# Hermitian Operators and Isometries on Banach Algebras of Continuous Maps with Values in Unital Commutative C*-Algebras.

1. IntroductionIn this paper an isometry means a complex-linear isometry. de Leeuw [1] probably initiated the study of isometries on the algebra of Lipschitz functions on the real line. Roy [2] studied isometries on the Banach space Lip(K') of Lipschitz functions on a compact metric space K, equipped with the max norm [mathematical expression not reproducible], where L(f) denotes the Lipschitz||/||M = max{||/||TO, L(f)}, where L(f) denotes the Lipschitz constant of f. Cambern [3] has considered isometries on spaces of scalar-valued continuously differentiable functions [C.sup.1]([0,1]) with norm given by [mathematical expression not reproducible] and determined a representation for the surjective isometries supported by such spaces. Rao and Roy [4] proved that surjective isometries on Lip([0,1]) and [C.sup.1]([0,1]) with respect to the norm [mathematical expression not reproducible] are of canonical forms in the sense that they are weighted composition operators. They asked whether a surjective isometry on Lip(K') with respect to the sum norm [mathematical expression not reproducible] is induced by an isometry on [mathematical expression not reproducible] induced by an isometry on [mathematical expression not reproducible] for every f [member of] Lip([0,1]). The reason is as follows. Let f [member of] Lip([0,1]). Then f is absolutely continuous. Hence the derivative /7 exists almost everywhere on [0,1], and it is integrable by the theory of the absolutely continuous functions. Furthermore the equality

[mathematical expression not reproducible] (1)

holds. As L(f) < [infinity]we see that f is essentially bounded. In fact,

[mathematical expression not reproducible] (2)

assures that [mathematical expression not reproducible]. By (1) we have

[mathematical expression not reproducible] (3)

It follows that [mathematical expression not reproducible]. We conclude that [mathematical expression not reproducible]. Thus [mathematical expression not reproducible] Jarosz [5] and Jarosz and Pathak [6] studied ||/ ||o([0>1]).) Jarosz [5] and Jarosz and Pathak [6] studied a problem when an isometry on a space of continuous functions is a weighted composition operator. They provided a unified approach for certain function spaces including [C.sup.1](K), Lip(K), [lip.sub.[alpha]](K"), and AC[0,1]. In particular, Jarosz [5, Theorem] proved that a unital isometry between unital semisimple commutative Banach algebras with natural norms is canonical. By a theorem of Jarosz [5] a surjective unital isometry on Lip(K) is an algebra isomorphism when the norm is either the max norm or the sum norm. The situation is very different without assuming the unitality for the isometry with respect to the max norm. There is a simple example of a surjective isometry which is not canonical [7, p.242]. On the other hand, Jarosz and Pathak exhibited in [6, Example 8] that a surjective isometry on Lip(F) with respect to the sum norm is canonical. After the publication of [6] some authors expressed their suspicion about the argument there and the validity of the statement there had not been confirmed until quite recently. Hence the problem on isometries with respect to the sum norm has not been well studied.

Jimenez-Vargas and Villegas-Vallecillos in [8] have considered isometries of spaces of Lipschitz maps on a compact metric space taking values in a strictly convex Banach space, equipped with the norm [parallel]f[parallel] = max{[parallel]f[[parallel].sub.[infinity]], L(f); see also[ 9]. Botelho and Jamison [10] studied isometries on [C.sup.1] ([0,1],E) with max [mathematical expression not reproducible]. See also [11-27]. Refer also to a book of Weaver [28].

We propose a unified approach to the study of isometries with respect to the sum norm on Banach algebras Lip(K,C(Y)), [lip.sub.[alpha]](FT,C(Y)), and [C.sup.1](K,C(y)), where K is a compact metric space, [0,1], or T (T denotes the unit circle on the complex plane), and Y is a compact Hausdorff space. We study isometries without assuming that they preserve unit. As corollaries of a general result we describe isometries on Lip(K,C(Y)), [lip.sub.[alpha]](K,C(Y)), [C.sup.1]([0,1], C(Y)), and [C.sup.1] (T, C(Y)), respectively.

The main result in this paper is Theorem 14, which gives the form of a surjective isometry U with respect to the sum norm between certain Banach algebras with the values in a commutative unital C*-algebra. The proof of the necessity of the isometry in Theorem 14 comprises several steps. The crucial part of the proof of Theorem 14 is to prove that U(1) = 1 [cross product]h for an h [member of] C([Y.sub.2]) with [absolute value of h] = 1 on [Y.sup.2] (Proposition 15). To prove Proposition 15 we apply Choquet's theory (cf. [29]) with measure theoretic arguments. A proof of Proposition 15 is completely the same as that of [30, Proposition 9]. Please refer to it. By Proposition 15 we have that [U.sup.0] = (1 [cross product] [bar.h])\U is a surjective isometry fixing the unit. Then by applying a theorem of Jarosz [5] (Theorem 1 in this paper) we see that [U.sub.0] is also an isometry with respect to the supremum norm. By the Banach-Stone theorem [U.sub.0] is an algebra isomorphism. Then by applying Lumer's method (cf. [30]) we see that [U.sub.0] is a composition operator of type BJ (cf. [31]).

Our proofs in this paper make substantial use of the theorem of Jarosz [5, Theorem]. The author believes that it is convenient for the readers to show a precise proof because there need to be some ambitious changes in the original proof by Jarosz.

2. Preliminaries

Let Y be a compact Hausdorff space. Let E be a real or complex Banach space. The space of all E-valued continuous maps on Y is denoted by C(Y,E). When E = C (resp. R), C(Y, E) is abbreviated by C(Y) (resp. [C.sub.R](Y)). For a subset S of Y, the supremum norm of F on S is [mathematical expression not reproducible] for F [member of] C(Y, E). When no confusion will result we omit the subscript S and write only [parallel] * [parallel]L. Let K be a compact metric space and 0< [alpha] [less than or equal to] 1. For F [member of] C(FC, E), put

[mathematical expression not reproducible] (4)

Then [L.sub.[alpha]] is called an [alpha]-Lipschitz number of F, or just a Lipschitz number of F. When [alpha] = 1 we omit the subscript [alpha] and write only L(F). The space of all F [member of] C(Th F) such that [L.sub.[alpha]](F) < [infinity] is denoted by [Lip.sub.[alpha]](K, E). When [alpha] =1 the subscript is omitted and it is written as Lip(K, E).

