On local isometries between algebras of C(Y)-valued differentiable maps

Let K be either the real unit interval [0, 1] or the complex unit circle T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}$$\end{document} and let C(Y) be the space of all complex-valued continuous functions on a compact Hausdorff space Y. We prove that the isometry group of the algebra C1(K,C(Y))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1(K,C(Y))$$\end{document} of all C(Y)-valued continuously differentiable maps on K, equipped with the Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document}-norm, is topologically reflexive and 2-topologically reflexive whenever the isometry group of C(Y) is topologically reflexive.


Introduction
Surjective linear isometries on the space C 1 ([0, 1], E) of all continuously differentiable maps on the real unit interval [0, 1] with values in a Banach space E, equipped with the -norm: 1] F(x) E + max have been studied for some concrete Banach spaces E. For E = C, such isometries were described first by Rao and Roy [28], and, later, by Jarosz and Pathak [12] within a more general study of the surjective isometries of some classic function spaces. Miura and Takagi [18] extended the result of Rao  Hatori and Oi [8] (see also [7]) proposed a unified approach to the study of isometries on Banach algebras of vector-valued maps by means of the notion of admissible quadruples and described isometries on Banach algebras C 1 (K , C(Y )), where K is either [0, 1] or T (the complex unit circle) and C(Y ) is the Banach algebra of all complex-valued continuous functions on a compact Hausdorff space Y with the supremum norm: Another usual norm considered on the linear space C 1 ([0, 1], E) is the C-norm given by Cambern [4] gave a representation for the surjective linear isometries of C 1 ([0, 1], C) with the C-norm. Botelho and Jamison [2] extended this result to spaces C 1 ([0, 1], H ), where H is a finite-dimensional real Hilbert space. Recently, Ranjbar-Motlagh [27] has characterized the surjective linear isometries of C 1 ([0, 1], E) whenever E is a strictly convex real Banach space. Equipped with other norms, Li and Wang [15] investigated surjective linear isometries between spaces C (n) where is an open subset of Euclidean space and E is a reflexive strictly convex space. The case in which is an open subset of R and E is a strictly convex Banach space with dimension greater than 1 was addressed by Li et al. [13].
Reflexivity and 2-reflexivity of the set of surjective linear isometries between Banach spaces are properties that are closely related to the study of isometries. Algebraical and topological reflexivity of the group of surjective isometries on Banach spaces were introduced by Molnár and Zalar in [22]. The paper [3] by Cabello Sánchez and Molnár is concerned with the algebraical and topological reflexivity of the isometry group and the automorphism group of some important metric linear spaces and algebras.
The research on 2-local isometries between Banach spaces was initiated by Molnár [19], motivated by the paper [30] of Šemrl who obtained the first results on 2-local automorphisms and 2-local derivations between Banach algebras. The study of 2-local isometries of C(X )spaces was raised by Molnár [20]. In [6], Győry gave a description of 2-local isometries of C 0 (L, C)-spaces.
Fleming and Jamison proposed the research on these topics in their monograph [5]. The study of reflexivity and 2-reflexivity of the sets of isometries, derivations and automorphisms on operator algebras and function algebras is a problem which follows attracting the attention of numerous researchers.
We briefly recall these notions.
where 2-ref alg (Iso(E, F)) and 2-ref top (Iso(E, F)) are the sets defined, respectively, by The members of 2-ref alg (Iso(E, F)) and 2-ref top (Iso(E, F)) are known as 2-local isometries and approximate 2-local isometries of E to F, respectively. The main purpose of this article is to prove that the isometry group of the algebra C 1 (K , C(Y )) for K = [0, 1] or K = T, equipped with the -norm, is topologically reflexive and 2-topologically reflexive whenever the isometry group of C(Y ) is topologically reflexive.
Apparently, this last condition is too restrictive since Molnár and Zalar [22] proved that the isometry group of C(Y ) is algebraically reflexive if Y is a first countable compact Hausdorff space, and Cabello-Sánchez and Molnár [3] gave an example where reflexivity may fail even if Y lacks first countability at only on point. However, the isometry group of C(Y ) is topologically reflexive whenever the homeomorphism group of Y is a finite group or a compact group, and it is known the abundance of such spaces Y in the literature (see [1,10,17]).
Our result finds a first motivation in the work [25] by Oi on the algebraic reflexivity of the isometry group of algebras of C(Y )-valued Lipschitz maps. The study of 2-local isometries and 2-local automorphisms without assuming linearity is a hard problem initiated by Molnár [21] in the algebra of all bounded linear operators on a Hilbert space. The 2-locality problem for surjective isometries on C 1 ([0, 1], C), without assuming linearity, was addressed by Hatori and Oi [9]. The algebraic reflexivity of the isometry group of C 1 ([0, 1], C) with the -norm was stated first in [26]. For more results on this subject in the setting of Banach spaces of differentiable maps equipped with other norms, we refer the reader to [11,16].
We have divided this paper into two sections. Section 2 gathers some known properties of C(K , C(Y ))-algebras. The type BJ representation of the isometry group of C 1 (K , C(Y )) with the -norm, stated by Hatori and Oi [8], is essential in our arguments. We complete this section with descriptions of the maximal ideal space of the Banach algebra C 1 (K , C(Y )) and of the algebra homomorphism group between C 1 (K , C(Y ))-algebras, which will be needed later. Section 3 contains the main results of this work. Using a spherical variant of the Gleason-Kahane-Żelazko theorem [14], the Gelfand theory and the Arzelá-Ascoli theorem, we prove first that the group Iso(C 1 (K )) is topologically reflexive. This fact joint to the Banach-Stone theorem are applied to state that Iso(C 1 (K , C(Y ))) is topologically reflexive whenever so is Iso(C(Y )). As a consequence, but applying now a spherical variant of the Kowalski-Słodkowski theorem [14], we deduce that Iso(C 1 (K , C(Y )) is also 2-topologically reflexive under the same condition on Iso(C(Y )). We finish with some observations about nice operators on Banach spaces that motivate new open problems.

