We present here a multiplicative version of the classical Kowalski-Słodkowski Theorem which identifies the characters among the collection of all functionals on a complex and unital Banach algebra A. In particular, we show that, if A is a C -algebra, and if φ : A → C is a continuous function satisfying φ(x)φ(y) ∈ σ(xy) for all x, y ∈ A (where σ denotes the spectrum), then either φ a character of A or −φ is a character of A.

Abstract. Let A be a complex unital Banach algebra. Since the set of invertible elements is open, there is an open ball around every invertible element. In this article, we investigate the Banach algebras for which the radius given by the Neumann series is optimal.Mathematics subject classification (2010): Primary 46H05, secondary 46H20.

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Let A be a complex unital Banach algebra and M be a left A-module. Let Λ : M → C be a map that is not necessarily linear. We establish conditions for Λ to be linear and of multiplicative kind, from its behavior on a small subset of M. We do not assume Λ to be continuous throughout. As an application, we give a characterization of weighted composition operators on the Hardy space H ∞ .