CLUSTER VALUES FOR ALGEBRAS OF ANALYTIC FUNCTIONS

DANIEL GALICER; DANIEL CARANDO; PABLO SEVILLA PERIS; MURO SANTIAGO

Londres

Congreso; Analysis / Stochastic Analysis; 2017

Imperial College

The {\it Cluster Value Theorem} is known for being a weak version of the classical \emph{Corona Theorem}.Given a Banach space $X$, we study the \emph{Cluster Value Problem} for the ball algebra $A_u(B_X)$, theBanach 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 $X$. 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 spectrumis formed only by evaluation functionals) and for the infinite dimensional Hilbert space.If time permits, we will present some structural results for the spectrum of these algebras. \bigskipJoint work with Daniel Carando, Santiago Muro and Pablo Sevilla-Peris.