On the spectra of algebras of analytic functions Daniel Carando, Domingo Garc´ ıa, Manuel Maestre, and Pablo Sevilla-Peris Abstract. In this paper we survey the most relevant recent developments on the research of the spectra of algebras of analytic functions. We concen- trate mainly in three algebras, the Banach algebra H ∞ (B) of all bounded holomorphic functions on the unit ball B of a complex Banach space X, the Banach algebra of the ball Au(B), and the Fr´ echet algebra H b (X) of all entire functions bounded on bounded sets. 1. Introduction Complex analysis permeates almost any aspect of mathematics since the early nineteenth century when it was first developed by Cauchy. But it appears that, around 1955, S. Kakutani was the first one to study the space H ∞ (D) of all bounded holomorphic functions on D, the open unit disk of the complex plane, as a Banach algebra. In the immediately following years a group of mathematicians (Singer, Wermer, Kakutani, Buck, Royden, Gleason, Arens and Hoffman) under a joint pseudonym (I. J. Schark [62]) and in individual papers used this functional analytic point of view to develop the study of many aspects of the theory. If we consider the space H ∞ (D) as a Banach algebra, a key element is to describe the spectrum M(H ∞ (D)), i.e. the set of all multiplicative linear functionals on H ∞ (D) which is a compact set when endowed with the weak-star topology. The biggest milestone of this early period is the Corona Theorem, given by Newman (in a weak form) and Carleson [26] in 1962, that states that the evaluations at points of D form a dense subset of the spectrum of H ∞ (D). 2010 Mathematics Subject Classification. Primary 30H50, 46E50; Secondary 46J15, 46G20. Key words and phrases. Banach algebras, homomorphisms, analytic functions, spectrum. The first author was partially supported by CONICET-PIP 11220090100624 and UBACyT Grant X038. The second, third and fourth authors were supported by MICINN and FEDER Project MTM2008-03211. The second author were also supported by Prometeo 2008/101. 1