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Cluster values for algebras of analytic functions. (arXiv:1705.05697v1 [math.FA])
来源于:arXiv
The Cluster Value Theorem is known for being a weak version of the classical
Corona Theorem. Given a Banach space $X$, we study the Cluster Value Problem
for the ball algebra $A_u(B_X)$, the Banach algebra of all uniformly continuous
holomorphic functions on the unit ball $B_X$; and also for the Fr\'echet
algebra $H_b(X)$ of holomorphic functions of bounded type on $X$ (more
generally, for $H_b(U)$, the algebra of holomorphic functions of bounded type
on a given balanced open subset $U \subset X$). We show that Cluster Value
Theorems hold for all of these algebras whenever the dual of $X$ has the
bounded approximation property. These results are an important advance in this
problem, since the validity of these theorems was known only for trivial cases
(where the spectrum is formed only by evaluation functionals) and for the
infinite dimensional Hilbert space.
As a consequence , we obtain weak analytic Nullstellensatz theorems and
several structural results for the spectrum of these alg 查看全文>>