We establish spherical variants of the Gleason-Kahane-Zelazko and Kowalski-S lodkowski theorems, and we apply them to prove that every weak-2-local isometry between two uniform algebras is a linear map. Among the consequences, we solve a couple of problems posed by O. Hatori, T. Miura, H.

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Under the right conditions on a compact metric space X and on a Banach space E, we give a description of the 2-local (standard) isometries on the Banach space Lip(X, E) of vector-valued Lipschitz functions from X to E in terms of a generalized composition operator, and we study when every 2-local (standard) isometry on Lip(X, E) is both linear and surjective.

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A pair of functions defined on a set X with values in a vector space E is said to be disjoint if at least one of the functions takes the value 0 at every point in X. An operator acting between vector-valued function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. We characterize compact and weakly compact disjointness preserving operators between spaces of Banach space-valued differentiable functions.2010 Mathematics Subject Classification. Primary 46E40, 46E50, 47B33, 47B38. Key words and phrases. Disjointness preserving operators, spaces of vector-valued differentiable functions, compact and weakly compact operators.