Local and 2-local isometries between absolutely continuous function spaces

Operators and Matrices Volume 15, Number 4 (2021), 1461–1468 doi:10.7153/oam-2021-15-91 LOCAL AND 2–LOCAL ISOMETRIES BETWEEN ABSOLUTELY CONTINUOUS FUNCTION SPACES MALIHEH HOSSEINI AND JUAN J. FONT (Communicated by L. Moln´ar) Abstract. In this paper we give a complete description of local and 2-local isometries defined between spaces of scalar-valued absolutely continuous functions on arbitrary (not necessarily compact) subsets of the real line with at least two points. 1. Introduction In the last years considerable work has been done on local maps, an area whose main problem of research is whether the local actions of some important classes of transformations (like derivations, automorphisms, isometries) on a given space determine the class under consideration completely. We refer the reader to [13] for more information. In particular, given two Banach spaces Aand B, the space of all bounded linear operators from Ato B,L(A,B)and S⊆L(A,B), a linear map T:A−→ Bis said to be locally in Sif for each a∈A, there exists a map Ta∈Ssuch that Ta =Taa. Similarly, a map T:A−→ B(which is not assumed to be linear) is said to be 2-locally in Sif for any pair a,a′∈A, there is a map Ta,a′∈Ssuch that Ta =Ta,a′(a)and Ta′=Ta,a′(a′). Local isometries are an active research area centered in the study of the algebraic reflexivity of certain function spaces. Let us recall that a Banach space Ais algebraically reflexive if any linear map on Abelonging locally to the group of all surjective linear isometries is surjective. For example, it is known (see [14] and [2]) that C(X,C)(resp. C0(X,C)) is algebraically reflexive provided Xis a first countable compact space (resp. Xis locally compact space whose one-point compactification is metrizable). Besides, C(X,E)is algebraically reflexive if Xis a first countable compact space and Eis a finite-dimensional complex Banach space or Eis a uniformly convex and algebraically reflexive Banach space (see [9]). One can also find recent results related to the algebraic reflexivity of Banach algebras of Lipschitz functions in Mathematics subject classification (2020): Primary 47B38; Secondary 46J10, 47B33. Keywords and phrases: Linear surjective isometry, absolutely continuous functions, local isometry, 2-local isometry. This work was partially supported by a grant from the IMU-CDC. J. J. Font was supported by Spanish AEI Project PID2019-106529GB-I00/AEI/10.13039/501100011033 and by Universitat Jaume I (Projecte UJI-B2019-08). c  D l , Zagreb Paper OaM-15-91 1461 1462 M. HOSSEINI AND J. J. FONT [15]. It is worth mentioning that little is known concerning spaces of real-valued functions since the main tool for the above results, the Gleason-Kahane- ˙ Zelazko theorem, only applies to complex algebras. On the other hand, 2-local maps were introduced by ˇ Semrl in [16] when trying to drop the linearity assumption for certain local maps. Subsequently, many authors have worked on these maps. Namely Gy¨ory ([4]) proved that every 2-local isometry on C0(X,C)is a surjective linear isometry provided Xis a first countable, σ -compact, locally compact space. Later, in [1], Al-Halees and Fleming generalized Gy¨ory’s results to C0(X,E)for σ -compact metric spaces