38ISOMETRIES ON BANACH ALGEBRAS OF C(Y)‐VALUED MAPSOSAMU HATORIABSTRACT. 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 notionof natural C(Y) ‐valuezations that take values in unital commutative C^{*} ‐algebras.

1. INTRODUCTION

The study on isometries on Banach algebras dates back to the classical Banach‐Stone
theorem. After that there are many literature on the study of isometries not only on Banach
algebras but also Banach spaces of functions and operators. In this paper we study isometries
on the algebra of Lipschitz functions and continuously differentiable functions with values
in unital commutative C^{*} ‐algebras. We propose a unified approach to such a study byconsidering natural C(Y) ‐valuezations.

The study on the space of Lipschitz functions is probably initiated by de Leeuw [10] forfunctions on the real line. Roy [41] considered isometries on the Banach space Lip (K)of Lipschitz functions on a compact metric space K , equipped with the norm \Vert f\Vert_{M}=\max\{\Vert f\Vert_{\infty}, L(f)\} , where L(f) denotes the Lipschitz constant. On the other hand, Cam‐bern [9] studied isometries on spaces of continuously differentiable functions C^{1}([0,1]) withnorm given by \Vert f\Vert=\max_{x\in[0,1]}\{|f(x)|+|f'(x)|\} for f\in C^{1}([0,1]) and exhibited theforms of the surjective isometries supported by such spaces. Rao and Roy [40] proved thatsurjective isometries between Lip ([0,1]) and C^{1}([0,1]) with respect to the norm \Vert f\Vert_{L}=\Vert f\Vert_{\infty}+\Vert f'\Vert_{\infty} are of the canonical forms in the sense that they are weighted composition
operators. Jiménez‐Vargas and Villegas‐Vallecillos in [22] considered isometries of spaces ofvector‐valued Lipschitz maps on a compact metric space taking values in a strictly convex

Banach space, equipped with the norm \Vert f\Vert=\max\{\Vert f\Vert {}_{\infty}L(f)\} , see also [21]. Botelho andJamison [3] studied isometries on C^{{\imath}}([0,1], E) with \max_{x\in[0,1]}\{\Vert f(x)\Vert_{E}+\Vert f'(x)\Vert_{E}\} . See also
[32, 23, 1, 2, 28, 6, 39, 5, 33, 24, 25, 26, 29, 27, 30, 20]. Refer also a book of Weaver [44].

In this paper an isometry means a complex‐linear isometry. Isometries on algebras of
Lipschitz maps and continuously differentiable maps have often been studied independently.

Jarosz [18] and Jarosz and Pathak [19] studied a problem when an isometry on a space ofcontinuous functions is a weighted composition operator. They provided a unified approachfor function spaces such as C^{1}(K), Lip(K), 1ip_{\alpha}(K) and AC[0,1] . In particular, Jarosz [18,Theorem] proved that a unital surjective isometry between unital semisimple commutative
Banach algebras with natural norms is canonical.

2010 Mathematics Subject Classification. 46E40,46B04,46J10,46J15 .
Key words and phrases. isometries, vector‐valued maps, admissible quadruples, vector‐valued Lipschitz
algebras, continuously differentiabıe maps .

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We consider a Banach algebra of continuous maps defined on a compact Hausdorff space
whose values are in a unital C^{*} ‐algebra. It is an abstraction of Lip (K, C(Y)) and C^{1}(K, C(Y)) .
We propose a unified approach to the study of isometries on algebras Lip (K, C(Y)), 1ip_{\alpha}(K, C(Y))

and C^{1}(K, C(Y)) , where K is a compact metric space, [0,1] or \Gamma (in this paper \Gamma denotes theunit circle on the complex plane), and Y is a compact Hausdorff space. We study isometries

without assuming that they preserve unit. We prove that the form of isometries between
such algebras are of a canonical form. As corollaries of the resuıt, we describe isometries on
Lip (K, C(Y)), 1ip_{\alpha}(K, C(Y)), C^{{\imath}}([0,1], C(Y)) , and C^{1}(\Gamma, C(Y)) respectively.

The main result Theorem 14 in [13] with a detailed proof is recaptured as Theorem 9 inthis paper. It gives the form of a surjective isometry U between certain Banach algebraof continuous maps with values in unital C^{*} ‐algebras. Verifying that U(1)=1\otimes h for anh\in C(Y_{2}) with |h|=1 on Y_{2} due to Choquet’s theory (Proposition 10), the Lumer’s method(cf. [16]) works very well. We see that U_{0} is a composition operator of type BJ (cf. [17]).

Then 9 is proved.

This paper surveys recent papers [16] by Hatori and Oi, and [13] by Hatori.
2. PRELIMINARIESLet Y be a compact Hausdorff space and E a real or complex Banach space. The space of
all E ‐valued continuous maps on Y is denoted by C(Y, E) . When E=\mathbb{C} (resp. \mathbb{R} ), C(Y, E)
is abbreviated by C(Y) (resp. C_{R}(Y) ). The supremum norm on S\subset Y is \Vert F\Vert_{\infty(S)}=\sup_{x\in S}\Vert F(x)\Vert_{E} for F\in C(Y, E) . We may omit the subscript S and write only \Vert . \Vert_{\infty} . Let
K be a compact metric space and 0<\alpha\leq 1 .

