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

Abstract

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.

2-local isometries on function spaces

“…, then by (15) we infer that w ′ 1 m − t = 0. By a calculation, for every positive integer k there exists a unique kπ < s k < kπ + π/2 such that w ′ 1…”

“…Let Iso C (M 1 , M 2 ) denote the set of all surjective complex-linear isometries. There exists an extensive literature on 2-local isometries in Iso C (M 1 , M 2 ) and 2-iso-reflexivity of Iso C (M 1 , M 2 ) (see, for example, [1,2,4,7,11,12,15,20,21]). Note that Hosseini showed that a 2-local real-linear isometry is in fact a surjective real-linear isometry on the algebra of n-times continuously differentiable functions on the interval [0, 1] with a certain norm [9, Theorem 3.1].…”

Isometries on Banach algebras of vector-valued maps

“…for every G ∈ B 2 . Thus we have (25) G(x, y) = U 0 (U −1 0 (G))(x, y) = U −1 0 (G)(ϕ(x, y), τ (y)) = G(ϕ 1 (ϕ(x, y), τ (y)), τ 1 (τ (y))), (x, y) ∈ X 2 × Y 2 for every G ∈ B 2 and…”

“…Jiménez-Vargas and Villegas-Vallecillos in [17] have considered isometries of spaces of vector-valued Lipschitz maps on a compact metric space taking values in a strictly convex Banach space, equipped with the norm f = max{ f ∞ , L(f )}, see also [16]. Botelho and Jamison [3] studied isometries on C 1 ([0, 1], E) with max x∈[0, 1] [32,26,18,1,2,23,6,31,5,27,19,20,21,24,22,25,15] From now on, and unless otherwise mentioned, α will be a real scalar in (0, 1). Jarosz and Pathak [14] studied a problem when an isometry on a space of continuous functions is a weighted composition operator.…”

“…There seem to be a confusion of the status of the result and it would be appropriate to clarify the current situation. After the publication of [14] some authors expressed their suspicion about the argument there and the validity of the statement there had not been confirmed when the authors of [25] pointed out a gap by referring the comment of Weaver [34, p. 243]. While Weaver in [34] pointed out that the argument of [14] failed on p.200 in which the norm max{ · ∞ , L(·)} was studied, he did not seem to have stated explicitly that the argument in the Example 8 contained a flaw.…”