Xunder certain conditions on the Banach space E. Similar results have been achieved for other function spaces such as, for instance, (pointed) Lipschitz spaces ([10,11,12]), uniform algebras ([5,12]) and spaces of functions of bounded variation ([7]). In this paper we give a complete description of local and 2-local isometries defined between spaces of scalar-valued absolutely continuous functions on arbitrary (not necessarily compact) subsets of the real line with at least two points. In particular, we extend some previous results in [6,7] to a noncompact framework. We would like to remark that, in most of the papers mentioned above, the compacity of the underlying spaces (or local compacity with functions vanishing at infinity) plays a crucial role. 2. Preliminaries Let Xbe a subset of the real line Rwith at least two points. Let us recall that a scalar-valued function fon Xhas bounded variation if the total variation V(f)of f is finite, that is, V(f):=sup(n ∑ i=1 |f(xi)−f(xi−1)|:n∈N,x0,x1,...,xn∈X,x0<x1< .. . < xn)<∞. Moreover, a scalar-valued function fon Xis said to be absolutely continuous if given ε >0, there is a δ >0 such that n ∑ i=1 |f(bi)−f(ai)|< ε , for each finite family of non-overlapping open intervals {(ai,bi):i=1,···,n}whose extreme points belong to Xand ∑n i=1(bi−ai)< δ . We denote by ACb(X)the space of all scalar-valued absolutely continuous functions of bounded variation on X, endowed with the norm k · k =max{k · k∞,V(·)}, where k · k∞stands for the supremum norm of a function. Note that when Xis bounded, each absolutely continuous function is automatically of bounded variation. In the sequel, by Xwe denote the closure of Xin R. Also, for any f∈ACb(X), let fbe the unique absolutely continuous extension of fto the closure Xof X, which exists by [8, Lemma 3.1]. Given two Banach spaces Aand B, we shall denote by Iso(A,B)the set of all surjective linear isometries from Aonto B. When A=B, we shall write Iso(A)instead of Iso(A,A). LOCAL ISOMETRIES BETWEEN ABSOLUTELY CONTINUOUS FUNCTION SPACES 1463 3. Local isometries of absolutely continuous function spaces In the sequel, Xand Ywill be two arbitrary (not necessarily closed nor bounded) subsets of Rwith at least two points. By Iso1(ACb(X),ACb(Y)) we denote the set of all surjective linear isometries T: ACb(X)−→ ACb(Y)such that T1 is bounded away from zero, i.e., there exists t>0 such that, for each y∈Y, we have |T1(y)|⩾t. Indeed, by [8, Theorem 3.13], we know that Iso1(ACb(X),ACb(Y)) is the set of all surjective linear isometries T:ACb(X)−→ ACb(Y)of the form of a weighted composition operator. The following result, which is used several times in our proofs, describes the form of these isometries. THEOREM 3.1. ([8, Theorem 3.13]) If T ∈Iso1(ACb(X),ACb(Y)), then there exist a unimodular scalar λ and a monotonic homeomorphism ϕ :Y−→ X such that T f = λ f◦ ϕ for all f ∈ACb(X). Moreover, it is worth pointing out that, according to Corollary 3.15 in [8], for the case X(and so Y) is connected we have Iso1(ACb(X),ACb(Y)) = Iso(ACb(X),ACb(Y)). Let us next adapt the concepts mentioned in the introduction to our context: DEFINITION 3.2. A linear map T:ACb(X)−→ ACb(Y)which is locally in Iso1(ACb(X),ACb(Y)) is called a local isometry. In fact, Tis a local isometry if for each f∈ACb(X)there