Put

L_{\alpha}(F)= \sup_{x\neq y}\frac{\Vert F(x)-F(y)\Vert_{E} {d(x,y)^{\alpha} .

for F\in C(K, E) . The number L_{\alpha} is called the \alpha‐Lipschitz number of F . When \alpha=1 weomit the subscript \alpha and write only L(F) and call it the Lipschitz number. We denote

Lip_{\alpha}(K, E)=\{F\in C(K, E) : L_{\alpha}(F)<\infty\}.
When E=\mathbb{C}, Lip(K, \mathbb{C}) is abbreviated to Lip (K) . When \alpha=1 the subscript is omitted and
it is written as Lip (K, E) and Lip (K) . When 0<\alpha<1 the subspace
1ip_{\alpha}(K, E)= { F\in Lip_{\alpha}(K, E) :\lim_{xar ow x_{0} \frac{\Vert f(x_{0})-f(x)\Vert_{E} {d(x_{0},x)^{\alpha} =0 for everyx_{0}\in K }of Lip_{\alpha}(K, E) is called a little Lipschitz space. For E=\mathbb{C}, 1ip_{\alpha}(K, \mathbb{C}) ) is abbreviated to
1ip_{\alpha}(K) . A variety of complete norms on Lip_{\alpha}(K, E) and 1ip_{\alpha}(K, E) exist. The norm \Vert . \Vert_{L}of Lip_{\alpha}(K, E) (resp. 1ip_{\alpha}(K, E) ) is defined by\Vert F\Vert_{L}=\Vert F\Vert_{\infty(K)}+L_{\alpha}(F) ,F\in Lip_{\alpha}(K, E) (resp. 1ip_{\alpha}(K, E) ),which is often called the \ell^{1} ‐norm or the sum norm.

The norm, which is called the \max norm,\Vert . \Vert_{M} of Lip_{\alpha}(K, E) (resp. 1ip_{\alpha}(K, E) ) is defined by\Vert F\Vert_{M}=\max\{\Vert F\Vert_{\infty}, L_{\alpha}(F)\} ,F\in Lip_{\alpha}(K, E) (resp. 1ip_{\alpha}(K, E) ).

Note that Lip_{\alpha}(K, E) (resp. 1ip_{\alpha}(K, E) ) is a Banach space with respect to \Vert . \Vert_{L} and \Vert\Vert_{M} respectively. If E is a unital (commutative) Banach algebra, then the norm \Vert . \Vert_{L} is
sub‐multiplicative. Hence Lip_{\alpha}(K, E) (resp. 1ip_{\alpha}(K, E) ) is a unital (commutative) Banach

40
algebra with respect to the norm \Vert . \Vert_{L} if E a unital (commutative) Banach algebra. The
norm \Vert . \Vert_{M} needs not be sub‐multiplicative even if E is a Banach algebra. With the norm
\Vert\cdot\Vert_{M}, Lip_{\alpha}(K) and 1ip_{\alpha}(K) need not be Banach algebras. In this paper we mainly concernswith \Vert . \Vert_{L} and E=C(Y) . Then Lip_{\alpha}(K, C(Y)) and hp_{\alpha}(K, C(Y)) are unital semisimple
commutative Banach algebras with \Vert\cdot\Vert_{L}.
Let K=[0,1] or \Gamma . We say that F\in C(K, E) is continuously differentiable if there exists
G\in C(K, E) such thatK \ni tar ow t_{0}1\dot{ \imath} m\Vert\frac{F(t_{0})-F(t)}{t_{0}-t}-G(t_{0}) \Vert_{E}=0

for every t_{0}\in K . We denote F'=G . Put

Cı (K, E)= { F\in C(K, E) : F is continuously differentiable}.
Then C^{1}(K, E) with norm 1F\Vert=\Vert F\Vert_{\infty}+\Vert F'\Vert_{\infty} is a Banach space and it is unital (commuta‐tive) Banach algebra provided that E is a unital (commutative) Banach algebra. We mainly
consider the case where E=C(Y) with the supremum norm for a compact Hausdorff space
Y . In this case Cı (K, C(Y)) with the norm \Vert F\Vert=\Vert F\Vert_{\infty}+\Vert F'\Vert_{\infty} for F\in C^{1}(K, C(Y)) is a
unital semisimple commutative Banach algebra. We may suppose that C(Y) is isometricallyisomorphic to \mathb {C} if Y is a singleton, and we abbreviate C^{1}(K, C(Y)) by C^{1}(K) when Y is asingleton.

By identifying C(K, C(Y)) with C(K\cross Y) we may assume that Lip (K, C(Y)) (resp.
1ip_{\alpha}(K, C(Y))) is a subalgebra of C(K\cross Y) by the correspondence

F\in Lip(K, C(Y))rightarrow((x, y)\mapsto(F(x))(y))\in C(K\cross Y).

Under this identification we may suppose that Lip(X, C(Y) ) \subset C(X\cross Y), 1ip_{\alpha}(X, C(Y))\subset
C(X\cross Y) , and C^{1}(K, C(Y))\subset C(K\cross Y) .
Let \emptyset\neq Q\subset C(Y) . We say that Q is point separating or Q separates the points of Y if
for every pair x and y of distinct points in Y , there corresponds a function f\in Q such that
f(x)\neq f(y) . In this paper, unity of a Banach algebra B is denoted by 1. The maximal ideal
space of B is denoted by M_{B}.
3. A THEOREM OF JAROSZ ON ISOMETRIES WHICH PRESERVE 1

In most cases the form of an isometry between Banach algebras depends not only on thealgebraic structure, but also on the norms on theses algebras. Jarosz [18] introduced natural

norms on spaces of continuous functions. He proved that isometries between a variety ofspaces of continuous functions equipped with the natural norms are of canonical forms. See[18] for precise notations and terminologies. The following is a theorem of Jarosz on surjective
unital isometries [18].

