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Geometry and analytic boundaries of Marcinkiewicz sequence spaces. (English) Zbl 1211.46009

Let Ψ=(Ψ(k)) k=0 be an increasing sequence of non-negative real numbers with Ψ(0)=0 and Ψ(k)>0 if k1. The Marcinkiewicz sequence space m Ψ is the space of all bounded sequences z=(z k ) k such that z=sup k j=1 k z j * /Ψ(k)<, and its subspace m Ψ 0 contains all elements z such that lim k j=1 k z j * /Ψ(k)=0. The space m Ψ equipped with the norm · is rearrangement invariant and m Ψ 0 is a predual of a Lorentz space. Given a complex Banach space E, denote by 𝒜 b (B E ) the Banach algebra of all functions which are continuous and bounded on B E , the closed unit ball of E, and holomorphic on the interior of B E . By 𝒜 u (B E ) denote the Banach algebra of functions in 𝒜 b (B E ) which are uniformly continuous on B E . A subset B of B E is said to be a boundary for 𝒜 u (B E ) if f=sup zB |f(z)| for all f𝒜 u (B E ). The Šilov boundary of 𝒜 u (B E ) is the minimal closed boundary. A point xB E is a peak point of 𝒜 u (B E ) if there is f𝒜 u (B E ) such that |f(y)|<f(x) for all yB E {x}.

Characterizations of extreme, complex extreme and exposed points of m Ψ 0 are given. For instance, it is proved that zB m Ψ 0 is complex extreme if and only if z is a peak point of 𝒜 u (B m Ψ 0 ). Applying those characterizations, a condition is found which is necessary and sufficient for a subset of B m Ψ 0 to be a boundary for 𝒜 u (B m Ψ 0 ). It is also shown that it is possible that a set of peak points of 𝒜 u (B m Ψ 0 ) is a boundary for 𝒜 u (B m Ψ 0 ), yet it is not the Šilov boundary for 𝒜 u (B m Ψ 0 ).

MSC:
46B20Geometry and structure of normed linear spaces
46J15Banach algebras of differentiable or analytic functions, H p -spaces
46E50Spaces of differentiable or holomorphic functions on infinite-dimensional spaces