is a surjective linear isometry Tf∈Iso1(ACb(X),ACb(Y)) (depending on f) such that T f =Tff. We can now provide a complete description of local isometries defined between spaces of absolutely continuous functions in a noncompact framework, which is a generalization of [6, Theorem 2.1]. THEOREM 3.3. If T :ACb(X)−→ ACb(Y)is a local isometry, then there exist a monotonic homeomorphism ϕ :Y−→ X , and a scalar λ with | λ |=1such that T f (y) = λ f( ϕ (y)) (f∈ACb(X),y∈Y). Proof. By [8, Lemma 3.1], each absolutely continuous function has a unique absolutely continuous function extension to the closure of the underlying space with the same total variation. This allows us to consider the map S:ACb(X)−→ ACb(Y)defined by S(f) = T f for all f∈ACb(X). It is easy to see that Sis a local isometry. Therefore we can assume, without loss of generality, that Xand Yare closed subsets of the real line. Moreover, since Tis a local isometry, we infer from Theorem 3.1 that T1 is a unimodular constant function. Hence, by considering T T1instead of T, we can assume, without loss of generality, that T1=1. We now continue the proof through several steps. Step 1. For each f∈ACb(X),kT f k∞=kfk∞and kT f k=kfk. Since Tis a local isometry, it is an immediate consequence of Theorem 3.1. 1464 M. HOSSEINI AND J. J. FONT REMARK. Before continuing with the proof, let us notice that if we had assumed compacity, we could have exploited the density of ACb(X)in C(X)and would have obtained the representation of Tfrom the famous Holszty´nski Theorem (see, e.g., [3, Theorem 2.3.10]). However, in this noncompact framework, such density is not valid (see [8, Remark 3.4]) and this forces us to use another approach in order to describe T. Before stating the next step, we need to fix some notation. For each x∈X, we set Fx:={f∈ACb(X):kfk∞=1=f(x)} which is clearly non-empty. Moreover, we also define Ix:=\{MT f :f∈Fx}, where MT f :={y∈Y:|T f (y)|=1=kT f k∞}. Step 2. Given x∈X,Ixis a non-empty subset of Y. Let us define e Ix:=\{Mf T f :f∈Fx}, where f T f is the unique extension of T f to the Stoneˇ Cech compactification, β Y, of Y. And, as expected, Mf T f ={y∈ β Y:|f T f (y)|=1=kf T f k∞}, which is obviously a non-empty compact subset of β Y. We now prove, by means of a standard technique in a compact framework (see, e.g., [8, Lemma 3.8]), that e Ixis a nonempty subset of β Y. To this end, let f1,..., fn∈ Fx. Define f=∑n i=1 fi n. Clearly, f∈ACb(X)with kf T f k∞=kT f k∞=kfk∞=1, by Step 1. Hence there is a point y∈ β Ysuch that |f T f (y)|=1, and so 1=|f T f (y)|= n ∑ i=1f T fi(y) n⩽ n ∑ i=1 |f T fi(y)| n⩽ n ∑ i=1 kf T fik∞ n= n ∑ i=1 kfik∞ n=1, which implies that |f T fi(y)|=1 for every i∈ {1,...,n}. Thus y∈n T i=1 Mf T fi. Therefore, the family {Mf T f :f∈Fx}has the finite intersection property and then e Ix6=/0, as desired. Finally, let us check that Ix=e Ix. Take f∈Fxwith compact support and {t∈ X:|f(t)|=1}={x}. From the representation given by Theorem 3.1, it follows that T f has compact support in Y. As a consequence, e Ix⊆Ybecause it is clear that e Ix⊆Supp(T f ). Therefore, Ix=e Ixis a non-empty subset of Y, as required. Step 3. If x∈Xand f∈ACb(X)with f(x) = 0, then T f (y) = 0 for all y∈Ix. This step is verified by a argument similar to the proof of Lemma 3.10 in [8]. Contrary to