When 0 < [alpha] < 1 the closed subspace

[mathematical expression not reproducible] (5)

of [Lip.sub.[alph]](K, E) is called a little Lipschitz space. There are a variety of complete norms on [Lip.sub.[alph]] (K,E) and [lip.sub.[alpha]](K,E). In this paper we are mainly concerned with the norm [parallel]*[[parallel].sub.L] of [Lip.sub.[alpha]] (K,E) (resp., [Lip.sub.[alph]] (K,E)) which is defined by

[mathematical expression not reproducible] (6)

The norm [mathematical expression not reproducible] is defined by

[mathematical expression not reproducible] (7)

Note that Lip(K, E) (resp., [lip.sub.[alpha]](K, E)) is a Banach space with respect to [parallel]*[[parallel].sub.L] and [parallel]*[[parallel].sub.M, respectively. If E is a Banach algebra, the norm [parallel]*[[parallel].sub.L is multiplicative. Hence Lip(K, E) (resp., [lip.sub.[alpha](K, E)) is a (unital) Banach algebra with respect to the norm [parallel]*[[parallel].sub.L] if E is a (unital) Banach algebra. The norm [parallel]*[[parallel].sub.M] fails to be submultiplicative even if E is a Banach algebra. For a metric d(*,*) on K,, the Holder metric is defined by [d.sup.[alpha]] for 0 < [alpha] < 1. [Lip.sub.[alpha]]((K,d),E) is isometrically isomorphic to Lip((K, [d.sup.[alpha]]),E).

We are mainly concerned with E = C(Y) in this paper. Then [Lip.[alpha]](K, C(Y)) and [lip.[alpha]](K C(Y)) are unital semisimple commutative Banach algebras with [parallel] * [[parallel].sub.L], when E = CLip(Th C) (resp., [lip.sub.[alpha]](K,C)) is abbreviated to Lip(K) (resp. [lip.sub.[alpha]](K)).

Let F [member of] C(F, C(Y)) for K = [0,1] or T. We say that F is continuously differentiable if there exists G [member of] C(Th C(Y)) such that

[mathematical expression not reproducible] (8)

for every [t.sub.0] [member of] K. We denote F' = G. Put

[mathematical expression not reproducible] (9)

Then [C.sub.1](K,C(Y)) with norm [mathematical expression not reproducible] is a unital semisimple commutative Banach algebra. If Y is singleton we may suppose that C(Y) is isometrically isomorphic to C and we abbreviate [C.sup.1](K,C(Y)) by [C.sup.1](K).

By identifying C(K, C(Y)) with C(K x Y) we may assume that Lip(K',C(y)) (resp., [lip.[alpha]](K,C(Y))) is a subalgebra of C(K x Y) by the correspondence

[mathematical expression not reproducible]. (10)

Throughout the paper we may suppose that

[mathematical expression not reproducible] (11)

We say that a subset Q of C(Y) is point separating if Q separates the points of Y. The unit of commutative Banach algebra B is denoted by 1. The maximal ideal space of B is denoted by [M.sub.B]. Suppose that B is a unital point separating subalgebra of C(Y) equipped with a Banach algebra norm. Then B is semisimple because [f [member of] B : f(x) = 0} is a maximal ideal of B for every x [member of] X and the Jacobson radical of B vanishes.

3. A Theorem of Jarosz Revisited:

Isometries Preserving Unit

Whether an isometry between unital semisimple commutative Banach algebras is of the canonical form depends not only on the algebraic structures of these algebras, but also on the norms in these algebra in most cases. A simple example is a surjective isometry on the Wiener algebra, which need not be canonical. Jarosz [5] defined natural norms and provided a theorem that isometries between a variety of algebras equipped with natural norms are of canonical forms. For the sake of completeness we outline the notations and the terminologies which are due to [5]. The set of all norms on[ R.sup.2] with p(1,0) = 1 is denoted by P. For p [member of] P we put

[mathematical expression not reproducible] 12)

Recently Tanabe pointed out by a private communication that D(p) exists and it is finite for every p [member of] P. (In fact, it is easy to see that (p(1,t) - 1)/t is increasing since p(1,t) is convex. We also see that [inf.sub.t>0]((p(1, t) - 1)/t) > -[infinity].) Let X be a compact Hausforff space and A a liner subspace of C(X) which contains constant functions. A seminorm [mathematical expression not reproducible] on A is called one-invariant (inthesense of Jarosz) if [mathematical expression not reproducible] for all f [member of] A. Let p [member of] P. A norm [parallel]*[parallel] on A is called a p-norm if there is a one-invariant seminorm [mathematical expression not reproducible] on A such that [mathematical expression not reproducible]. A natural norm is a p-norm for some p [member of] P.

Theorem 1 (Jarosz [5]). 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 [member of] P. Assume A and B contain constant functions, and let [mathematical expression not reproducible] be a p-norm and q-norm on A and B, respectively. Assume next that there is a linear isometry T from (A, [parallel]*[[parallel].sub.A]) onto (B, [parallel]*[[parallel].sub.A]) with T1 = 1. Then if D(p) = D(q) = 0, or if A and B are regular subspaces of C(X) and C(Y), respectively, then T is an isometry from (A, [parallel]*[[parallel].sub.[infinity]]) onto (B, [parallel]*[[parallel].sub.[infinity]]).

In the sequel a unital semisimple commutative Banach algebra A is identified via the Gelfand transforms with a subalgebra of C([M.sub.A]). A unital semisimple commutative Banach algebra is regular (in the sense of Jarosz [5]). Hence we have by a theorem of Nagasawa [32] (cf. [33]) that the following holds.

Corollary 2. Let A and B be unital semisimple commutative Banach algebras. Assume they have natural norms, respectively. Suppose that T : A [right arrow] B is a surjective complex-linear isometry with T1 = 1. Then there exists a homeomorphism [phi] : [M.sub.B] [right arrow] [M.sub.A] such that

[mathematical expression not reproducible] (13)

Proof. A unital semisimple commutative Banach algebra is regular by Proposition 2 in [5]. Then Theorem 1 ensures that T is a surjective linear isometry from (A, A, [parallel]*[[parallel].sub.[infinity]]) onto (B, [parallel]*[[parallel].sub.[infinity]]).It is easy to see that T is extended to a surjective linear isometry [??] from the uniform closure [bar.A] of A onto the uniform closure [bar.B] of B. Then a theorem of Nagasawa asserts that there exists a homeomorphism [mathematical expression not reproducible] we have the conclusion.

Corollary 3. Let [K.sub.j] be a compact metric space for j = 1,2. Suppose that T : Lip([K.sub.1]) [right arrow] Lip([K.sub.2]) is a surjective complex-linear isometry with respect to the norm A, [parallel]*[[parallel].sub.L]. Assume T1 = 1. Then there exists a surjective isometry [phi] : [X.sub.2] [right arrow] [X.sub.1] such that

[mathematical expression not reproducible]14)

Conversely if T : Lip([K.sub.1]) [right arrow] Lip([K.sub.2]) is of the form as (14), then T is a surjective isometry with respect to both of [parallel] * [[parallel].sub.M] and [parallel] * [[parallel].sub.L] such that T1 = 1. Proof. As (Lip(Kj), [parallel] * [[parallel].sub.L]) is a unital semisimple commutative Banach algebra with maximal ideal space [K.sub.j], Corollary 2 asserts that there is a homeomorphism [phi] : [K.sup.2] [right arrow] [K.sub.1] such that

[mathematical expression not reproducible]. (15)

Then by a routine argument we see that [phi] is an isometry.