Preliminaries
We first present the algebras of continuously differentiable maps which are studied in this paper. Given two Hausdorff spaces X and Z , denote When X is a compact Hausdorff space and Z is a complex Banach space, we consider the linear space C(X , Z ) equipped with the supremum norm: In the case Z = C, we write C(X ) instead of C(X , C). Given any set X , the symbols id X and 1 X stand for the identity map on X and the function constantly 1 on X , respectively. For a metric space X , Iso(X ) denotes the set of all isometries of X onto itself. All elements in Iso(X ) are assumed to be linear when X has a vector space structure.
for every x 0 ∈ K . We denote F = G. Consider the set , equipped with the -norm given by is a unital semisimple commutative Banach algebra with unit Following to Hatori and Oi [8], we identify Taking into account that and where · denotes the complex conjugation, we can gather in a unique statement the following representations, obtained and called of type BJ in [8], for the surjective linear isometries of the spaces C 1 (K , C(Y )) endowed with the -norm.
In what follows, given a function ϕ : K × Y → K , for each y ∈ Y , we denote by ϕ y : K → K the function defined by ϕ y (x) = ϕ(x, y) for all x ∈ K .

Theorem 1 (see [8,Corollaries 18 and 19]) Let K be either
For our proofs, we also will need the following results. The first one shows that the maximal ideal space of C 1 (K , C(Y )) can be identified with K × Y . The second one provides a description of unital algebra homomorphisms between C 1 (K , C(Y ))-algebras.
It is clear that We next prove that this embedding is surjective. Here we apply an argument similar to the proof of Proposition 11 in [24]. Suppose that is an open subset of Y . From the compactness of Y , we conclude that there exist y 1 , . . . , We have and Put Next, by using the above functions, we define a map G ∈ C 1 (K , C(Y )) such that 1/G ∈ C 1 (K , C(Y )). First consider the case where K = [0, 1]. Set Obviously, is differentiable at x and its derivative is H x 0 (x)(z). Then, according to the mean value Theorem, for each x ∈ K , there exists t x ∈ K such that This discussion yields x 0 ∈ Int(V x 0 ). From the above argument, It is clear that G ∈ C 1 (K , C(Y )) and G(x)(z) ≥ δ 0 > 0 for all x ∈ K and z ∈ Y , where δ 0 = min{δ x 1 /2, . . . , δ x m /2}. Now, one can easily check that Obviously, t 0 ∈ V t 0 . Similarly to the previous case, we prove that Similarly to above, the function P z H t 0 is differentiable at each t ∈ [0, 2π] and its derivative is ie it H x 0 (e it )(z), and so from the mean value Theorem, we obtain Then G ∈ C 1 (K , C(Y )), and if x ∈ T, then there exists j ∈ {1, . . . , m} such that t ∈ V t j and x = e it , which implies that Finally, as observed in both cases, G ∈ C 1 (K , C(Y )) and χ(G) = m j=1 χ(H x j ) = 0. However, 1/G ∈ C 1 (K , C(Y )), and so 1 = χ(G · (1/G)) = χ(G)χ(1/G) which yields χ(G) = 0, a contradiction. This completes the proof.
