422‐local isometries and the reflexivity property ofcertain spaces of continuous maps
Hiroko Meguro,

Master’s Program in Fundamental Sciences
Graduate School of Science and Technology,
Niigata University
This work was supported by the Research Institute for Mathematical Sciences,a Joint Usage/Research Center located in Kyoto University.
1

Introduction
The studies about local maps were started by Larson, Kadison and Sourour. In

1988, Larson[9] studied local automorphisms of Banach algebra and obtained thefirst results concerning to local maps. In 1990, Kadison[8] exhibited the results
concerning to local derivations on von neumann algebras. Larson and Sourour[10]got the results of local derivations of B(X) for a Banach space X.

The studies of 2‐local maps were initiated by Šemr1[13]. He got the results about
2‐local automorphisms and 2‐local derivations in 1997. Inspired by his results,

Molnár[12] started the sudies about 2‐local isometries in 2002. He considered thegroup of all surjective complex linear isometries. If X is locally compact Hausdorrf

space, Gyóry[3] studies that 2‐local isometries are complex linear isometries on
the ste of all continuous functions vanishing at infinity C_{0}(X) . Hatori, Miura,Oka and Takagi[4] got the results in the case of the uniform algebras in 2007.

C^{(n)}[0,1] denotes the set of all ‐times continuously differentiable functions onn[0,1] with

\Vert f\Vert_{C}=\sup_{t\in[0,1]}\sum_{k=0}^{n}|f^{(k)}(t)|/k! . In 2018, Kawamura, Koshimizu

and Miura[7] studied about C^{(n)}[0,1]They got the results that 2‐local isometries

are surjective complex linear isometries on each space. In recent years, the case43of surjective real linear isometries are studied. Hosseini[5] studied C^{(n)}[0,1] with

\Vert f\Vert_{n}=\max\{|f(0)|, |f'(0)|, |f^{(2)}(0)|, |f^{(n-1)}(0)|, \Vert f^ {(n)}\Vert_{\infty}\} in 2017. The results
about 2‐local isometries in the case of real linear isometries is fewer than the caseof complex linear isometries. I get the result about surjective real linear isometries.
I will prove it.

2

Fundamental definitions
In this paper, \mathbb{R} stands for the set of all real numbers. The symbol \mathb {C} stands for

all complex numbers.

Definition 2.1 (isometry). Let (X, d_{X}), (Y, d_{Y}) be metric spaces. Let T be a map
Xinto Y. If d_{X}(x_{1}, x_{2})=d_{Y}(T(x_{1}), T(x_{2})) for all points x_{1}, x_{2}\in X , thenT iscalled an isometry.
Note that T is injective if T is an isometry.
Definition 2.2. LetX

be a Banach space.

The set of all surjective complex

linear isometries on X is denoted by Iso_{\mathbb{C}}(X) . The set of all surjective real linear
isometries on X is denoted by Iso_{\mathbb{R}}(X) .

Definition 2.3 (2‐local isometry). Let X be a Banach space. Let T be a map
on X. If for each pair of elements f, g\in X there exists T_{f,g}\in Iso_{\mathbb{C}}(X) (or\in Iso_{\mathbb{R}}(X)) such that T_{f,g}(f)=T(f) and T_{f,g}(g)=T(g) depending on f and g,then T is called a 2‐local isometry

We note that no continuity, surjectivity nor linearity are assumed for T.
Definition 2.4. Let C[0,1] denote the set of all complex‐valued functions f on
the closed interval endowed with the supremum norm

\Vert f\Vert_{\infty}=\sup\{|f(t)| : t\in[0,1]\}.
Then (C[0,1], \Vert\cdot\Vert_{\infty}) is a Banach algebra.

Definition 2.5 (Choquet boundary). Let X be a locally compact Hausdorff space.
Let A be a uniform algebra on X. Define a subset E of X by E=\{t\in X : f(x)=1\} for some f\in A. Then E is called a peak set for A. For every x\in X , E_{\alpha}44
is a peak set for A. If \{x\}=\bigcap_{\alpha}E_{\alpha},x

is called a weak peak point of A. Define

Ch(A) by Ch(A)= { x\in X : x is a weak peak point for A }. Then Ch(A) is called
the Choquet boundary of A.

Definition 2.6 (reflexivity). Let X be a Banach space. We say that Iso_{\mathbb{R}}(X) is
2‐local reflexive if every 2‐local isometry is in Iso_{\mathbb{R}}(X) .

Surjective real linear isometries on C[0,1]3

In this section, we consider the form of surjective real linear isometries (Theorem

3.1)

This theorem was essentially proved by Ellis[2] or Miura[11] . We note that

the Chouque boungary and the Shilov boudary of C[0,1] corresponds to the closed
interval[0,1].