Converse statement is trivial.

Without assuming T1 = 1, we have that T is a weighted composition operator. We exhibit a general result as Theorem 14 (see also [30]).

(Lip(K), [parallel] * [[parallel].sub.M]) need not be a Banach algebra since [parallel] * [[parallel].sub.M] need not be submultiplicative. On the other hand, [parallel] * [[parallel].sub.M] is a natural norm in the sense of Jarosz (see [5]) such that [mathematical expression not reproducible] ((max{1, t} - 1)/t) = 0. Then by Theorem 1 we have the following.

Corollary 4. Let [K.sub.j] be a compact metric space for j = 1,2. Suppose that T : Lip([K.sub.1]) [right arrow] Lip([K.sub.2]) is a surjective complexlinear isometry with respect to the norm [parallel] * [[parallel].sub.M]. Assume T1 = 1. Then there exists a surjective isometry p : [X.sub.2] [right arrow] [X.sub.1] such that

Tf = f O [phi], f [member of] Lip([K.sub.1]). (16)

Conversely if T : Lip([K.sub.1]) [right arrow] Lip(K2) is of a similar form as (16), then T is a surjective isometry with respect to both of [parallel] * [[parallel].sub.M]|| and [parallel] * [[parallel].sub.L] such that T1 = 1.

Proof. As [parallel] * [[parallel].sub.M] is a natural norm, we have by Corollary 2 that there is a homeomorphism [phi] : [K.sub.2] [right arrow] [K.sub.1] such that

[mathematical expression not reproducible] (17)

Then by a routine argument we see that [phi] is an isometry.

Converse statement is trivial.

Without the assumption that T1 = 1 in Corollary 4, one may expect that T is a weighted composition operator. But it is not the case. A simple counterexample is given by Weaver [7, p.242] (see also [28]).

As is pointed out in [34] the original proof of Theorem 1 needs a revision in some part and a proof when A and B are algebras of Lipschitz functions is revised [34, Proposition 7]. Although a revised proof for a general case is similar to that of Proposition 7 in [34], we exhibit it here for the sake of completeness of this paper. To prove Theorem 1 we need Lemma 2 in [5] in the same way as the original proof ofJarosz. The following is Lemma 2 in [5].

Lemma 5 (Jarosz [5]). Assume A is a regular subspace of C(X) with 1 [member of] A and let [x.sub.0] [member of] Ch(A). Then for any [epsilon] >0 and any open neighborhood U of [x.sub.0], there isan f [member of] A such that

[mathematical expression not reproducible] (18)

Proof. The proof is essentially due to the original proof of Lemma 2 in [5]. Several minor changes are needed. We itemize them as follows.

(i) Five [epsilon]/2's between 11 lines and 5 lines from the bottom of page 69 read as [epsilon]/3.

(ii) Next x [epsilon] X\[U.sub.1] reads as x [epsilon] [U.sub.1] on the bottom of page 69.

(iii) We point out that the term [mathematical expression not reproducible] which appears on the first line of the first displayed inequalities on page 70 reads 0 if [k.sub.0] = 1.

(iv) The term 1 + [epsilon] on the right hand side of the second line of the same inequalities reads as 1 + [epsilon]/3.

(v) Two [epsilon]/2's on the same line read as [epsilon]/3.

(vi) On the next line ((n + 1)/n)([epsilon]/2) reads as [epsilon]/3.

(vii) For any 1 [less than or equal to] [k.sub.0] [less than or equal to] n we infer that

[mathematical expression not reproducible](19)

Hence we have [absolute value of [f(x)] [less than or equal to] 1 + [epsilon] if x [member of] [U.sub.1] by the first displayed inequalities of page 70.

(viii) The inequality [parallel]f[parallel].sub.[infinity]] [less than or equal to] [epsilon] on the fifth line on page 70 reads as [parallel]f[parallel].sub.[infinity]] [less than or equal to] [epsilon]

Let K be a nonempty convex subset of the complex plane and [phi] [epsilon][0,2[pi]). Put

[mathematical expression not reproducible] (20)

Note that we may write

[mathematical expression not reproducible]. (21)

Let A be a subspace of C(X) for a compact Hausdorff space. For f [member of] Awe put [mathematical expression not reproducible], where co(*) denotes the closed convex-hull. We define the functions

[mathematical expression not reproducible] (22)

Proof of Theorem 1. Let f [member of] A. First we note that

[mathematical expression not reproducible] (23)

since [??](f) is the closed convex-hull of a compact set [sigma](f) = f(X). We prove the inequalities

[mathematical expression not reproducible] (24)

which appear on p. 68 in [5]. Put [mathematical expression not reproducible] is compact, there exists be [member of] such that [mathematical expression not reproducible]. Hence

[mathematical expression not reproducible] (25)

As

[mathematical expression not reproducible] (26)

we have

t + [c.sub.A] (f,[phi])[less than or equal to] [r.sub.A] (f,t,[[phi]] (27)

Let x [member of] X. By the definition of s = [c.sub.A](f,<p), we infer that [Re([e.sup.-1[phi]], f(x)) [less than or equal to] s, hence we have [Re([e.sup.-1[phi]], f(x)) [less than or equal to] t s for every t [greater than or equal to] 0. Then

[mathematical expression not reproducible]. (28)

Letting M = max {[absolute value of t + s], [Re([e.sup.-1[phi]], f(x)}, we have

[mathematical expression not reproducible] (29)

As x [member of] X is arbitrary we have

[mathematical expression not reproducible] (30)

It follows that (24) holds. In the same way we have

[mathematical expression not reproducible] (31)

for every g [member of] B. By (24) and (31) we infer that

[mathematical expression not reproducible] (32)

As [mathematical expression not reproducible] is 1-invariant we have

[mathematical expression not reproducible] (33)

As T is an isometry, T(1) = 1, and [mathematical expression not reproducible] is 1-invariant, we Have

[mathematical expression not reproducible] (34)

Thus

[mathematical expression not reproducible] (35)

It follows that

[mathematical expression not reproducible] (36)

Recall that[mathematical expression not reproducible]

[mathematical expression not reproducible] (37)

Suppose that D(p) = D(q) = 0. Then we have by (37) that [c.sub.B] (Tf, [phi]) = [c.sub.A] (f, [phi]) for every f [member of] A and [phi] [member of] [0,2[pi]). By Lemma 1 in [5] we infer that [mathematical expression not reproducible]. Thus we have [mathematical expression not reproducible]. We have proved that T is an isometry from [mathematical expression not reproducible]

