Algebraic Reflexivity of Non-Canonical Isometries on Lipschitz Spaces

Let Lip([0, 1]) be the Banach space of all Lipschitz complex-valued functions f on [0, 1], equipped with one of the norms: ‖ f ‖σ = | f (0)|+ ‖ f ‖L∞ or ‖ f ‖m = max{| f (0)|, ‖ f ‖L∞}, where ‖·‖L∞ denotes the essential supremum norm. It is known that the surjective linear isometries of such spaces are integral operators, rather than the more familiar weighted composition operators. In this paper, we describe the topological reflexive closure of the isometry group of Lip([0, 1]). Namely, we prove that every approximate local isometry of Lip([0, 1]) can be represented as a sum of an elementary weighted composition operator and an integral operator. This description allows us to establish the algebraic reflexivity of the sets of surjective linear isometries, isometric reflections, and generalized bi-circular projections of Lip([0, 1]). Additionally, some complete characterizations of such reflections and projections are stated.


Introduction
A function f : [0, 1] → C is said to be Lipschitz if there exists a positive constant K such that The infimum of such constants K is called the Lipschitz constant of f and it is denoted by Lip( f ).
On the other hand, a measurable function f : [0, 1] → C is said to be essentially bounded if there is a positive constant K such that Namely, the derivative map f → f is an isometric isomorphism from the space Lip([0, 1]) (with the Lipschitz seminorm Lip(·)) onto the space L ∞ ([0, 1]) (with the essential supremum norm · L ∞ ). On L ∞ ([0, 1]), we take into account the usual convention about identifying functions equal almost everywhere.
The isometry group of Lip([0, 1]) has been studied under the following equivalent norms: Indeed, for any f ∈ Lip([0, 1]), we have Furthermore, (Lip([0, 1]), · Σ ) is a Banach algebra as it satisfies the Banach algebra law: but Lip([0, 1]), equipped with any of the other norms, is only a complete normed algebra in the sense that there exists a positive constant K (not necessarily equal to 1) such that Surjective linear isometries of Lip([0, 1]), with both the σ-norm or the m-norm, were characterized as a sum of a weighted composition operator and an integral operator by Koshimizu [1,2], in contrast with the isometry groups of Lip([0, 1]), with both the Σ-norm or the M-norm, whose members have a canonical form in the sense that they can be represented as a weighted composition operator [3][4][5][6].
Indeed, the prominent part of the representation of the isometries on Lip([0, 1]) with the σ-norm or the m-norm lies on the integral operator as the involved weighted composition operator is elementary in the sense that it is the evaluation at the point 0 multiplied by a unimodular constant.
To present our results, we recall the concepts of reflexivity studied in this paper. Let E be a Banach space, B(E) be the space of all bounded linear operators from E into E, and S be a nonempty subset of B(E). For each e ∈ E, let S(e) be {L(e) : L ∈ S} and let S(e) denote the norm-closure of S(e) in E. Define the algebraic reflexive closure and the topological reflexive closure of S, respectively, by We say that S is algebraically reflexive (topologically reflexive) if ref alg (S) = S (respectively, ref top (S) = S).
In the case S = Iso(E) (the set of all linear isometries from E onto E), the elements of ref alg (S) and ref top (S) are known as local isometries and approximate local isometries of E, respectively.
The consideration of approximate local isometries instead of local isometries is more general and allows us to deal with the problems of topological reflexivity and algebraic reflexivity at the same time.
The study of algebraic and topological reflexivity of the sets of isometries, derivations, and automorphisms on operator algebras and function algebras is a classical problem which follows attracting the attention of numerous researchers. Molnár's monograph [7] can give a complete account of these developments. Reflexivity problems have been addressed on spaces of vector and scalar-valued Lipschitz functions defined on metric spaces, equipped with norms of type · Σ and · M [8][9][10][11][12]. In the last three references, the canonical form of the isometries of such spaces allowed the application of the Gleason-Kahane-Żelazko theorem (or of some of its generalizations [11,13]) in their arguments.
In this paper, we deal with the reflexivity of the isometry group of Lip([0, 1]), equipped with the σ-norm or the m-norm. We provide the form of approximate local isometries of Lip([0, 1]). With the aid of this description, we prove that the isometry group of Lip([0, 1]), with each one of these norms, is algebraically reflexive.
Although the non-canonical form of the isometries on such spaces added initially a little more difficulty to the problem, another application of a spherical variant of Gleason-Kahane-Żelazko theorem, stated in [11], allows us to show that every approximate local isometry of Lip([0, 1]) admits a representation as a sum of an elementary weighted composition operator and an integral operator. Compare this fact with [11], p. 250, where an example shows that the cited generalization of Gleason-Kahane-Żelazko theorem can not be applied when the isometry group is not canonical.
We prove also that the sets of isometric reflections and generalized bi-circular projections on Lip([0, 1]) are algebraically reflexive. Our approach requires to characterize these types of maps on Lip([0, 1]), endowed with the σ-norm or the m-norm. This kind of projections was introduced in [14] and they have been characterized in various settings (see, for example, in [10,15] and the references therein). Namely, it is known (see, for example, Theorem 1.36 and Corollary 1.39 in [16]) that if f is function from [0, 1] to C, then the following are equivalent:
In what follows, σ(·) denotes the spectrum. To simplify, we introduce the following notations: From now on, the symbols 1 and ι stand for the function with the constant value 1 and the identity function on [0, 1], respectively.
Koshimizu [1,2] gave the following characterizations of surjective linear isometries on the space Lip([0, 1]), equipped with the σ-norm or the m-norm given by for f ∈ Lip([0, 1]). These descriptions provide a key tool in our study on the reflexivity of some subsets of linear maps on such spaces.
From Theorem 1, we deduce immediately the following.
Proof. In view of the expression of T as in Theorem 1, we infer that T( f )(0) = λ f (0) and 1] with the norm: and thus we may consider that L ∞ ([0, 1]) is equipped with the weak* topology. A result due to Sikorski and von Neumann (see in [18], Theorem 1) shows that every weak* continuous algebra homomorphism Φ : is a measurable function. In particular, every algebra automorphism of L ∞ ([0, 1]) has this form as it is weak* continuous by Lemmas 1 and 2 in [18] (observe that every algebra automorphism of L ∞ ([0, 1]) is a local algebra automorphism and it is continuous by Corollary 2.1.10 in [17]). This last fact can also be proven as in Theorem 1 of [20].