Theorem 3.1. A map

T

is a surjective real linear isometry on C[0,1] if andonly if there exist a continuous function T(1) : [0,1]arrow\{z\in \mathbb{C} : |z|=1\} and ahomeomorphism \varphi : [0,1]arrow[0,1] such that one of the following equalities\{\begin{ar ay}{l } T(f)(t)=T(1)f\circ\varphi(t) (f\in C[0,1], t\in[0,1]) T(f)(t)=T(1)f\circ\varphi(t) (f\in C[0,1], t\in[0,1]) . \end{ar ay}
Proof. First, we assume that a map T:C[0,1]arrow C[0,1] is a surjective real linear
isometry on C[0,1] . The Couquet boundary of C[0,1] coincides with the closed
interval [0,1] . By a theorem of Miura[11] and the connectivity of [0,1] , one of thefollowing equalities\{\begin{ar ay}{l } Tf(t)=T(1)f\circ\varphi(t) (f\in C[0,1], t\in[0,1]) Tf(t)=T(1)f\circ\varphi(t) (f\in C[0,1], t\in[0,1]) . \end{ar ay}
Next, we assume that there exist a continuous function T(1) : [0,1]arrow\{z\in \mathbb{C} :|z|=1\} and a homeomorphism \varphi : [0,1]arrow[0,1] such that one of the following
equalities\{\begin{ar ay}{l } T(f)(t)=T(1)f\circ\varphi(t) (f\in C[0,1], t\in[0,1]) T(f)(t)=T(1)f\circ\varphi(t) (f\in C[0,1], t\in[0,1]) . \end{ar ay}
We infer that T is a surjectve real linear isometry on C[0,1].

45\square42‐local isometries inC[0,1]The studies about 2‐local isometries were started by Molnár[12]. If there exists

T_{f,g}\in Iso_{\mathbb{R}}(C[0,1]) such that Tf =T_{f,g}f and Tg =T_{f,g}g for every pair ofelements f, g\in C[0,1] , then T is called 2‐local isometry.
The following is the main result in this paper.
Theorem 4.1. Let

T be a 2‐local isometry on

C[0,1] . Then T is a 2‐local isom‐etry. Thus Iso_{\mathbb{R}}(C[0,1]) is 2‐local reflexive.

To prove Theorem 4.1, we can reduce the case of T(1)=1 (Proposition 4.1).
When we assume that T(1)=1 , for every element f\in C[0,1] there exists an
isometry T_{1,f} such that T(1)=T_{1,f}(1) . Since T(1)=1 , we get T_{1,f}(1)=1 . By
Theorem 3.1,

T satisfies one of the following equalities\{\begin{ar ay}{l} Tf(t)=T_{1,f} (t)=T_{1,f}(1)fo\varphi_{1,f}(t)=fo\varphi_{1,f}(t) (f\in C[0,1], t\in[0,1]) Tf(t)=T_{1f} (t)=T_{1,f}(1)fo\varphi_{1,f}(t)=fo\varphi_{1,f}(t) (f\in C[0,1], t \in[0,1]) , \end{ar ay}
where \varphi_{1,f} is a homeomorphism. When we put t_{0} such that \varphi_{1,f}(t)=t_{0} , one of

the following equalities\{_ Tf(t)=}^{Tf(t)=}\frac{f(t_{0}) {f(t_{0}) .
Proposition 4.1. Let

T be a 2‐local isometry on

C[0,1] . When T(1)=1, T is a2‐local isometry.

Proof. Let Id be the identity map of C[0,1] . Since

T is a 2‐local isometry,forevery f\in C[0,1] there exists T_{f^{Id}},\in Iso_{\mathbb{R}}(C[0,1]) such that T(f)=T_{f^{Id}},(f) and

TId=T_{f^{Id}},(Id) , also there exists T_{1,Id}\in Iso_{\mathbb{R}}(C[0,1]) such that T(1)=T_{1,Id}(1)and T(Id)=T_{1,Id}(Id) . By Theorem 3.1, T_{f^{Id}}, and T_{1,Id} are represented by\{ begin{ar y}{l T_{f,Idg(t)}=T_{f},Id(1)g\cir \varphif,Id(t) (g\inC[0,1] t\in[0,1]) or T_{f,Idg(t)}=T_{f},Id(1)g\cir \varphi_{f},Id(t) (g\inC[0,1] t\in[0,1]) \end{ar y}

(1)