Suppose that A and B are regular subspaces of C(X) and C(Y), respectively. Let f [member of] A. Put

[mathematical expression not reproducible]. (38)

Suppose that [DELTA]f [greater than or equal to] 0. For any r [greater than or equal to] 0 and any nonempty compact convex subset K [subset] C, we have that

c(K + K(r), [phi]) = c(K, [phi]) + r (39)

for all ([phi] [member of] [0, 2[pi]), where K(r) = {z [member of] C :[absolute value of z] [less than or equal to] r}. Then by (37) we have

[mathematical expression not reproducible] (40)

for all [phi] [member of] [0,2[pi]). It follows by Lemma 1 in [5] that

[mathematical expression not reproducible], (41)

and therefore

[mathematical expression not reproducible](42)

If [DELTA]f < 0, then a similar calculation shows that

[mathematical expression not reproducible] (43)

and

[mathematical expression not reproducible] (44)

It follows that in any case ([DELTA] f [greater than or equal to] 0, [DELTA]f [less than or equal to] 0) we obtain

[mathematical expression not reproducible] (45)

We will prove that

[mathematical expression not reproducible] (46)

for all f [member of] A. Once it is proved, applying the same argument for [T.sup.-]1 instead of T, we see that [mathematical expression not reproducible] for every g [member of] B. As T is abijection, it follows that [mathematical expression not reproducible] for every f [member of] A. It will follow that [mathematical expression not reproducible] for every f [member of] A. A proof of (46) is the following. For every [epsilon] >0, denote

[mathematical expression not reproducible] (47)

The inequality in (46) is deduced by the following assertions which appear in the proof of [5, Theorem]:

(1) T is a continuous mapping from (A, [parallel] *[[parallel].sub.[infinity]] onto (B, [parallel] *[[parallel].sub.[infinity]]).

(2) For each [epsilon] > 0, the set [A.sub.e] is dense in (A, [parallel] *[[parallel].sub.[infinity]]).

(3) For each [epsilon] > 0 and each f [member of] [A.sub.e], it holds that [mathematical expression not reproducible]

Suppose that these assertions are proved. Let f [member of] A. By (2), for any [member of] > 0, there is a sequence {[f.sub.n]} of functions in [A.sub.[epsilon] such that [mathematical expression not reproducible]. By (3) we have

[mathematical expression not reproducible] (48)

for every n. Letting n [right arrow] [infinity] we have

[mathematical expression not reproducible] (49)

by (1). As [epsilon] > 0 is arbitrary, we have that r/L -ii/l >0 (50)

We show proofs of three assertions (1), (2), and (3) above precisely. The proof of (1) is slightly different from the corresponding one in [5, p. 70]. This change is rather ambitious. We also point out that the terms -[pi]/2 and [pi]/2 which appear in the formulae (7) and (8) in [5] seem inappropriate; they read, for example, as 3[pi]/4 and [pi]/4, respectively.

We now proceed to prove the first statement. Aiming for a contradiction, suppose that T is not continuous from ([mathematical expression not reproducible]. Let [epsilon] be a positive real number less than 1/100. Then there is a function [y.sub.0] [member of] A such that [mathematical expression not reproducible] Then there exist [y.sub.0] [member of] Ch(B) such that [absolute value of ]T([f.sub.0])([y.sub.0])| = 1 by [29, Proposition 6.3]. Since T is complex-linear we may suppose that T([f.sub.0])([y.sub.0]) = 1.

By (41) and (45), we deduce that [mathematical expression not reproducible], we infer that [mathematical expression not reproducible]. Thus

[mathematical expression not reproducible]. (51)

Hence [mathematical expression not reproducible]. (52)

Consider the open neighborhood [U.sub.0] of [y.sub.0] in [X.sub.2] given by

[mathematical expression not reproducible] (53)

We infer that [U.sub.0] is a proper subset of X2 by (52). Then, by [5, Lemma 2], there exists g [member of] S such that [mathematical expression not reproducible] for every y [member of] [X.sub.2] [U.sub.0] and |Im g(y)| < [epsilon] for all y [member of] [X.sub.2]. If H denotes the closed rectangle whose vertices are the four points [not equal to](1 + [epsilon]) [+ or -] ei, we have

[??] (g) [subset] H (54)

Consider now the set

[mathematical expression not reproducible]. (55)

We claim that T([f.sub.0])([X.sub.2])[intersection] L [not equal to] [theta]. Suppose that T([f.sub.0])([X.sub.2]) [intersection] L = [theta]. As T ([f.sub.0])([X.sub.2]) is compact, there exists a positive integer n such that [mathematical expression not reproducible]. Then (52) gives [mathematical expression not reproducible] is the closed convex-hull of T([f.sub.0])([X.sub.2]), it is contained in the closed convex set X(1) \ [L.sub.n]. On the other hand, [mathematical expression not reproducible]. As [mathematical expression not reproducible], this contradicts [mathematical expression not reproducible], and this proves our claim. Hence there is [y.sub.1] [subset] [X.sub.2] with T([f.sub.0]))([y.sub.1]) [member of] L. As [epsilon] [less than or equal to] 1/100, it follows that [absolute value of T([f.sub.0])([y.sub.1]) 1] [greater than or equal to] [epsilon] and so [y.sub.1] [member of] [X.sub.2] \ [U.sub.0]. Hence [absolute value of g([y.sub.1]) +1|] < [epsilon]. Thus g([y.sub.1]) + T([f.sub.0])([y.sub.1]) is in L -1 + K([epsilon]). Thus we have

[mathematical expression not reproducible]. (56)

We claim that

[mathematical expression not reproducible], (57)

where [mathematical expression not reproducible]. Let y [member of] [X.sub.2]. Suppose first that [absolute value of ]T(f0)(y)-1] < [epsilon]. Since g([X.sub.2]) [subset] H by (54), we have

T ([f.sub.0])(y) + g (y) [member of] K (1, [epsilon]) + Th = (Th +1) + K ([epsilon]). (58)

Suppose next that [absolute value of T ([f.sub.0])(y) - 1] [greater than or equal to] [epsilon]. Then y [member of] [X.sub.2] \ [U.sub.o] and so [absolute value of g(y) + 1]0 < [epsilon]. Moreover, [absolute value of T([f.sub.0])(y)] < 1. Therefore we have

[mathematical expression not reproducible] (59)

It follows from (58) and (59) that

[mathematical expression not reproducible], (60)

and hence

[mathematical expression not reproducible] (61)

as is claimed. Therefore we have

[c.sub.B](g + T ([f.sub.0]), [pi]/4) [less than or equal to] [square root of 2] + 3[epsilon]. (62)

Put [f.sub.1] = [T.sup.-1](g). We claim that [DELTA][f.sub.1] [greater than or equal to] [epsilon]. If Af1 < 0, there is nothing to prove. Suppose that [DELTA][f.sub.1] [greater than or equal to] 0. Then, by (41), we have

[mathematical expression not reproducible] (63)

Since [mathematical expression not reproducible] (54), we have

[mathematical expression not reproducible] (64)

As H does not include a closed disk with the radius greater than [epsilon], we conclude that [DELTA] [f.sub.1] [less than or equal to] [epsilon].