Theorem 1 (Jarosz [18]). Let X and Y be compact Hausdorff spaces, let A and B be complex
linear subspaces of C(X) and C(Y) , respectively, and let p, q\in \mathcal{P} . Assume A and B contain
constant functions, and let \Vert . \Vert_{A_{f} \Vert . \Vert_{B} be a p ‐norm and q ‐norm on A and B , respectively.
Assume next that there is a linear isometry T from (A, \Vert . \Vert_{A}) onto (B, \Vert . \Vert_{B}) with Tl=
1.
Then if D(p)=D(q)=0 , or ifA and B are regular subspaces of C(X) and C(Y) ,respectively, then T is an isometry from (A, \Vert \Vert_{\infty}) onto (B, \Vert . \Vert_{\infty}) .

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41
We provide a precise proof of a theorem of Jarosz in [13] by making an ambitious revisionof one in [18].
In the following a unital semisimple commutative Banach algebra A is identified throughthe Gelfand transform with a unital subalgebra of C(M_{A}) for maximal ideal space M_{A} of A.
Hence we see that the uniform closure of a unital semisimple commutative Banach algebrain C(M_{A}) is a uniform algebra on M_{A} . A unital semisimple commutative Banach algebra isregular in the sense of Jarosz [18]. Applying a theorem of Nagasawa [34] (cf. [ı1]) we havethe following.

Corollary 2 (Corollary 2 [13]). Let A and B be unital semisimple commutative Banach

algebras with natural norms. Suppose that T:Aarrow B is a surjective complex‐linear isometry
with T1=1 . Then there exists a homeomorphism \varphi : M_{B}arrow M_{A} such that

T(f)(x)=fo\varphi(x) , f\in A, x\in M_{B}.
In particular,

T is an algebra isomorphism.

We omit a proof (see the proof of Corollary 2 in [13]).

Corollary 3 (Corollary 3 [13]). Let K_{j} be a compact metric space for j=1,2 . Supposethat T:Lip(K_{1})arrow Lip(K_{2}) is a surjective complex‐linear isometry with respect to the norm
\Vert\cdot\Vert_{L} . Assume Tl=1 . Then there exists a surjective isometry \varphi : X_{2}arrow X_{1} such that
(3.1)

Tf(x)=f\circ\varphi(x) , f\in Lip(K_{1}), x\in K_{2}.

Conversely if T : Lip(Kı)

arrow Lip(K2) is of the form as (3.1), then T is a surjective isometry\Vert_{M} and \Vert . \Vert_{L} such that T1=1.

with respect to both of \Vert

We exhibit a proof which is a little bit precise than one given in [13].
Proof. It is well known that (Lip (K_{j}), \Vert\cdot\Vert_{L} ) is a unital semisimple commutative Banach alge‐

bra with maximal ideal space K_{j} . Hence Corollary 2 asserts that there is a homeomorphism\varphi :K_{2}arrow K_{1} such that

(3.2)

Tf(x)=fo\varphi(x) , f\in Lip(K_{1}), x\in K_{2}.

Let y_{0}\in K_{2} . Define f : K{\imath}arrow \mathbb{C} by f(x)=d_{1}(x, \varphi(y_{0})) . Then by a simple calculation we
infer that L(f)=1 . Since T is an isometry with respect \Vert . \Vert_{L} , so is for \Vert . \Vert_{\infty} by Corollary
2. Hence L(Tf)=1 . By the definition of L(\cdot) we have

1=L(Tf) \geq\frac{d_{1}(\varphi(x),\varphi(y_{0})}{d_{2}(x,y_{0})}for every x\in K_{2}\backslash \{y_{0}\} . Thus we have that d_{2}(x, y_{0})\geq d_{1}(\varphi(x), \varphi(y_{0})) for every x\in K_{2} . As
y_{0}\in K_{2} is arbitrary, we have that(3.3)

d_{2}(x, y)\geq d{\imath} (\varphi(x), \varphi(y))

for every pair x and y in K_{2} . By (3.2) we haveT^{-1}g(x)=g\circ\varphi^{-1}(x) , g\in Lip(K_{2}), x\in K_{1}.
Since T^{-1} is an isometry we have the same argument as above that(3.4)

d_{1}(x, y)\geq d_{2}(\varphi^{-1}(x), \varphi^{-1}(y))

42for every pair x and y in K_{1} . By (3.3) and (3.4) we have that \varphi is an isometry.

\squareWe omit a proof of the converse statement since it is trivial.

It is natural to ask what is the form of a surjective isometry between Lip (K) without the
hypothesis of T1=1 . Rao and Roy [40] proved that it is a weighted composition operator if
K=[0,1] . They asked whether a surjective isometry on Lip (K) with respect to the metricinduced by P^{1} ‐norm is induced by an isometry on K . Jarosz and Pathak [19, Example 8]
exhibited a positive solution. After the publication of [19] some authors expressed theirsuspicion about the argument there and the vaıidity of the statement there had not been

confirmed. In [16] we proved that Example 8 in [19] is true. In this paper we exhibit a slight
general result (see also [13]).
Since the \max norm \Vert . \Vert_{M} is not sub‐mutiplicative in general, (Lip (K), \Vert . \Vert_{M} ) need nots\dot{{\imath}}mp1eca1cu1\dot{{\imath}}t\dot{{\imath}}sseethat\Vert\cdot\Vert_{M}isanaturalnorm
wehavet(see[18])
following. Refer the proof of Corollary 4 in [13].

inthesenseofJaroszbeaBanachalgebra. heBy
a such t hat1\dotation{ \imath} m_{tar ow+0}\frac{\max\{1,t\}-ieasyto}{t}=0.ByTheorem lCorollary 4 (Corollary 4 [13]). Let K_{j} be a compact metric space for j=1,2 . Supposethat T:Lip(K_{1})arrow Lip(K_{2}) is a surjective complex‐linear isometry with respect to the norm
\Vert . \Vert_{M} . Assume T1=1 . Then there exists a surjective isometry \varphi : X_{2}arrow X_{1} such that
(3.5)

Tf=fo\varphi, f\in Lip(K_{1}) .