Results
In the rest of the paper, we will consider that the linear space Lip([0, 1]) is equipped with the σ-norm or the m-norm. As the proofs of the results are similar for both norms, we only will prove them when Lip([0, 1]) is provided with the σ-norm.
We first give a representation of the elements of the topological reflexive closure of the isometry group of Lip([0, 1]). Theorem 2. Every approximate local isometry T of Lip([0, 1]) is a linear isometry having the form . We establish some properties to prove the theorem.
This implies that .
From on now, Property 1 will be frequently applied without any explicit mention along the paper.
we obtain the equalities of Property 2.
Property 2 yields |T(1)(0)| = 1. Take λ = T(1)(0) and define the functional Clearly, T 0 is linear and unital. Let us recall that (Lip([0, 1]), · σ ) is a complete normed algebra, but it is not a Banach algebra. To see that T 0 is multiplicative, define S 0 from Lip([0, 1]) to C by As S 0 is linear and As λ f ,n f (0) ∈ Tσ( f ) for all n ∈ N, it follows that Applying a spherical variant of the Gleason-Kahane-Żelazko theorem, stated in Proposition 2.2 of [11], we conclude that T 0 = S 0 (1)S 0 is multiplicative. It is easy to see . Furthermore, it is obvious that the unital Banach function algebra (Lip([0, 1]), · σ ) is self-adjoint. Now, from Proposition 4.1.5 (ii) in [21] we infer that the maximal ideal space of that algebra is homeomorphic to [0, 1].
Define the isometric linear embedding S of L ∞ ([0, 1]) into itself by .

Definition 2.
Let E be a Banach space. A projection of E is a map P : E → E such that P 2 = P. A generalized bi-circular projection of E is a linear projection P : E → E such that P + τ(Id − P) is a linear surjective isometry for some τ ∈ T \ {1}. We denote by GBP(E) the set of all generalized bi-circular projections of E.

Discussion
Given a Banach space E, an operator T ∈ B(E) is a local isometry of E whenever for every e ∈ E, there exists a T e ∈ Iso(E), possibly depending on e, such that T e (e) = T(e), and T is an approximate local isometry of E if for every e ∈ E, there is a sequence {T e,n } n∈N in Iso(E) such that lim n→∞ T e,n (e) = T(e).
One of the main questions addressed in the studies on local isometries is for which Banach spaces E, every local isometry of E is a surjective isometry or, equivalently, which Banach spaces E have an algebraically reflexive isometry group. The topological variant of this question can also be considered, that is, when every approximate local isometry of E is a surjective isometry or, with other words, when Iso(E) is topologically reflexive.
To address this type of problem, it is necessary to have convenient descriptions of the isometries on the involved spaces. However, Koshimizu [1,2] showed that the surjective linear isometries of such spaces do not have a canonical representation as a weighted composition operator but instead they admit a representation as integral operators.
Although this fact added some initial difficult to the problem, we have been able to apply a spherical variant of the Gleason-Kahane-Żelazko theorem [11] to establish our results.
Our main theorem states that every approximate local isometry of Lip([0, 1]) can be represented as a sum of an elementary weighted composition operator and an integral operator. Applying this description, we obtain some important consequences: the groups of surjective linear isometries and isometric reflections and the set of generalized bi-circular projections of Lip([0, 1]) are algebraically reflexive. In the process, we give complete descriptions of such reflections and projections.
Besides, the advantage of considering approximate local maps rather than local maps is that they are more general and they allow us to state these results more easily.
The problems studied in this paper are closely related to the research on 2-local isometries between Banach spaces, which was raised by Molnár [24]. Let us recall that given a Banach space E, a set S ⊆ B(E) is called 2-algebraically reflexive if 2-ref alg (S) = S, where 2-ref alg (S) = ∆ ∈ E E : ∀e, u ∈ E, ∃S e,u ∈ S | S e,u (e) = ∆(e), S e,u (u) = ∆(u) .
In the case S = Iso(E), the members of 2-ref alg (S) are called 2-local isometries. A complete information on 2-local maps can also be found in [7].
It would be interesting to study the 2-algebraic reflexivity of the sets of surjective linear isometries, isometric reflections and generalized bi-circular projections on Lip([0, 1]), equipped with the σ-norm or the m-norm.
We believe that results of this type have strong potential for further applications, fitting into a quickly growing area of international research.
Author Contributions: Investigation, A.J.-V. and M.I.R. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
Funding: This research was partially supported by Junta de Andalucía grant FQM194.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.