In the following we will consider two cases: 0 < Af1 < e and [DELTA] [f.sub.1] [less than or equal to] 0. Suppose first that 0 [less than or equal to] [DELTA][f.sub.1] [less than or equal to] [epsilon]. Then (64) yields [mathematical expression not reproducible] we deduce that [mathematical expression not reproducible]. Hence we have

[mathematical expression not reproducible] (65)

Since H + K ([epsilon]) is convex we have

[mathematical expression not reproducible]. (66)

From (39) we infer that

[mathematical expression not reproducible] (67)

Since T([f.sub.1] + [f.sub.0]) = g + T([f.sub.0]), from (56) and (67) we obtain that 1

[mathematical expression not reproducible] (68)

By (63) and 1 = g([y.sub.0]), we deduce that [mathematical expression not reproducible]. Thus there is z [mathematical expression not reproducible] such that [absolute value of z - 1] [less than or equal to] [DELTA][f.sub.1]. It follows that [mathematical expression not reproducible]); hence we have

[mathematical expression not reproducible] (69)

[mathematical expression not reproducible]. We get by (62) and (69) that

[mathematical expression not reproducible] (70)

On the other hand, [c.sub.B](T(f), [phi]) - [c.sub.A](f, [phi]) is invariant for any [phi] by (37). From (68) and (70) we deduce that [epsilon] [greater than or equal to] (2 - [square root of 2])/2(9 + [square root of 2]) and this contradicts that [epsilon] < 1/100.

For the second case, suppose next that [DELTA][f.sub.1] [less than or equal to] 0. Then, by (43), we have

[mathematical expression not reproducible] (71)

and, by (54), it follows that [mathematical expression not reproducible]. Moreover, [mathematical expression not reproducible]. Then

[mathematical expression not reproducible] (72)

Hence, [mathematical expression not reproducible]. Using (39), we infer that

[mathematical expression not reproducible] (73)

By (71), we obtain that [mathematical expression not reproducible], and, as g([y.sub.0]) = 1, we infer that [mathematical expression not reproducible]. Hence [mathematical expression not reproducible, so that

[mathematical expression not reproducible] (74)

as [mathematical expression not reproducible , we obtain by (56) and (73) that

[mathematical expression not reproducible] (75)

We also obtain by (62) and (74) that

[mathematical expression not reproducible] (76)

Since [c.sub.B](T(f), [c.sub.A]) (f, [phi]) is invariant for any [phi] by (37), from (75) and (76) we deduce that [epsilon] [greater than or equal to] (2 - [square root of 2])/2(8 + [square root of 2]) and this is impossible since [epsilon] [less than or equal to] 1/100.

Next we show a proof of the second assertion (2). Let f [member of] A. We prove that there exists a sequence {[f.sub.n]} c A which uniformly converges to f such that [mathematical expression not reproducible]. Without loss of generality we may assume that [parallel]f[[parallel].sub.[infinity] = 1. Then there exists [y.sub.0] [member of] Ch(A) such that [absolute value of ]f([y.sub.0])] = [parallel]f[[parallel].sub.[infinity] by [29, Proposition 6.3]. We may assume that f([y.sub.0]) = 1. Suppose that n [greater than or equal to] 4. Put

[mathematical expression not reproducible] (77)

and

[mathematical expression not reproducible] (78)

(In the following we identify [R.sup.2] and C; that is, we identify (x, y) and x + iy for every x, y [member of] R.) Since we assume that n [less than or equal to] 4we infer by a simple calculation that

[mathematical expression not reproducible] (79)

for [[delta].sub.n] with 0 < [[delta].sub.n] [less than or equal to] 1/[n.sup.2]. We assume that 0 < [[delta].sub.n] [less than or equal to] 1/[n.sup.2]. By [5, Lemma 2] there exists [h.sub.n] [member of] A such that [mathematical expression not reproducible]

[mathematical expression not reproducible] (80)

Hence

[mathematical expression not reproducible] (81)

and

[mathematical expression not reproducible] (82)

Hence

[mathematical expression not reproducible] (83)

It follows that we have

[mathematical expression not reproducible] (84)

for x [member of] [U.sub.n]. Suppose that x [member of] X \ Un. Then

[mathematical expression not reproducible] (85)

and hence

[mathematical expression not reproducible] (86)

for x [member of] X \ [U.sub.n]. Since 1 + 1/n = f([y.sub.0]) + [g.sub.n]([y.sub.0])/n, we have by combining (84) and (86) that

[mathematical expression not reproducible] (87)

As [OMEGA] is convex we obtain

[mathematical expression not reproducible]. (88)

Recall that for r [greater than or equal to] 0 and a complex number [z.sub.0]

[mathematical expression not reproducible] (89)

denotes the closed disk with center [z.sub.0] and radius r. We observe that [rho[([OMEGA], 1 + 1/n) = 1/n + 1/[n.sup.2]. Recall that

[mathematical expression not reproducible] (90)

Let [l.sub.1] be the line defined by the equation

[mathematical expression not reproducible] (91)

part of which is a part of the boundary of [OMEGA]. Let [l.sub.2] be the line defined by the equation

[mathematical expression not reproducible] (92)

By some calculation we have that the distance between 1 1/[n.sup.2] and 1 + 1/n is 1/n+1/[n.sup.2] and it coincides with the distance between the point 1 - 1/[n.sup.2] and the line [l.sub.1]. Hence we see that

[mathematical expression not reproducible]. (93)

Thus 1/n+ 1/[n.sup.2] < [rho]([OMEGA], 1 + 1/n).