what we claim, suppose that x∈Xand f∈ACb(X)with f(x) = 0, but T f (y)6=0 for some y∈Ix. Let r>kfk∞and choose h∈Fxsuch that 0 ⩽h⩽1 LOCAL ISOMETRIES BETWEEN ABSOLUTELY CONTINUOUS FUNCTION SPACES 1465 and k| f|+rhk∞=kf±rhk∞=r, by [8, Lemma 3.6 (1)]. Since y∈Ix,|Th(y)|=1, and so we have r=kf±rhk∞=kT(f±rh)k∞ ⩾max{|T f (y)+ rTh(y)|,|T f (y)−rTh(y)|} >r, which is impossible. This completes the proof of Step 3. Step 4. For any two distinct points xand x′in X,IxTIx′=/0. Assume, on the contrary, that there exists a point yin IxTIx′. Choose a function f∈Fxwith f(x′) = 0. By Step 3, T f (y) = 0 because y∈Ix′. On the other hand, |T f (y)|=1 since f∈Fxand y∈Ix, which is impossible. This argument yields IxTIx′=/0. We can now introduce the following nonempty subset of Y: Y0:={y∈Y:y∈Ixfor some x∈X}. Besides, we can define a mapping ϕ :Y0−→ Xsuch that ϕ (y) = xif y∈Ix. According to the preceding step, ϕ is well-defined. Meantime, it is clear that ϕ is surjective. Step 5. For each f∈ACb(X)and y∈Y0, we have T f (y) = f( ϕ (y)). Let f∈ACb(X)and y∈Y0. Since (f−f( ϕ (y)))( ϕ (y)) = 0, we have T(f− f( ϕ (y)))(y) = 0 by Step 3. Hence T f (y) = T(f( ϕ (y)))(y) = f( ϕ (y)), as desired. Step 6. Y0=Y. Let us first define the function h:X−→ Rby h(x) = (2+1 x−1if x∈(−∞,0)∩X, 1 x+1if x∈[0,+∞)∩X. Clearly, his an injective function which belongs to ACb(X)with V(h)⩽2. Since T is a local isometry, there exist a monotonic homeomorphism ϕ h:Y−→ X, and a scalar λ hwith | λ h|=1 such that Th =Thh= λ h(h◦ ϕ h). Hence, from Step 5, we get h( ϕ (y)) = λ hh( ϕ h(y)) (y∈Y0), which taking into account that | λ h|=1 and h⩾0, implies that h( ϕ (y)) = h( ϕ h(y)) (y∈Y0). Hence ϕ (y) = ϕ h(y)for all y∈Y0because his injective. Now we can prove that Y0=Y. Otherwise, there would exist a point y0∈Y\Y0. Set x0= ϕ h(y0). Since ϕ is surjective, there is a point y∈Y0such that x0= ϕ (y). Hence, from the above discussion, we conclude that ϕ h(y0) = ϕ h(y), which contradicts the injectivity of ϕ h. Therefore, Y0=Y. Then, as observed above, ϕ = ϕ his a monotonic homeomorphism from Yonto X, and also we have T f (y) = λ f( ϕ (y)) (f∈ACb(X),y∈Y), which completes the proof of the theorem.  1466 M. HOSSEINI AND J. J. FONT 4. 2-Local isometries of absolutely continuous function spaces In this section we present a complete description of 2-local isometries between absolutely continuous function spaces. First let us recall the definition of 2-local isometries in the following. DEFINITION 4.1. A map T:ACb(X)−→ ACb(Y)(no linearity nor surjectivity are assumed) is called a 2-local isometry if it belongs 2-locally to Iso1(ACb(X),ACb(Y)), i.e., for every f,g∈ACb(X)there exists a Tf,g∈Iso1(ACb(X),ACb(Y)) such that T f = Tf,gfand T g =Tf,gg. THEOREM 4.2. For each 2-local isometry T :ACb(X)−→ ACb(Y), there exist a monotonic homeomorphism ϕ :Y−→ X , and a scalar λ with | λ |=1such that T f (y) = λ f◦ ϕ for all f ∈ACb(X). Proof. We will use a modification of the proof provided in [7, Theorem 2.5]. As in the beginning of the proof of Theorem 3.3, we can assume, without loss of generality, that Xand Yare closed subsets of R. Since T1 is a unimodular constant function, we can assume, without loss of generality that Tis unital, i.e., T1=1. Then, following the proof of [7, Theorem 2.5] (see also [11, Theorem 2.1]), one