Conversely if T : Lip(Kı) arrow Lip(K2) is of the form as (3.5), then T is a surjective isometry

with respect to both of | \cdot\Vert_{M} and \Vert\cdot\Vert_{L} such that

When

Tl=1

T1=1.

is not assumed in Corollary 4, a simple counterexample such that K_{j} isa two‐point‐set is given by Weaver[ 43 , p.242] (see also [44]) shows that

T neednot be aweighted composition operator,

We have already pointed out [14] that the original proof of Theorem 1 need a revison andmade an ambitious revision in [14, 16]. Although the revised proof for a general case [16]is similar to that of Proposition 7 in [14], a detailed revision is exhibited in [13]. Note that
Tanabe [42] pointed out that \lim_{tarrow+0}(p(1, t)-1)/t always exists and it is finite for every
p-‐norm. To prove Theorem 1 we need Lemma 2 in [18] in the same way as the original proofof Jarosz. We note minor points in the original proof of Lemma 2. Note first that five \varepsilon/2 ’s
between 11 lines and 5 ıines from the bottom of page 69 read as \varepsilon/3 . Next x\in X\backslash U_{1}reads as x\in U_{1} on the bottom of page 69. We point out that the term

\sum_{J^{=1} ^{k_{0-1} (f_{J}(x)-1)

which

appears on the first line of the first displayed inequalities on page 70 reads 0 if k_{0}=1 . The
term 1+\varepsilon on the right hand side of the second line of the same inequalities reads as 1+ \frac{\varepsilon}{3}.
Two \frac{\varepsilon}{2}s on the same line read as \frac{vrepsilon}{3 . On the next line \frac{n+1}{n}\frac{\varepsilon}{2} reads as \frac{\Xi}{3 . For any 1\leq k_{0}\leq n
we infer that

1 \geq 1-2\frac{k_{0}-1}{n}\geq 1-2\frac{n-1}{n}>−ı.

Hence we have |f(x)|\leq 1+\varepsilon if x\in U{\imath} by the first displayed inequalities of page 70. The
inequality \Vert f\Vert_{\infty}\leq\varepsilon on the fifth line on page 70 reads as \Vert f\Vert_{\infty}\leq 1+\varepsilon.

A precise revision of the proof of a theorem of Jarosz (Theorem 1) is given in [13].

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4. BANACH ALGEBRAS OF

C(Y) ‐VALUED MAPSLet X be a compact Hausdorff space and B a unital point separating subalgebra of C(X)equipped with a Banach algebra norm. Then B is semisimple since \{f\in B:f(x)=0\} is a
maximal ideal of B for every x\in X and the Jacobson radical of B vanishes. The inequality\Vert f\Vert_{\infty}\leq\Vert f\Vert_{B} for every f\in B is well known. We say that B is natural Banach algebra if
the map e : Yarrow M_{B} defined by y\mapsto\phi_{y} , where \phi_{y}(f)=f(y) for every f\in B , is bijective.
We say that B is self‐adjoint if B is natural and conjugate‐closed in the sense that f\in B
implies that \overline{f}\in B for every f\in B , where: denotes the complex conjugation on Y.
Let X and Y be compact Hausdorff spaces. For functions f\in C(X) and g\in C(Y) , let
f\otimes g\in C(X\cross Y) be the function defined by f\otimes g(x, y)=f(x)g(y) for (x, y)\in XxY , andfor a subspace E_{X} of C(X) and a subspace E_{Y} of C(Y) , let

E_{X} \otimes E_{Y}=\{\sum_{j=1}^{n}f_{j}\otimes g_{\dot{j} :n\in \mathb {N}, f_{j}\in E_{X}, g_{j}\in E_{Y}\},and

1\otimes E_{Y}=\{1\otimes g:g\in E_{Y}\}.
A natural C(Y) ‐valuezation is introduced in [13].
Definition 5 (Definition ı2 in [13]). Let X and Y be compact Hausdorff spaces. Supposethat B is a unital point separating subalgebra of C(X)equ\underline{i}pped with a Banach algebra
norm \Vert . \Vert_{B} . Suppose that B is self‐adjoint. Sup‐pose that B is a unital point separating
subalgebra of C(X\cross Y) such that B\otimes C(Y)\subset B ‐equipped with a Banach algebra norm
\Vert \Vert_{\overline{B} . Suppose that \overline{B} is self‐adjoint. We say that B is a natural C(Y) ‐valuezation of B if
there exists a compact Hausdorff space \mathfrak{M} and a complex‐linear map D : \overline{B}ar ow C(\mathfrak{M}) suchthat kerD=1\otimes C(Y) and D(C_{R}(X\cross Y)\cap\overline{B})\subset C_{R}(\mathfrak{M}) which satisfies\Vert F\Vert_{\overline{B} =\Vert F\Vert_{\infty(X\cross y)}+\Vert D(F) \Vert_{\infty(\mathfrak{M})} , F\in\overline{B}.
Note that the norm \Vert\cdot\Vert_{\overline{B} is a natural norm in the sense of Jarosz [18].
Let B be a Banach algebra which is a unital separating subalgebra of C(X) for a compact

Hausdorff space X . Then if (X, C(Y), B,\overline{B}) is an admissible quadruple of type L definedin [16], then \overline{B} is a natural -C(Y) ‐valuezation of B due to Definition 5. On the other handfor a C(Y) ‐valuezation of . B of B,(X, C(Y), B,\overline{B}) need not be an admissible quadruple

defined by Nikou and O’Farrell [35] (cf.