Proof Since T is a unital algebra homomorphism, it is easy that to see for each (x, y) ∈ K × Y 2 , the functional T (x,y) : belongs to the maximal ideal space of C 1 (K , C(Y 1 )). Then, taking into account Theorem 2, this allows us to define the functions ϕ : Hence it follows that We now prove that τ and ϕ are continuous. For this purpose, assume that {(x α , y α )} α is a net in K × Y 2 converging to (x 0 , y 0 ) ∈ K × Y 2 .
Finally, let y ∈ Y 2 . For each x ∈ K , we have

Results
We first show that every approximate local isometry of C 1 (K ) is a surjective isometry. By [11,Corollary 2], this fact is known for C 1 ([0, 1]) equipped with the norm:

Theorem 4 For K = [0, 1] or K = T, the group Iso(C 1 (K )) is topologically reflexive.
Proof We first prove that Iso(C 1 (K )) is algebraically reflexive. It holds for K = [0, 1] (see the last paragraph in [26]) but, apparently, the property is unknown for K = T. For that reason, we include here a proof that is valid for both cases.
Clearly, for each x ∈ K , the functional S x : is linear and unital. To show its multiplicativity, define the functional T x :

Clearly, T x is linear and continuous. Given any
where σ ( f ) denotes the spectrum of f . Applying [14,Proposition 2.2], which is a spherical variant of the Gleason-Kahane-Żelazko theorem, we conclude that S x = T x (1 K )T x is multiplicative. By above-proved, the map S : is a unital algebra homomorphism. Since C 1 (K ) is a semisimple commutative Banach algebra and the maximal ideal space of C 1 (K ) is homeomorphic to K , it is well known [29] that S is automatically continuous and induces a map φ ∈ C(K , K ) such that We now prove that φ is an isometry of K . Take any f ∈ C 1 (K ). By hypothesis, we get that is strongly separating. This means that for any pair of distinct points x 1 , From this expression of T as a weighted composition operator, we deduce that T ∈ Iso(C 1 (K )) by Theorem 1, and this proves that Iso(C 1 (K )) is algebraically reflexive.
We now prove that Iso(C 1 (K )) is topologically reflexive. For it, let T ∈ ref top (Iso(C 1 (K )). By Theorem 1, for each f ∈ C 1 (K ), we can take two sequences {λ f ,n } n∈N in T and {φ f ,n } n∈N in Iso(K ) such that Since T is compact in C, and Iso(K ) is compact in C(K ) by the Arzelá-Ascoli theorem, we may take subsequences {λ f ,n k } k∈N and {φ f , Since Iso(C 1 (K )) is algebraically reflexive as first proved, we conclude that T ∈ Iso(C 1 (K )).
We are ready to state the topological reflexivity of the isometry group of C 1 (K , C(Y ))algebras under a convenient condition on the isometry group of C(Y )-algebras.
Proof Let T be an approximate local isometry of C 1 (K , C(Y 1 )) to C 1 (K , C(Y 2 )). We are going to show that T has a representation of type BJ as in Theorem 1, and therefore T will be a linear isometry of C 1 (K , C(Y 1 )) onto C 1 (K , C(Y 2 ))).
For each n ∈ N, we can write with h F,n , ϕ F,n and τ F,n being as in Theorem 1. An easy calculation yields Since U F,n (G) = G , we also have U F,n (G) ∞ = G ∞ . Now the claim follows easily.
The following fact will be used repeatedly without any explicit mention in our proof.

