Isometries on Banach algebras of vector-valued maps

Abstract

We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of some admissible quadruples that take values in unital commutative C *algebras. As a consequence we confirm the statement of [14, Example 8] on Lipschitz algebras and show that isometries on such algebras indeed take the canonical form.2010 Mathematics Subject Classification. 46E40,46B04,46J10,46J15 .

2-local isometries on function spaces

“…, there exists λ f,c ∈ E 2 and α f,c ∈ C of modulus 1, a homeomorphism π f,c : X 2 → X 1 , and ǫ f,c ∈ {±1} such that (7) T (8) we infer that λ f,c is a constant. By comparing (4) for f with (7) we get…”

“…Jarosz and Pathak had exhibited in [10, Example 8] the form of a surjective complex-linear isometry on the Banach algebra Lip(K j ) with the norm · Σ of the Lipschitz functions on a compact metric space K j by answering the question posed by Rao and Roy [22]. After the publication of [10] some authors expressed their suspicion about the argument there and the validity of the statement there had not been confirmed until the correction [8,Corollary 15] was published by Hatori and Oi. In this section by applying [8,Lemmas 10,11] and [5,Proposition 7] we exhibit the form of a surjective real-linear isometry between the Banach algebras of Lipschitz functions.…”

“…Proof. As the same way as in the proof of [8,Proposition 9] we apply Choquet's theory. Let j = 1, 2.…”

“…By a simple calculation we have b 0 ∈ Lip(K 2 ), 0 ≤ b 0 ≤ 1 on K 2 , and b 0 (x) = 1 if and only if x = x 0 . Then by [8,Lemma 10] there exists a pair (m 0 , γ 0 ) ∈ M 2 ×T such that (x 0 , m 0 , γ 0 ) ∈ Ch B 2 . By [8, Lemma 11] we have that p θ = (x 0 , m 0 , e iθ γ 0 ) ∈ Ch B 2 for every…”

“…It follows that U 0 is a complex-linear map if U 0 (i) = i and U 0 is a complex-linear map if U 0 (i) = −i. Applying [8,Corollary 15] there exists a surjective isometry π : K 1 → K 2 such that…”