[17]). This is because we do not assume that\{F (., y) : F\in\overline{B}, y\in Y\}\subset B , which is a requirement for the admissible quadruple.
The following Examples 6, 7 and 8 are exhibited in [16].

Example 6. Let (K, d) be a compact metric space and Y a compact Hausdorff space. Let
0<\alpha\leq 1 . Suppose that B is a closed subalgebra of Lip ((K, d^{\alpha})) which contains theconstants and separates the points of K , where d^{\alpha} is the Hölder metric induced by d . For a
metric d(\cdot, \cdot) on E , the Hölder metric is defined by d^{\alpha} for 0<\alpha< ı. Then Lip_{\alpha}((K, d), E)
is isometrically isomorphic to Lip ((K, d^{\alpha}), E) . Suppose that \overline{B} is a closed subalgebra ofLip ((K, d^{\alpha}), C(Y)) which contains the constants and separates the points of K\cross Y . Suppose

44
that

B and \overline{B} are self‐adjoint.

Let\mathfrak{M} be the Stone‐Čech compactification of

Suppose that

B\otimes C(Y)\subset\overline{B}.
\{(x, x')\in K^{2} : x\neq x^{I}\}\cross Y . For F\in\overline{B},

D(F) be the continuous extension to \mathfrak{M} of the function (F(x, y)-F(x', y))/d^{\alpha}(x, x') on\{(x, x')\in K^{2} : x\neq x'\}-\cross Y . Then D : \overline{B}ar ow C(\mathfrak{M}) is well defined. We have \Vert D(F)\Vert_{\infty}=
L_{\alpha}(F) for every F\in B . Hence \overline{B} is a natural C(Y) ‐valuezation of B.
There are two typical example of \overline{B} above. The algebra Lip ((K, d^{\alpha}), C(Y)) is one. The
algebras Lip ((K, d^{\alpha})) and Lip ((K, d^{\alpha}), C(Y)) are self‐adjoint (see [16, Corolıary 3]). The

letinclusions

Lip ((K, d^{\alpha}))\otimes C(Y)\subset Lip ((K, d^{\alpha}), C(Y))

is obvious. Another example of a natural C(Y) ‐valuezation is 1ip_{\alpha}(K, C(Y)) for

0<\alpha<

In fact 1ip_{\alpha}(K) (resp. 1ip_{\alpha}(K, C(Y)) ) is a closed subalgebra of Lip ((K, d^{\alpha})) (resp.
Lip ((K, d^{\alpha}), C(Y)) which contains the constants. In this case Corollary 3 in [16] assertsthat 1ip_{\alpha}(K) separates the points of K . As 1ip_{\alpha}(K)\otimes C(Y)\subset 1ip_{\alpha}(K, C(Y)) we see that\overline{B}=1ip_{\alpha}(K, C(Y)) separates the points of K\cross Y . By Corollary 3 in [16] 1ip_{\alpha}(K) and
1ip_{\alpha}(K, C(Y)) are self‐adjoint. The inclusions
1.

1ip_{\alpha}(K)\otimes C(Y)\subset 1ip_{\alpha}(K, C(Y))
is obvious.

Example 7. Let Y be a compact Hausdorff space. Then C^{1}([0,1], C(Y)) is a natural C(Y)valuezation of Cı ([0,1]) , where the norm of f\in C^{1}([0,1]) is defined by \Vert f\Vert=\Vert f\Vert_{\infty}+\Vert f'\Vert_{\infty}
and the norm of F\in C^{1}([0,1], C(Y)) is defined by \Vert F\Vert=\Vert F\Vert_{\infty}+\Vert F'\Vert_{\infty} . It is easy to see thatC^{1} ([0, ı]) \otimes C(Y) \subset C^{1}([0,1], C(Y)) . Let \mathfrak{M}=[0,1]\cross Y and D : C^{1}([0,1], C(Y))arrow C(\mathfrak{M})be defined by D(F)(x, y)=F'(x, y) for F\in C^{1}([0,1], C(Y)) . Then \Vert F'\Vert_{\infty}=\Vert D(F)\Vert_{\infty} for
F\in C^{1}([0,1], C(Y)).

Example 8. Let Y be a compact Hausdorff space. Then C^{1}(\Gamma, C(Y)) ) is a natural C(Y)-valuezation of C^{1}(\Gamma) , where the norm of f\in C^{1}(\Gamma) is defined by \Vert f\Vert=\Vert f\Vert_{\infty}+\Vert f'\Vert_{\infty} andthe norm of F\in C^{1}(\Gamma, C(Y)) is defined by \Vert F\Vert=\Vert F\Vert_{\infty}+\Vert F\Vert_{\infty} . It is easy to see that

Cı (\Gamma)\otimes C(Y)\subset C{\imath} (\Gamma, C(Y)) . Let \mathfrak{M}=\Gamma\cross Y and D : C^{1}(\Gamma, C(Y))arrow C(\mathfrak{M}) be defined by
D(F)(x, y)=F'(x, y) for F\in C^{1}(\Gamma, C(Y)) . Then \Vert F'\Vert_{\infty}=\Vert D(F)\Vert_{\infty} for F\in C{\imath} (\Gamma, C(Y)) .
5. ISOMETRIES ON NATURAL

C(Y) ‐VALUEZATIONS

The following theorem is exhibited in [13, Theorem 14]. It slightly generalize a similar
result for admissible quadruples of type L [ 16 , Theorem 8]. The proof of Theorem 8 in [ı6]applies Proposition 3.2 and the following comments in [17]. Instead of this we can prove
Theorem 9 by Lumer’s method, with which a proof is simpler than one given in [16, Theorem
8]. Refer the detailed proof of Theorm 14 in [13] for a proof of Theorem 9.
Theorem 9 (Theorem 14 in [13]). Suppose that \overline{B_{j} is a natural C(Y_{j}) ‐valuezation of B\subset
C(X_{j}) for j=1,2 . We assume that