Next we prove that [mathematical expression not reproducible]. It will follow that 1/n + 1/[n.sup.2] [greater than or equal to] [rho]([OMEGA], 1 + l/n) and the equality will hold. Let [mathematical expression not reproducible]. We prove the case where [y.sub.p] [greater than or equal to] 0. A proof for the case where [y.sub.p] [less than or equal to] 0 is the same and we omit it. We divide [mathematical expression not reproducible] into two parts:

[mathematical expression not reproducible] (94)

and

[mathematical expression not reproducible] (95)

Suppose that p [member of] [[OMEGA].sub.1] and r = [absolute value of p-(1 + 1/n)] > 1/n + l/[n.sup.2]. The distance between [l.sub.1] and [l.sub.2] is 1/n + 1/[n.sup.2]. Hence K(p, r) [intersection] [[OMEGA].sup.c] [not equal to] [empty set]. Suppose that [mathematical expression not reproducible]; that is, p [not equal to] 1-1/[n.sup.2]. Let l' be the line passing through p which is parallel to l2. Let p' be the unique point in the intersection of 1 and the v-axis. Then p' = 1 - 1/[n.sup.2] - u for some u[less than or equal to] 0. Then the distance between [l.sub.1] and l is 1/ n + 1 / [n.sup.2] + u /(1 + 1/n), which is equal to the distance between the point p and the line [l.sup.1]. On the other hand

[mathematical expression not reproducible] (96)

It follows that K(p, r) [intersection] [[OMEGA].sup.c] [not equal to] [theta]. We conclude that if p [member of] [OMEGA] satisfies [mathematical expression not reproducible]. Thus we have

[mathematical expression not reproducible] (97)

Since [mathematical expression not reproducible], we have

[mathematical expression not reproducible] (98)

On the other hand, [mathematical expression not reproducible] ensures that [mathematical expression not reproducible]. It follows that, for every [epsilon] > 0, [A.sub.[epsilon]] is dense in (A, [parallel]*[[parallel].sub.[infinity]]).

Finally we show a proof of the third assertion (3). As is pointed out in the proof of [5, Theorem], [rho](co(K) + K(c)) [greater than or equal to] c for any K [subset] C and any c [greater than or equal to] 0. Let [epsilon] >0 and f [member of] [A.sub.[epsilon]]. Suppose that [DELTA]f [less than or equal to] 0. Then by (43) we have [mathematical expression not reproducible]. Hence we have

[mathematical expression not reproducible] (99)

As [mathematical expression not reproducible], we conclude by (45) that

[mathematical expression not reproducible] (100)

This completes the proof of the theorem.

4. Hermitian Operators on a Banach Algebras of Continuous Maps Whose Values Are in a Uniform Algebras

Let X and Y be compact Hausdorff spaces. Let B be a unital subalgebra of C(X) which separates the points of X. Throughout this section we assume B is a Banach algebra with the norm [parallel]*[[parallel].sub.B] and A is a uniform algebra on Y. Recall that a uniform algebra on Y is a uniformly closed subalgebra of C(Y) which contains constants and separates the points of Y. For functions f [member of] C(X) and g [member of] C(Y), let f [cross product] g [member of] C(X x Y) be the function defined by f [cross product] g(x,y) = f(x)g(y) for (x,y) [member of] X x Y, and for a subspace [E.sub.X] of C(X) and a subspace [E.sub.Y] of C(Y), put

[mathematical expression not reproducible] (101)

and

[mathematical expression not reproducible] (102)

Throughout the section [??] is a unital subalgebra of C(X x Y) with a Banach algebra norm [mathematical expression not reproducible]. We assume that B [cross product] A [subset] [??]. Note that [??] separates the points of XxY since A separates the points of Y and B separates the points of X. We assume that there exists a compact Hausdorff space M and a complex-linear map [mathematical expression not reproducible] such that ker D = 0 A. We assume that [mathematical expression not reproducible]. Hence D is continuous. Defining

[mathematical expression not reproducible], (103)

[mathematical expression not reproducible] is a one-invariant seminorm in the sense of Jarosz; [mathematical expression not reproducible] is a seminorm on [??] such that [mathematical expression not reproducible] for every F [member of] [??]. Hence the norm [mathematical expression not reproducible] is a natural norm (see [5, p.67]) Note that [??] is a regular subspace of C(Xx Y) in the sense of Jarosz [5, Proposition 2].

Lumer's seminal paper [35] opened up a useful method of finding isometries which is often referred to as Lumer's method. It involves the notion of Hermitian operators and the fact that [UHU.sup.-1] must be Hermitian if H is Hermitian and U is a surjective isometry.

Definition 6. Let A be a unital Banach algebra. We say that e [member of] A is a Hermitian element if

[mathematical expression not reproducible] (104)

for every t [member of] R. The set of all Hermitian elements of A is denoted by H(A).

If A is a unital C*-algebra, then H(A) is the set of all self-adjoint elements of A. Hence H([M.sub.n](C)) is the set of all Hermitian matrices, and H(C(Y)) = [C.sub.R](Y).

Definition 7. 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 [member of] B(E) is a Hermitian operator if T [member of] H(B(E)).

Note that a Hermitian element of a unital Banach algebra and a Hermitian operator are usually defined in terms of numerical range or semi-inner product. Here we define them by an equivalent form (see [36]). By the definition of a Hermitian operator we have the following.

Proposition 8. Let [E.sub.j] be a complex Banach space for j = 1,2. Suppose that V : [E.sub.1] [right arrow] [E.sub.2] is a surjective isometry and H : [E.sub.1] [right arrow] [E.sub.1] is a Hermitian operator. Then [VHV.sup.-1]: [E.sub.2] [right arrow] [E.sub.2] is a Hermitian operator.

Proposition 9. An element F [member of] [??] is Hermitian if and only if there exists f [member of] A [intersection] [C.sub.R](Y) such thatF = 1 [cross product] f.

Proof. Suppose that F [member of] [??] is a Hermitian element. Then

[mathematical expression not reproducible] (105)

for every t [member of] R. Suppose that there exists a point (x, y) [member of] X x Y with Im F(x, y) [not equal to] 0, where Im denotes the imaginary part of a complex number. Suppose that Im F(x, y) > 0. Then

[mathematical expression not reproducible] (106)

Suppose that Im F(x, y) < 0. Then

[mathematical expression not reproducible] (107)

In any case we have there exists t [member of] R such that

[mathematical expression not reproducible] (108)

which contradicts our assumption. We have that

F [member of] [C.sub.R] (X x Y). (109)

Thus for every (s,t) e X x Y and 1 [member of] R, [absolute value of ]exp(itF(s,t))] = 1. Hence [mathematical expression not reproducible] for every t [member of] R. By (105) we have [mathematical expression not reproducible], which ensures that D(exp(itF)) = 0 for every i [member of] R. Thus exp(itF) [member of] 1 [[cross product]] A for every i [member of] R. We have

[mathematical expression not reproducible] (110)

and hence for every t [member of] R with [absolute value of ] < 1 we have

[mathematical expression not reproducible] (111)

It follows that

[mathematical expression not reproducible] (112)

as t [right arrow] 0. Since (exp(itF) - 1)/t [member of] 1 [direct sum]] A, for each t e R there exists [g.sub.t] [member of] A such that

[mathematical expression not reproducible] (113)

By (112) we have

[mathematical expression not reproducible] (114)

as n [right arrow] [infinity]. We have that {1 [direct sum] [g.sub.1/n]} is a Cauchy sequence in C(X x Y); thus we infer that{[g.sub.1/n]} is a Cauchy sequence in C(Y). Since A is uniformly closed as it is a uniform algebra, there exists g [member of] A such that

[mathematical expression not reproducible] (115)

and hence

[mathematical expression not reproducible] (116)

as n [right arrow] [infinity]. It follows by (114) that iF = 1 [R] g; thus

F = 1 [[direct sum]] (-ig) [member of] 1 [cross product] A. (117)

By (109) we see that -ig [member of] [C.sub.R](Y); thus we have f = -ig [member of] A [intersection] [C.sub.R](Y) and F = 1 [cross product] f.