Claim 3 There exists a function h
On a hand, we obtain that T (F) ∞ = F ∞ = 0 by Claim 1. Hence T (F) is a constant function from K to C(Y 2 ) and therefore T (F) = 1 K ⊗ h for some h ∈ C(Y 2 ). On the other hand, we have Since {h F,n } n∈N ⊆ C(Y 2 , T), it follows that h is a unimodular function and this proves the claim.

Claim 4
For each (x, y) ∈ K × Y 2 , the functional S (x,y) : is linear, unital and multiplicative.
Fix (x, y) ∈ K × Y 2 . Clearly, S (x,y) is linear with S (x,y) (1 K ⊗ 1 Y 1 ) = 1 by Claim 3. To prove its multiplicativity, define T (x,y) : Since T (x,y) is linear and we have that T (x,y) is continuous. Pick now any F ∈ C 1 (K , C(Y 1 )). We have We infer that Using Claim 4, it is easily deduced that S : is a unital algebra homomorphism. By Theorem 3, there exist two maps ϕ ∈ C(K × Y 2 , K ), with ϕ y ∈ C 1 (K ) for each y ∈ Y 2 , and τ ∈ C(K × Y 2 , Y 1 ) such that for all F ∈ C 1 (K , C(Y 1 )).
Fix y ∈ Y 2 and define T y : By Claim 5, we have .
Consequently, we obtain For each n ∈ N, define T y, f ,n : In the light of Theorem 1, T y, f ,n ∈ Iso(C 1 (K )) because h f ⊗1 Y 1 ,n (y) ∈ T and ϕ y f ⊗1 Y 1 ,n ∈ Iso(K ). Therefore T y ∈ ref top (Iso(C 1 (K ))). Hence T y ∈ Iso(C 1 (K )) by Theorem 4. By Theorem 1 again, we can find a number α y ∈ T and a map φ y ∈ Iso(K ) such that In addition, α y = T y (1 K )(x) = h(y) where x is any point in K , and thus Therefore we can write Since C 1 (K ) separates the points of K , we conclude that and so ϕ y = φ y ∈ Iso(K ), as required.

Claim 7
There exists a map β ∈ Homeo(Y 2 , Y 1 ) such that where x is any point of K .
Let x ∈ K be fixed and define T x :
In fact, α(y) = T x (1 Y 1 )(y) = h(y) for all y ∈ Y 2 , and therefore Consequently, we have Since C(Y 1 ) separates the points of Y 1 , we infer that This proves Claim 7 and the proof of Theorem 5 is finished. Notice that Theorem 5 contains Theorem 4 as a special case (consider the case that Y 1 and Y 2 are singletons). Note, however, that Theorem 4 is used in its proof.
We next apply Theorem 5 to study the 2-topological reflexivity of the set of surjective linear isometries between algebras C 1 (K , C(Y )).
This proves our claim. By the arbitrariness of (x, y), we infer that is linear. Consequently, is an approximate local isometry of C 1 (K , C(Y 1 )) to C 1 (K , C(Y 2 )), hence is a linear isometry of C 1 (K , C(Y 1 )) onto C 1 (K , C(Y 2 )) by Theorem 5, and the proof of the corollary is finished.
We close the paper with an application on nice operators. Let T : E → F be a continuous linear operator between Banach spaces and T * : F * → E * be its adjoint operator. Let Ext(B E ) denote the set of all extreme points of the unit closed ball B E of E. Then T is said to be nice if T * (Ext(B F * )) ⊆ Ext(B E * ). Surjective linear isometries are nice operators but the converse is not certain in general.
In [23], the authors show that C 1 ([0, 1]) is an example of an infinite-dimensional Banach space for which each nice operator on C 1 ([0, 1]) is an isometric isomorphism. As an immediate consequence of Theorem 4 and Corollary 1, we deduce the following.

Corollary 2
The set of all nice operators on C 1 ([0, 1]) is topologically reflexive and 2topologically reflexive.
The search of a Banach-Stone type representation for nice isomorphisms has been approached by some authors (see Chapter 7 in [5]). It is a natural question to ask what can be stated on the algebraic and topological reflexivity (and 2-reflexivity) of the sets of nice isomorphisms of the classical Banach spaces.