\Vert(1\otimes h)F\Vert_{\overline{B,} =\Vert F\Vert_{\overline{B_{J} }45
for every F\in\overline{B_{J} and h\in C(Y_{j}) with |h|=1 on Y_{j} for j=1,2 . Suppose that U : \overline{B_{ \imath} }ar ow\overline{B_{2}is a surjective complex‐linear isometry. Then there exists h\in C(Y_{2}) such that |h|=1 on Y_{2},a continuous map \varphi : X_{2}\cross Y_{2}arrow X_{1} such that \varphi(\cdot, y) : X_{2}arrow X_{1} is a homeomorphism foreach y\in Y_{2} , and a homeomorphism \tau : Y_{2}arrow Y_{1} which satisfy

U(F)(x, y)=h(y)F(\varphi(x, y), \tau(y)) , (x, y)\in X_{2}\cross Y_{2}for every F\in\overline{B_{1} .

The weighted composition operator which appears in Theorem 9 has a peculiar form in thesense that the second variable of the composition part depends only on the second variable.

A composition operator induced by such a homeomorphism is said to be of type BJ in [ı5, 17]
after the study of Botelho and Jamison [7].
6. THE FORM OF

U(1_{\overline{B}_{1} )Throughout this section we assume that \overline{B_{j} is a natural C(Y_{j}) ‐valuezation of B\subset C(X_{j})for j=1,2 and that U : \overline{B_{1} ar ow\overline{B_{2} is a surjective complex‐linear isometry. We assume
that X_{2} is not a singleton in this section. Our main purpose in this section is to show anessence of the proof of Proposition 10, which is a crucial part of proof of Theorem 9. Similar

p—roposition and lemmata for admissible quadruples of type L are proved in [16]. Although
B_{j} in this paper need not be an admissible quadruple of type L , proofs for Proposition 10

and Lemmata 11 and 12 are completely the same as that in [ı6]. Please refer proofs in [16].
Proposition 10. There exists h\in C(Y_{2}) with |h|=1 on Y_{2} such that U(1_{\overline{B}_{1}})=1_{B_{2}}\otimes h.

To prove Pr\underline{op} osition 1 0 we apply Lemma 12. To state Lemma 12 we first define an
isometry from B_{j} into a uniformly closed space of complex‐valued continuous functions. Let
j=1,2 . Define a map

I_{j}:\overline{B_{j} arrow C(X_{j}\cross Y_{j}\cross \mathfrak{M}_{j} \cross\Gamma)
by I_{j}(F)(x, y, m, \gamma)=F(x, y)+\gamma D_{j}(F)(m) for F\in\overline{B_{j} and (x, y, m, \gamma)\in X_{j}\cross Y_{j}\cross M_{j}\cross\Gamma, where \Gamma is the unit circle in the complex plane. For simplicity we just write I and D insteadof I_{j} and D_{j}res\underline{pe} ctively. Scince D is a complex linear map, so is I . Put S_{j}=X_{j}\cross Y_{j}\cross M_{j}\cross\Gamma.
For every F\in B_{j} the supremum norm \Vert I(F)\Vert_{\infty} on S_{j} of I(F) is written as\Vert I(F)\Vert_{\infty}=\sup\{|F(x, y)+\gamma D(F)(m)| : (x, y, m, \gamma)\in S_{j}\}= \sup\{|F(x, y)| : (x, y)\in X_{j}\cross Y_{j}\}+ \sup\{|D(F)(m)| : m\in \mathfrak{M}_{j}\}=\Vert F\Vert_{\infty(X_{j}\cross Y_{j})}+\Vert D(F)\Vert_{\infty(\mathfrak{M}) }.
The second equality holds since \gamma runs through the whole \Gamma . Therefore we have

\Vert I(F)\Vert_{\infty}=\Vert F\Vert_{\infty}+\Vert D(F)\Vert_{\infty}=\Vert F \Vert_{\overline{B_{j} }for every F\in\overline{B_{j} . Since 0=\Vert D(1)\Vert_{\infty} , we have D(\underline{1)}=0 and I(1)=1 . Hence I is a complex‐
linear isometry with I(1)=1 . In particular, I(B_{j}) is a complex‐linear closed subspace ofC(S_{j}) which contains 1. In general I(\overline{B_{j} ) needs not separate the points of S_{j}.

46
By the definition of the Choquet boundary Ch I(\overline{B_{2} \underline{)} of I(\overline{B_{2} ) (see [38]), we see that apoint p=(x, y, m, \gamma)\in X_{2}\cross Y_{2}\cross \mathfrak{M}\cross\Gamma is in Ch I(B_{2}) if the point evaluation \phi_{p} at p is
an extreme point of the state space, or equivalently \phi_{p} is an extreme point of the closed unit
ball (I(\overline{B_{2} ) _{ \imath} ^{*} of the dual space (I(\overline{B_{2} ) ^{*} of I(\overline{B_{2} ) .

Lemma 11. Suppose that (x_{0}, y_{0})\in X_{2}\cross Y_{2} and \mathfrak{U} is an open neighborhood of (x_{0}, y_{0}) .
Then there exists functions b_{0}\in B_{2} and f_{0}\in C(Y_{2}) such that 0\leq F_{0}\leq 1=F_{0}(x_{0}, y_{0}) on
X_{2}\cross Y_{2} and F_{0}<1/2 on X_{2}\cross Y_{2}\backslash \mathfrak{U} , where F_{0}=b_{0}\otimes f_{0} . Furthermore there exists a
point (x_{c}, y_{c}, m_{c}, \gamma_{c}) in the Choquet boundary for I_{2}(\overline{B_{2} ) such that (x_{c}, y_{c})\in \mathfrak{U}\cap F_{0}^{-{\imath}}(1) and\gamma_{c}D(F_{0})(m_{c})=\Vert D(F_{0})\Vert_{\infty}\neq 0.
Note that \gamma_{c}=1 if

D(F_{0})(m_{c})>0 and \gamma_{c}=-1 if D(F_{0})(m_{c})<0.