Suppose conversely that [mathematical expression not reproducible] . We infer that F [member of] [C.sub.R](X x Y) and [absolute value of exp(it(F(x, y))] = 1 for every t [member of] R and (x,y) [member of] X x Y. Hence [mathematical expression not reproducible] for every t [member of] R. Since

[mathematical expression not reproducible]

for every t [member of] R. We conclude that F is a Hermitian element in [??].

Note that f [member of] A is Hermitian if and only if [member of] A[intersection][C.sub.R] (Y) by [37, Proposition 5]. Hence Proposition 9 asserts that F is a Hermitian element in [??] if and only if F = 1[cross product]f for a Hermitian element f in A.

Proposition 10. Suppose that U : [mathematical expression not reproducible] is a surjective unital isometry. Then U is an algebra isomorphism.

Proof. As we have already mentioned, [??] is a regular subspace (in the sense of Jarosz) with a natural norm. Then by Theorem 1 U is also an isometry with respect to the supremum norm on XxY. Then U is uniquely extended to a surjective isometry, with respect to the supremum norm, [??], from the uniform closure [??] onto itself. Since [??] is a uniform algebra, a theorem of Nagasawa [32] asserts that [??] is an algebra isomorphism since [??](1) = 1. Thus U is an algebra isomorphism from B onto itself. ?

Theorem 11. A bounded operator T : [mathematical expression not reproducible] is a Hermitian operator if and only if T(1) is a Hermitian element in [??] and T = [M.sub.t(1)], the multiplication operator by T(1).

Proof. By Proposition 10, every surjective unital isometry on [??] is multiplicative. Then by [37, Theorem 4], we have the conclusion.

5. Banach Algebras of C(Y)-Valued Maps

Suppose that X is a compact Hausdorff space. Suppose that B is a unital point separating subalgebra of C(Y) equipped with a Banach algebra norm. Then B is semisimple because {f [member of] B : f(x) = 0} is a maximal ideal of B for every x [member of] X and the Jacobson radical of B vanishes. The inequality [mathematical expression not reproducible] for every f [member of] Bis well known. We say that B is natural if the map e : Y [right arrow] [M.sub.B] defined by [mathematical expression not reproducible] for every f [member of] B, is bijective. We say that B is self-adjoint if B is natural and conjugate-closed in the sense that f [member of] B implies that [bar.f] [member of] B for every f [member of] B, where 7 denotes the complex conjugation on Y.

Definition 12. Let X and Y be compact Hausdorff spaces. Suppose that B is a unital point separating subalgebra of C(X) equipped with a Banach algebra norm [parallel] * [[parallel].sub.B]. Suppose that B is self-adjoint. Suppose that [??] is a unital point separating subalgebra of C(X x Y) such that B [cross product] C(Y) [subset] [??] equipped with a Banach algebra norm [mathematical expression not reproducible] Suppose that [??] is self-adjoint. We say that [??] is a natural C(Y)-valuezation of B if there exists a compact Hausdorff space M and a complex-linear map [mathematical expression not reproducible] such that ker D = 1 [cross product] C(Y) and D([C.sub.R](X x Y) [intersection][??]) [subset] [C.sub.R](M) which satisfies

[mathematical expression not reproducible]. (120)

The term "a natural C(Y)-valuezation of B" comes from the natural norm defined by Jarosz [5]. In fact the norm [mathematical expression not reproducible] is a natural norm in the sense of Jarosz [5].

Note that (X,C(Y), B, [??]) need not be an admissible quadruple defined byNikou and O'Farrell [38] (cf. [31]) since we do not assume that [mathematical expression not reproducible], which is a requirement for the admissible quadruple. On the other hand if [mathematical expression not reproducible] is an admissible quadruple of type L defined in [30], then [??] is a natural C(Y)-valuezation of B due to Definition 12.

Example 13. Let B = [C.sup.1] ([0,1]) and [??] = Lip([0,1],C(Y)) for Y = fp}, a singleton. Then Lip([0,1]) is algebraically isomorphic to Lip([0,1], C(Y)). Suppose that M is the maximal ideal space of [L.sup.[infinity]]([0,1]) and [mathematical expression not reproducible] is defined by f [??] r(f'), where r denotes the Gelfand transform in [L.sup.[infinity]]([0,1]). Then [??] is a natural C(Y)-valuezation of B. The Banach algebra Lip([0,1]) with the norm [mathematical expression not reproducible] is isometrically isomorphic to [??].

Let Y be a compact Hausdorff space. Note that a closed subalgebra [mathematical expression not reproducible] which appears in Example 12 in [30] is an example of a natural C(Y)-valuezation of B. The Banach algebras [C.sup.1] ([0,1], C(Y)) and [C.sup.1](T, C(Y))) which appear in Examples 16 and 17 in [30], respectively, are also examples of natural C(Y)-valuezations of [C.sup.1]([0,1]).

6. Isometries on Natural C(Y)-Valuezations

The main theorem in this paper is the following.

Theorem 14. Suppose that [??] is a natural C([Y.sub.j])-valuezation of [B.sub.j] [subset] C([X.sub.j]) for j = 1,2. We assume that

[mathematical expression not reproducible] (121)

for every [mathematical expression not reproducible] for j = 1,2. Suppose that [mathematical expression not reproducible] is a surjective complexlinear isometry. Then there exists h [member of] C([Y.sub.2]) such that [absolute value of h] = 1 on [Y.sub.2], a continuous map [phi] : [X.sub.2] x [Y.sub.2] [right arrow] [X.sub.1] such that [phi](, y) : [X.sub.2] [right arrow] [X.sub.1] is a homeomorphism for each y [member of] [Y.sub.2], and a homeomorphism [tau] : [Y.sub.2] [right arrow] [Y.sub.1] which satisfies

U (F) (x, y) = h (y) F ([phi] (x, y), [tau] (y)), (x,y) [member of] [X.sub.2] x [Y.sub.2] (122)

for every [mathematical expression not reproducible].

In short a surjective isometry between C(Y)-valuezations is a weighted composition operator of a specific form: the homeomorphism [X.sub.2] x [Y.sub.2] [right arrow] [X.sub.1] x [Y.sub.1], (x,y) [??] ([phi](x,y), [tau](y)) has the second coordinate that depends only on the second variable y [member of] [Y.sub.2]. A composition operator induced by such a homeomorphism is said to be of type BJ in [31, 37] after the study of Botelho and Jamison [39].