can deduce that for each x∈X, the set Ix:=Tf∈ACb(X)Ex,fis a singleton, where Ex,f={z∈Y:T f (z) = f(x)}. By ψ (x)we denote the unique point in Ix. This allows us to define an injective map ψ :X−→ Ysuch that T f ( ψ (x)) = f(x)for all x∈Xand f∈ACb(X). Now, taking Y0:= ψ (X)and ϕ := ψ −1we will have the bijective map ϕ :Y0−→ Xsuch that T f (y) = f( ϕ (y)) (f∈ACb(X),y∈Y0). We now claim that Y0=Y. Let hbe defined as in the proof of Theorem 3.2 (Step 6). Since Tis a 2-local isometry, there exists Th,1∈Iso1(ACb(X),ACb(Y)) such that T1,h(1) = 1 and T h =T1,h(h). According to Theorem 3.1, there is a monotonic homeomorphism ϕ 1,h:Y−→ Xsuch that T1,h(h) = h◦ ϕ 1,h. Combining the latter equations, we get h( ϕ (y)) = h( ϕ 1,h(y)) (y∈Y0), which easily implies that ϕ = ϕ 1,hon Y0because of the injectivity of h. In order to prove that Y0=Y, let ybe an arbitrary point in Y. Since ϕ is surjective, there exists y0∈Y0with ϕ (y0) = ϕ 1,h(y). On the other hand, from the previous paragraph we have ϕ (y0) = ϕ 1,h(y0), which implies that ϕ 1,h(y0) = ϕ 1,h(y). Consequently, y=y0because ϕ 1,his injective, which yields y∈Y0. Therefore, we can conclude that Y0=Y. Gathering all the information, we infer that ϕ :Y−→ Xis a monotonic homeomorphism ( ϕ = ϕ 1,h) and also T f (y) = f( ϕ (y)) (f∈ACb(X),y∈Y), as required.  LOCAL ISOMETRIES BETWEEN ABSOLUTELY CONTINUOUS FUNCTION SPACES 1467 REMARK 4.3. It should be noted that, for the complex case, one can exploit the spherical version of the Kowalski-Słodkowski theorem provided in [12] to obtain the description of 2-local isometries (see the remark at the end of [7]). More precisely, let T:ACb(X)−→ ACb(Y)be a 2-local isometry. For each y0∈Y, define the map Ty0:(ACb(X),k · kΣ)−→ Cby Ty0f=T f (y0)(f∈ACb(X)), where k · kΣ=k · k∞+ V(·). Clearly, since Tis a 2-local isometry, Ty0is 1-homogeneous, and according to Theorem 3.1, for each pair f,g∈ACb(X), there exist a scalar λ f,g∈Tand a monotonic homeomorphism ϕ f,g:Y−→ Xsuch that Ty0f= λ f,gf( ϕ f,g(y0)) and Ty0g= λ f,gg( ϕ f,g(y0)). Then Ty0f−Ty0g= λ f,g(f( ϕ f,g(y0))−g( ϕ f,g(y0))) ∈T σ (f−g), where σ (f−g)is the spectrum of f−g. By Proposition 3.2 in [12] it follows that Ty0is linear. Consequently, Tis a linear map since y0was arbitrary, which especially implies that T:ACb(X)−→ ACb(Y)is a local isometry. Thus the result follows from Theorem 3.2. Acknowledgements. 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MOLN ´ AR,Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lecture Notes in Mathematics, 1895, Springer-Verlag, Berlin, 2007. [14] L. MOLN ´ AR, B. ZALAR,Reflexivity of the group of surjective isometries on some Banach spaces, Proc. Edinb. Math. Soc. 42 (1999), 17–36. 1468 M. HOSSEINI AND J. J. FONT [15] S. OI,Algebraic reflexivity of isometry groups of algebras of Lipschitz maps, Linear Algebra Appl. 566 (2019), 167–182. [16] P. ˇ SEMRL,Local automorphisms and derivations on B(H), Proc. Amer. Math. Soc. 125 (1997), 2677– 2680. (Received October 16, 2020) Maliheh Hosseini Faculty of Mathematics K. N. Toosi University of Technology Tehran, 16315-1618, Iran e-mail: [email protected] Juan J. Font Departamento de Matem´aticas Universitat Jaume I Campus Riu Sec 8029 AP, Castell´on, Spain e-mail: [email protected] Operators and Matrices www.ele-math.com [email protected]