Lemma 12. Suppose that (x_{0}, y_{0})\in X_{2}\cross Y_{2} and \mathfrak{U} is an open neighborhood of (x_{0}, y_{0}) .
Let F_{0}=b_{0}\otimes f_{0}\in\overline{B_{2}} be a function such that 0\leq F_{0}\leq 1=F_{0}(x_{0}, y_{0}) on X_{2}\cross Y_{2} , and
F_{0}< ı/2 on X_{2}\cross Y_{2}\backslash \mathfrak{U} . Let (x_{c}, y_{c}, m_{c}, \gamma_{c}) be a point in the Choquet boundary for I_{2}(\overline{B_{2} )such that (x_{c}, y_{c})\in \mathfrak{U}\cap F_{0}^{-1}(1) and \gamma_{c}D(F_{0})(m_{c})=\Vert D(F_{0})\Vert_{\infty}\neq 0 . (Such functions and apoint (x_{c}, y_{c}, m_{c}, \gamma_{c}) exist by Lemma 1\underline{1.)} Then for any 0<\theta<\pi/2, c_{\theta}=(x_{c}, y_{c}, m_{c}, e^{x\theta}\gamma_{c})is also in the Choquet boundary for I(B_{2}) .
By Lemma 12 we can prove Proposition 10 in the same way as the proof of Proposition 9
in [ı6].
7. AN APPLICATION OF LUMER’S METHOD FOR A PROOF OF THEOREM 9

To find isometries Lumer [31] introduced a useful method which is now called Lumer’s
method. It involves the notion of Hermitian operators and the fact that UHU‐ı must be
Hermitian if H is Hermitian and U is a surjective isometry. Hermitian operators are usually
defined in the notions of the semi inner product. We define it in an equivalent form. A
Hermitian element is defined for a unital Banach algebra.
Definition 13. Let \mathfrak{A} be a unital Banach algebra. We say that e\in \mathfrak{A} is a Hermitian element
if\Vert\exp(ite)\Vert_{\mathfrak{A}}=1
for every t\in \mathbb{R} . The set of all Hermitian element of \mathfrak{A} is denoted by H(\mathfrak{A}) .
The set of the Hermitian elements H(M_{n}(\mathbb{C})) in the matrix algebra M_{n}(\mathbb{C}) coincides withthe set of all Hermitian matrices, and H(C(Y))=C_{R}(Y) for the algebra C(Y) of all complex
valued continuous functions on a compact Hausdorff space Y . In general, for a unital C^{*} ‐algebra \mathfrak{A} , the space of all Hermitian elements H(\mathfrak{A}) is the space of all self‐adjoint elements of\mathfrak{A} . A Hermitian element of a unital Banach algebra and a Hermitian operator are sometimes
defined in terms of a numerical range, or a semi‐inner product. In this paper we define a
Hermitian operator by an equivalent form (see [12]).

Definition 14. Let E be a complex Banach space. The Banach algebra of all bounded
operators on E is denoted by B(E) . We say that T\in B(E) is a Hermitian operator if
T\in H(B(E)).

47
The following is a trivial consequence.
Proposition 15. Let E_{j} be a complex Banach space for j=1,2 . Suppose that V : E_{1}arrow E_{2}is a surjective isometry and H:E_{l} arrow Eı is a Hermitian operator. Then VHV^{-1} : E_{2}arrow E_{2}is a Hermitian operator.

Suppose that \overline{B} is a natural C(Y) ‐valuezation. Since \Vert| . \Vert| : \overline{B}ar ow \mathbb{R} defined by \Vert|F\Vert|=\Vert D(F)\Vert_{\underline{\infty}(\mathfrak{M})}, F\in\overline{B} is 1‐invariant seminorm (see the definition in \underline{[1}8] ) by the hypothesis

Barrow C(\mathfrak{M}) . Then \Vert F\Vert_{\overline{B} =\Vert F\Vert_{\infty(X\cross Y)}+\Vert D(F) \Vert_{\infty(\mathfrak{M})}, F\in B is a natural norm on\tilde{B} (see the definition in [18]). Then Theorem 1 asserts that a unital surjective isometry V
from \overline{B} onto \overline{B} is an isometry from (\overline{B}, \Vert . \Vert_{\infty}) onto (\overline{B}, \Vert- \cdot \Vert_{\infty}) , too. Hence it is extended
on D :

to a surjective isometry between the uniform closure of B on X\cross Y . The Stone‐Weierstrasstheorem asserts that the uniform closure of \overline{B} is C(X\cross Y) . By the Banach‐Stone theorem
it is an algebra isomorphism. Hence V is an algebra isomorphism. Hence we have
Proposition 16. Any surjective unital complex‐linear isometry on \overline{B} is an algebra isomor‐
phism.

Our method of proving Theorem 9 is to find the Hermitian operators. Applying Proposition

16 we have by Theorem 4 in [15] that
Proposition 17. A bounded operator T on \overline{B} is a Hermitian operator if and only if T(1) is
a Hermitian element in \overline{B} and T=M_{T(1)} , the multiplication operator by T(1) .

By the similar argument as that in the proof of Proposition 6 in [15] we have
Proposition 18. An element F\in\overline{B} is Hermitian if and only if there exists u\in C_{R}(Y) suchthat F=1\otimes u.