Quite recently the author of this paper and Oi [30, Theorem 8] proved a similar result of Theorem 14 for admissible quadruples of type L. To prove it we apply Proposition 3.2 and the following comments in [31]. Instead of this we prove Theorem 14 by Lumer's method, with which a proof is simpler than that in [30].

In the following in this section we assume that [??] is a natural C([Y.sub.j])-valuezation of B [subset] C([X.sub.j]) for j = 1,2. We assume that

[mathematical expression not reproducible] (123)

for every F[mathematical expression not reproducible]. Suppose that [mathematical expression not reproducible] is a surjective complex-linear isometry. A crucial part of a proof of Theorem 14 is to prove Proposition 15.

Proposition 15. Suppose that [X.sub.2] is not a singleton. There exists h [member of] C([Y.sub.2]) with [absolute value of h] = 1 on [Y.sub.2] such that [mathematical expression not reproducible].

A similar result for admissible quadruples of type L is proved in [30, Proposition 9]. If we assumed that

[mathematical expression not reproducible], (124)

then [mathematical expression not reproducible] were an admissible quadruple of type L. Although [mathematical expression not reproducible] in this paper need not be an admissible quadruple of type

L, a proof of Proposition 15 is completely the same as that in [30, Proposition 9] since we do not make use of the condition (124) in the proof of [30, Proposition 9]. The condition (124) is needed in [30] when we apply Proposition 3.2 and the following comments in [31].

7. Proof of Theorem 14: An Application of Lumer's Method

Proof of Theorem 14. A proof for the case where [X.sub.1] = {[x.sub.1] } and [X.sub.2] = {[x.sub.2]} are singletons is the same as the proof of Theorem 8 in [30].

Suppose that [X.sub.2] is not a singleton. By Proposition 15 there exists [mathematical expression not reproducible]. Letting [mathematical expression not reproducible], we see by the hypothesis [mathematical expression not reproducible] for every [mathematical expression not reproducible] is a surjective unital isometry from [mathematical expression not reproducible]. Then Corollary 2 asserts that [U.sub.0] is an algebra isomorphism. Let f [member of] [C.sub.R]([Y.sub.1]). By Proposition 9, 1 [cross product] f is a Hermitian element in [mathematical expression not reproducible]. Then by Theorem 11, [mathematical expression not reproducible] is a Hermitian operator on [??]. By Proposition [mathematical expression not reproducible] is a Hermitian operator on [??]. Then by Theorem 11 there exists [mathematical expression not reproducible]. Hence an operator S : [C.sub.R]([Y.sub.1]) [right arrow] [C.sub.R]([Y.sub.2]) is defined. Since [U.sub.0] is an algebra isomorphism, it is easy to see that S is a real algebra isomorphism from [mathematical expression not reproducible]. Then [??] : C([Y.sub.1]) [right arrow] C([Y.sub.2]) defined by [mathematical expression not reproducible] gives a complex algebra isomorphism. Gelfand theory asserts that there is a homeomorphism [tau] : [Y.sub.2] [right arrow] [Y.sub.1] such that [mathematical expression not reproducible]. It follows that

[mathematical expression not reproducible]. (125)

Since [U.sup.-1.sub.0] (1) = 1 we have

[mathematical expression not reproducible]. (126)

Define [mathematical expression not reproducible]. Since [U.sub.0] is an algebra isomorphism, the map [phi] is a unital homomorphism. Since the maximal ideal space of [B.sub.1] is [X.sub.1] and the maximal ideal space of [mathematical expression not reproducible], there is a continuous map [phi] : [X.sub.2] x [Y.sub.2] [right arrow] [X.sub.1] such that

[mathematical expression not reproducible] (127

It follows by (126) and (127) that

[mathematical expression not reproducible] (128)

for every a [member of] B1 and f [member of] C(Y1). Thus

[mathematical expression not reproducible] (129)

for every [mathematical expression not reproducible]. By the Stone-Weierstrass theorem [B.sub.1] [R] C([Y.sub.1]) is uniformly dense in C([X.sub.1] x [Y.sub.1]); hence any element in [??] is uniformly approximated by [cross product] [R] C([Y.sub.1]). As [U.sub.0] is also an isometry with respect to the uniform norm, we see that

[mathematical expression not reproducible] (130)

for every [mathematical expression not reproducible] and

[mathematical expression not reproducible] (131)

As [U.sub.0] is an algebra isomorphism, the map [X.sub.2] x [Y.sub.2] [right arrow] [X.sub.1] x [Y.sub.1] defined by [mathematical expression not reproducible] gives a homeomorphism. Therefore, for every y [member of] [Y.sub.2], the map

[phi](*,y): [X.sub.2] [right arrow] [X.sub.1] (132)

is a homeomorphism.

Suppose that [X.sub.1] is not a singleton. By the same way as in the last part of the proof of Theorem 8 in [30] we have that [X.sub.2] is not a singleton. Then we have the conclusion by the previous argument. ?

8. Application of Theorem 14

We exhibit applications of Theorem 14.

Corollary 16 ([4, Theorem 3.3]). Suppose that U : Lip ([0,1]) [right arrow] Lip([0,1]) is a surjective isometry

with respect to the norm defined by [mathematical expression not reproducible] for f [member of] Lip([0,1]). Then U(1) is a constant function of unit modulus such that

[mathematical expression not reproducible] (133)

[mathematical expression not reproducible] (134)

The converse statement also holds.

Proof. By Example 13 we may suppose that Lip([0,1]) is a Banach algebra of C(Y)-valuezation. Applying Theorem 14 we have that U(1) = 1 [cross product] h for h [member of] C(Y) with [absolute value of h] = 1. Since our Y is a singleton, U(1) is a constant function of unit modulus. We also see that the corresponding continuous map [phi] : [0,1] x Y [right arrow] [0,1] can be considered as a homeomorphism from [0,1] onto [0,1]; therefore we have that

[mathematical expression not reproducible] (135)

The rest is a routine argument to prove that [phi] is an isometry; hence [phi] (v) = v, x [member of] [0,1] or [phi] (v) = 1 - x, x [member of][0,1].

The converse statement is trivial.

Corollaries 14,15,18, and 19 in [30, Section 6] follow here with a similar proof.

https://doi.org/10.1155/2018/8085304

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This work was supported by JSPS KAKENHI Grants Numbers JP16K05172 and JP15K04921.

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Osamu Hatori (iD)

Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-2181, Japan

Correspondence should be addressed to Osamu Hatori; hatori@math.sc.niigata-u.ac.jp

Received 21 April 2018; Accepted 26 July 2018; Published 2 September 2018

Academic Editor: Miguel Martin

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Title Annotation: | Research Article |
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Author: | Hatori, Osamu |

Publication: | Journal of Function Spaces |

Date: | Jan 1, 2018 |

Words: | 10331 |

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