Bounded holomorphic functions attaining their norms in the bidual

DANIEL CARANDO; MARTÍN MAZZITELLI

PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES

KYOTO UNIV

Año: 2015

0034-5318

Under certain hypotheses on the Banach space $X$, we prove that the set of analytic functions in $mathcal{A}_u(X)$ (the algebra of all holomorphic and uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions attain their norms, is dense in $mathcal{A}_u(X)$. This Lindenstrauss type result holds also for functions with values in a dual space or in a Banach space with the so-called property $(eta)$. We show that the Bishop-Phelps theorem does not hold for $mathcal{A}_u(c_0,Z´´)$ for a certain Banach space $Z$, while our Lindenstrauss theorem does. In order to obtain our results, we first handle their polynomial cases.