Applying propositions above we can prove Theorem 9. Please refer the proof of Theorem

14 in [ı3].
8. APPLICATIONS OF THEOREM 9

We exhibit applications of Theorem 9. Corollaries 19,20,21,22 are exhibited in [16, Section
6]. We omit proofs (see [16, Section 6]).
Corollary 19 (Corollary 14 in [16]). Let (X_{j}, d_{j}) be a compact metric space and Y_{j} acompact Hausdorff space for j=1,2 . Then U : Lip (X_{1}, C(Y_{1}))arrow Lip(X_{2}, C(Y_{2})) (resp.
U : 1ip_{\alpha}(X_{{\imath}}, C(Y_{1}))arrow 1ip_{\alpha}(X_{2}, C(Y_{2}))) is a surjective isometry with respect to the norm
\Vert . \Vert=\Vert . \Vert_{\infty}+L(\cdot) ( resp. \Vert . \Vert=\Vert . \Vert_{\infty}+L_{\alpha}(\cdot)) if and only if there exists h\in C(Y_{2}) with|h|=1 on Y_{2} , a continuous map \varphi : X_{2}\cross Y_{2}arrow X_{1} such that \varphi(\cdot, y) : X2 arrow Xı is a surjective
isometry for every y\in Y_{2} , and a homeomorphism \tau : Y2 arrow Yı which satisfy that

U(F)(x, y)=h(y)F(\varphi(x, y), \tau(y)) , (x, y)\in X_{2}\cross Y_{2}for every

F\in Lip (X_{1}, C(Y_{1}))( resp. F\in 1ip_{\alpha}(X_{1}, C(Y_{1}))) .

Note that if Y_{j} is a singleton in Corollary 19, then Lip (X_{j}, C(Y_{j})) (resp. 1ip_{\alpha}(X_{g}, C(Y_{j})) )is naturally identified with Lip (X_{j}) (resp. 1ip_{\alpha}(X_{j}) ). In this case we have Example 8 of [19].

48
Corollary 20 (Corollary 15 in [16]). [19, Example 8] The map U:Lip(X_{1})arrow Lip(X_{2}) (resp.
U:1ip_{\alpha}(X_{1})arrow 1ip_{\alpha}(X_{2})) is a surjective isometry with respect to the norm \Vert\cdot\Vert=\Vert\cdot\Vert_{\infty}+L(\cdot)( resp. \Vert . \Vert=\Vert . \Vert_{\infty}+L_{\alpha}(\cdot)) if and only if there exists a complex number c with the unit
modulus and a surjective isometry \varphi : X_{2}arrow X_{1} such that

U(F)(x)=cF(\varphi(x)) , x\in X_{2}for every F\in Lip(X_{1}) ( resp. F\in 1ip_{\alpha}(X_{1})) .

Corollary 21 (Corollary 18 in [16]). Let Y_{j} be a compact Hausdorff space for j=1,2.
The norm \Vert F\Vert of F\in C^{1}([0,1], C(Y_{j})) is defined by \Vert F\Vert=\Vert F\Vert_{\infty}+\Vert F'\Vert_{\infty} . Then U :C^{1}([0,1], C(Y_{1}))arrow C^{1}([0, {\imath}], C(Y_{2})) is a surjective isometry if and only if there exists h\in
C(Y_{2}) such that |h|=1 on y_{2} , a continuous map \varphi : [0,1]xY_{2}arrow[0,1] such that for eachy\in Y_{2} we have \varphi(x, y)=x for every x\in [ 0 , ı] or \varphi(x, y)=1-x for every x\in [ 0 , ı], and ahomeomorphism \tau : Y_{2}arrow Y_{1} which satisfy that

U(F)(x, y)=h(y)F(\varphi(x, y), \tau(y)) , (x, y)\in[0,1]\cross Y_{2}for every F\in C^{1}([0,1], C(Y_{1})) .

Note that if Y_{j} is a singleton in Corollary 21, then C^{1} ([0, {\imath}], C(Y_{j})) is C^{1}([0,1], \mathbb{C}) . The

corresponding result on isometries was given by Rao and Roy [40].

Corollary 22 (Corollary 19 in [16]). Let Y_{j} be a compact Hausdorff space for j=1,2.
The norm \Vert F\Vert of F\in C^{1}(\Gamma, C(Y_{j})) is defined by \Vert F\Vert=\Vert F\Vert_{\infty}+\Vert F'\Vert_{\infty} . Suppose that
U : C^{1}(\Gamma, C(Y_{1}))arrow C^{1}(\mathbb{T}, C(Y_{2})) is a surjective isometry if and only if there exists h\in C(Y_{2})such that |h|=1 on Y_{2_{J} a continuous map \varphi : \Gamma\cross Y_{2}arrow\Gamma and a continuous map u:Y_{2}arrow\Gammasuch that for every y\in Y_{2}\varphi(z, y)=u(y)z for every z\in\Gamma or \varphi(z, y)=u(y)\overline{z} for everyz\in\Gamma ,and a homeomorphism \tau : Y_{2}arrow Y_{1} which satisfy that

U(F)(z, y)=h(y)F(\varphi(z, y), \tau(y)) , (z, y)\in\Gamma\cross Y_{2}for every F\in C^{1}(\Gamma, C(Y_{1})) .

Acknowledgements. This work was supported by JSPS KAKENHI Grant Numbers JP16K05172,
JPı5K04921. This work was supported by the Research Institute for Mathematical Sciences,a Joint Usage/Research Center located in Kyoto University
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ISBN: 981‐02‐3873‐8

DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, NIIGATA UNIVERSITY, NIIGATA 950‐2181,
JAPAN

Email address: hatori@math.sc.niigata‐u.ac.jp