Abstract

The article under review deals with relations between the notions hyperbolic, peak and antipeak plurisubharmonic function, taut and locally taut for domains in Banach spaces. After introducing the notions of weak tautness and locally weak tautness in this context, the authors first generalize a classical result of \textit{H. L. Royden} [Several Complex Variables II, Conf. Univ. Maryland 1970, 125–137 (1971; Zbl 0218.32012)], proving that a weakly taut Banach analytic manifold is hyperbolic (in the sense that the Kobayashi pseudo-distance is actually a distance defining the topology). Then the authors generalize two results of \textit{H. Gaussier} [Proc. Am. Math. Soc. 127, No.1, 105–116 (1999; Zbl 0912.32025)] using the same proof ideas as him: Let \(\Omega\) be a domain in a Banach space \(E\) admitting local peak and antipeak plurisubharmonic functions at infinity. First they prove that \(\Omega\) is hyperbolic. Then they prove that if, moreover, \(\Omega\) is locally weakly taut at every point in \(\partial \Omega\), then \(\Omega\) is weakly taut.