Weak-2-local isometries on uniform algebras and Lipschitz algebras

We establish spherical variants of the Gleason-Kahane-Zelazko and Kowalski-Slodkowski 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. Oka, and H. Takagi in 2007.L. Li was partly supported by NSF of China project no. 11301285. A. M. Peralta was partially supported by the Spanish Ministry of Economy and Competitiveness and European Re- gional Development Fund project no. MTM2014-58984-P and Junta de Andalucía grant FQM375. L. Wang was partly supported by NSF of China Grants No. 11371222, 11871303, and 11671133. Y.-S. Wang was partly supported by Taiwan MOST 104-2115-M-005-001-MY2