46

\{ begin{ar y}{l T_{1,Idg}(t)=T_{1},Id(1)go\varphi_{1},Id(t) (g\inC[0,1] t\in[0,1]) or T_{1,Idg}(t)=T_{1},Id(1)go\varphi_{1},Id(t) (g\inC[0,1] t\in[0,1]), \end{ar y}where \varphi_{f^{Id}}, and \varphi_{1,Id} are homeomorphisms on [0,1] respectively. Since T_{1,Id}(1)=T(1)=1, T_{1,Id} is represented by\{ begin{ar y}{l T_{1},Idg(t)=g\cir \varphi_{1},Id(t) (g\inC[0,1] t\in[0,1]) or T_{1,Idg}(t)=go\varphi1,Id(t) (g\inC[0,1] t\in[0,1]) \end{ar y}
We define a set E_{t_{0}f} by

E_{t_{0}f}= \{t\in[0,1] : Tf(t)=Tf(t)=\frac{f(t_{0})}{f(t_{0})}\}(2)for every f\inC[0,1], t_{0}\in[0,1] . Now, E_{t_{0}f} is a subset of [0,1] . By the definition of E_{t_{0}f}, E_{t_{0}Id}
is represented by E_{t_{0}Id}=\{t\in[0,1] : T(Id)(t)=Id(t_{0})\} . Since TId=T_{1,Id}Id

and (2), we get

TId=T_{1,Id}Id=Id\circ\varphi_{1,Id}=\varphi_{1,Id} .

(3)

We get E_{t_{0}Id}=\{t\in[0,1] : \varphi_{1,Id}(t)=t_{0}\} since (3) and Id(t_{0})=t_{0} . Since \varphi_{1,Id} isa homeomorphism, E_{t_{0}Id} is a singleton.
We take b_{t_{0}}\in[0,1] such that \{b_{t_{0}}\}=E_{t_{0}Id} . We have TId(b_{t_{0}})=Id(t_{0})=t_{0} byb_{t_{0}}\in E_{t_{0}Id} and the definition of E_{t_{0}Id} . Therefore we obtain\varphi_{1} , Id (b_{t_{0}})=t_{0}(4)TId(b_{t_{0}})=T_{f,Id}Id(b_{t_{0}})=T_{f^{Id}},(1)Id\circ\varphi_{f^{Id}},(b_{t_{0}})=T_{f,Id}(1)\varphi_{f,Id}(b_{t_{0}})(5)by (3). Forthermore we haveby TId=T_{f^{Id}},Id and (1). By (5) and T(Id)(b_{t_{0}})=t_{0} , we have T_{f^{Id}},(1)\varphi_{f^{Id}},(b_{t_{0}})=t_{0} .

Since

\varphi_{f^{Id}},(b_{t_{0}}) is in [0,1] and t_{0} is in [0,1], T_{f^{Id}}(1)(b_{t_{0}}) is a real number

which is a scalar of modulars 1. we get

T_{f^{Id}},(1)(b_{t_{0}})=1 .

(6)\varphi_{f,Id}(b_{t_{0}})=t_{0} .

(7)

Therefore we obtain

47
We consider

E_{t_{of} = \{t\in[0,1] : Tf(t)=Tf(t)=\frac{f(t_{0})}{f(t_{0})}\}

for every f\in C[0,1] . Since

Tf=T_{f^{Id}},f and (1), we get\{ begin{ar y}{l Tf(b_{t 0})=T_{f^Id},(1)b_{t 0})f\cir \varphi_{f^Id},(b_{t 0}) Tf(b_{t 0})=T_{f^Id},(1)b_{t 0})f\cir \varphi_{f^Id},(b_{t 0}). \end{ar y}By (6), we have\{ begin{ar y}{l Tf(b_{t 0})=f\cir \varphi_{f,Id}(b_{t 0}) Tf(b_{t 0})=f\cir \varphi_{f^Id},(b_{t 0}). \end{ar y}By (7), we have\{_ Tf(b_{t_{0} )=}^{Tf(b_{t_{0} )=}\frac{f(t_{0}) {f(t_{0}) .
Therefore b_{t_{0} is an element of E_{t_{of}} . Since f is an arbitrary element of C[0,1] , we
get E_{t_{0}Id}= \{b_{t_{0}}\}=\bigcap_{f\in C[0,1]}E_{t_{0}f}.

Let \psi be a map [0,1] into [0,1] such that\{\psi(t_{0})\}=\bigcap_{f\in C[0,1]}E_{t_{0}f} .

