [12]), who studied the existence and characterization of such generalized boundaries. These authors considered algebras of holomorphic mappings defined on the closed unit ball of concrete complex Banach spaces which were, by and large, sequence spaces.…”