Since\{b_{t_{0}}\}=\bigcap_{f\in C[0,1]}E_{t_{0}f} , we get \psi(t_{0})=b_{t_{0}} . By TId=T_{1,Id}Id and (2), we have
TId (\psi(t_{0}))=T_{1,Id}Id(\psi(t_{0}))=Id\varphi_{1,Id}(\psi(t_{0}))=\varphi_{1,Id}(\psi(t_{0}))=\varphi_{1,Id}(b_{t_{0}}).

By (4), we get
TId (\psi(t_{0}))=t_{0} .

(8)We will prove that a map \psi is bijective. Let x\in[0,1] be x=\varphi_{1,Id}(y) for every

y\in[0,1] . We obtain b_{\varphi_{1,Id}(y)}=\psi(\varphi_{1,Id}(y))\in E_{\varphi_{1,Id}(y)Id} . We get TId=\varphi_{1,Id}by (3). By TId=\varphi_{1,Id} and (8), we get \varphi_{1,Id}(\psi(\varphi_{1,Id}(y)))=TId(\psi(\varphi_{1,Id}(y)))=\varphi_{1,Id}(y) . Since \varphi_{1,Id} is a homeomorphism, we get \psi(\varphi_{1,Id}(y))=y . Byx=\varphi_{1,Id}(y), y is represented by \psi(x)=y . Therefore \psi is surjective.
We take t_{1},t_{2}\in[0,1] and assume that t_{1}\neq t_{2} . We notice \psi(t_{1})=b_{t_{1}}\in E_{t_{1}f}and \psi(t_{2})=b_{t_{2}}\in E_{t_{2}f}(f\in C[0,1]) . We get TId(\psi(t_{1}))=\varphi_{1,Id}(\psi(t_{1})) by (3).

Since we have TId(\psi(t_{1}))=t_{1}by(8) , we get \varphi_{1,Id}(\psi(t_{1}))=t_{1} . In the same way, we
get \varphi_{1,Id}\psi(t_{2})=t_{2} . By the assumption t_{1}\neq t_{2} , we get \varphi_{1,Id}(\psi(t_{1}))\neq\varphi_{1,Id}(\psi(t_{2})) .

We obtain \psi(t_{1})\neq\psi(t_{2}) . Therefore \psi is injective.

48
By (4) and (7), we get \varphi_{1,Id}(b_{t_{0}})=\varphi_{f,Id}(b_{t_{0}}) . Since b_{t_{0}}=\psi(t_{0})(t_{0}\in[0,1]) ,we have \varphi_{1,Id}(\psi(t_{0}))=\varphi_{f^{Id}},(\psi(t_{0})) . Since \psi is a bijection, for every t\in[0,1] we
represent \varphi_{1,Id}(t)=\varphi_{f,Id}(t) . We get(9)\varphi_{1}, Id=\varphi_{f,Id}.
Let i be a constant function : [0,1]arrow i . A map T is represented by\{ begin{ar y}{l Ti(\psi(t_{0})=i(t_{0})=i or Ti(\psi(t_{0})=\overline{(t_0})=-i \end{ar y}for every t_{0}\in[0,1] . Since \psi is bijective and [0,1] is connected,

T satisfies eitherof the cases(a) T satisfies

Ti=i for everyt\in[0,1]or(b) T satisfies

Ti=-i for everyt\in[0,1].

First, we consider the case (a). We get
TId=T_{f} , Id (1) Id 0\varphi_{f} , Id
=T_{f} , Id (1) \varphi_{f} , Id

for the identity map Id of C[0,1] . By the above equation and (3), we get \varphi_{1,Id}=

T_{f^{Id}},(1)\varphi_{f^{Id}},\cdot By (9), we get T_{f^{Id}},(1)=1 . Since (9) and T_{f^{Id}},(1)=1 , and we get
Tf=T_{f} , Id (1) f\circ\varphi_{f} Id=f\circ\varphi_{f} , Id
=fo\varphi_{1Id}.

Consequently, in the case (a),T is represented by

C[0,1] . Next, we consider the case (b). Let

U

Tf=f\circ\varphi_{1,Id} for every f\in

be a map :C[0,1]arrow C[0,1]such that U=\overline{T} . We notice U is a 2‐local isometry. For the constant functions
1, i\in C[0,1] we haveU(1)=\overline{T(1)}=1 and U(i)=\overline{T(i)}=\overline{-i}=i . we apply thecase (a) to U , we get \overline{Tf}=Uf =f\circ\varphi_{1,Id} . So we get Tf=f\circ\varphi_{1,Id} . Therefore
when T(1)=1 , one of the following equalities\{\begin{ar ay}{l } Tf(t)=f\varphi_{1,Id}(t) (f\in C[0,1], t\in[0,1]) Tf(t)=f\varphi_{1,Id}(t) (f\in C[0,1], t\in[0,1]) . \end{ar ay}
By Theorem 3.1,

T is a surjective real linear isometry on

C[0,1].

\square49
Proposition 4.2. Let|T(1)(t)|=1

T(t\in[0,1])

Proof. SinceT

be a 2‐local isometry on C[0,1] .

Then

T

satisfies.

is a 2‐local isoetry, for every f\in C[0,1] there exists T_{f,1}\in

Iso_{\mathbb{R}}(C[0,1]) such that

T_{f,1}(f)=T(f) and T_{f,1}(1)=T(1) . Since T_{f,1} is an elementof Iso_{\mathbb{R}}(C[0,1]) , there exists T_{f,1}(1) such that |T_{f,1}(1)|=1 . By T_{f,1}(1)=T(1) ,there exists

T(1) such that |T(1)(t)|=1(t\in[0,1]) .

Proposition 4.3. LetT\squarebe a 2‐local isometry on C[0,1] .

Define a map S by

S=\overline{T(1)}T . Then S is a 2‐local isometry on C[0,1]\mathcal{S}uch that S(1)=1.
Proof. Since T is a 2‐local isometry, for every pair of elements f, g\in C[0,1] there
exist T_{f,g}\in Iso_{\mathbb{R}}(C[0,1]) such that T_{f,g}f=Tf and T_{f,g}g=Tg. Define a map

S_{f,g} by S_{f,g}=\overline{T(1)}T_{f,g} . Since T_{f,g} is a real linear isometry, we get that for every\alpha, \beta\in \mathbb{R},u,v\in C[0,1]S_{f,g}(\alpha u+\beta v)=\overline{T(1)}T_{f,g}(\alpha u+\beta v)=\overline{T(1)}(\alpha T_{f,g}(u)+\beta T_{f,g}(v))=\alpha\overline{T(1)}T_{f,g}(u)+\beta\overline{T(1)}T_{f,g}(v))=\alpha S_{f,g}(u)+\beta S_{f,g}(v).

Consequently, S_{f,g} is a real linear map. We get that for every u\in C[0,1]\Vert S_{f,g}(u)\Vert_{\infty}=\Vert\overline{T(1)}T_{f,g}(u)\Vert_{\infty}=\Vert T_{f,g}(u)\Vert_{\infty}=\Vert u\Vert_{\infty}.
So S_{f,g} is an isometry. Since T_{f,g} is a surjective real linear isometry on C[0,1],

T_{f,g} is bijective. There exists a map T_{f,g}^{-1} which is an inverse of T_{f,g} . Define a mapv

by v=T_{f,g}^{-1}T(1)u for every u\in C[0,1] , then v is an element of C[0,1] . We get

S_{f,g}(v)=\overline{T(1)}T_{f,g}T_{f,g}^{-1}T(1)u=u . We notice S_{f,g} is surjentive. Therefore S_{f,g} isa surjective real linear isometry on C[0,1] . By the assumption,
We have

S_{f,g}f=\overline{T(1)}T_{f,g}f=\overline{T(1)}Tf=Sf.

S_{f,g}=\overline{T(1)}T_{f,g}.

50
By the same way, we get S_{f,g}g=Sg. Therefore S is a 2‐local isometry. For theconstant function 1\in C[0,1] we get

S(1)=\overline{T(1)}T(1)=1.

Proof of Theorem 4.1. Let S be a map

S=\overline{T(1)}T .

\squareBy Proposition 4.3, S is a2‐local isometry of C[0,1] such that S(1)=1 . We apply Proposition 4.1 to S,

S

satisfies that one of the following equalities\{_ Sf(t)=}^{Sf(t)=}\frac{f\circ\varphi(t)}{f\circ\varphi(t)} (t\in(t\in[0,1]) [0,1])
where\varphi

is a homeomorphism on [0,1] .

Since

,

S=\overline{T(1)}T , we get T(1)S=

T(1)\overline{T(1)}T=T . Therefore T satisfies that one of the following equalities\{\begin{ar ay}{l } Tf(t)=T(1)f\circ\varphi(t) (f\in C[0,1], t\in[0,1]) Tf(t)=T(1)f\circ\varphi(t) (f\in C[0,1], t\in[0,1]) . \end{ar ay}
By Theorem 3.1,

T is a surjective real linear isometry.

Therefore Iso_{\mathbb{R}}(C[0,1]) is

2‐local reflexive.

\squareAcknowledgements
I would like to thank professor O. Hatori. Without his guidance and persistent
help this paper would not have been possible.

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