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

In 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 ‖f‖M = max{‖f‖∞, L(f)}, where L(f) denotes the Lipschitz constant of f. Cambern [3] has considered isometries on spaces of scalar-valued continuously differentiable functions C1([0, 1]) with norm given by ‖f‖ = maxx∈[0,1]{|f(x)| + |f󸀠(x)|} for f ∈ C1([0, 1]) 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 C1([0, 1]) with respect to the norm ‖f‖ = ‖f‖∞ + ‖f󸀠‖∞ 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 ‖f‖L = ‖f‖∞ + L(f) for f ∈ Lip(K) is induced by an isometry on K (note that ‖f‖∞([0,1]) + L(f) = ‖f‖∞([0,1]) + ‖f󸀠‖∞([0,1]) for every f ∈ Lip([0, 1]). The reason is as follows. Let f ∈ Lip([0, 1]). Then f is absolutely continuous. Hence the derivative f󸀠 exists almost everywhere on [0, 1], and it is integrable by the theory of the absolutely continuous functions. Furthermore the equality


𝑓 (𝑏) − 𝑓 (𝑎) = ∫
It follows that () ≤ ‖  ‖ ∞([0,1]) .We conclude that () = ‖  ‖ ∞([0,1]) .Thus ‖‖ ∞([0,1]) + () = ‖‖ ∞([0,1]) + ‖  ‖ ∞([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  1 (), Lip(), lip  (), and [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() 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() 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.
The main result in this paper is Theorem 14, which gives the form of a surjective isometry  with respect to the sum norm between certain Banach algebras with the values in a commutative unital  * -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 (1) = 1 ⊗ ℎ for an ℎ ∈ ( 2 ) with |ℎ| = 1 on  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  0 = (1 ⊗ ℎ) is a surjective isometry fixing the unit.Then by applying a theorem of Jarosz [5] (Theorem 1 in this paper) we see that  0 is also an isometry with respect to the supremum norm.By the Banach-Stone theorem  0 is an algebra isomorphism.Then by applying Lumer's method (cf.[30]) we see that  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.

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 2 with (1, 0) = 1 is denoted by P. For  ∈ P we put In the sequel a unital semisimple commutative Banach algebra  is identified via the Gelfand transforms with a subalgebra of (  ).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  and  be unital semisimple commutative Banach algebras.Assume they have natural norms, respectively.Suppose that  :  →  is a surjective complex-linear isometry with 1 = 1.Then there exists a homeomorphism  :   →   such that Proof.A unital semisimple commutative Banach algebra is regular by Proposition 2 in [5].Then Theorem 1 ensures that  is a surjective linear isometry from (, ‖⋅‖ ∞ ) onto (, ‖⋅‖ ∞ ).It is easy to see that  is extended to a surjective linear isometry T from the uniform closure  of  onto the uniform closure  of .Then a theorem of Nagasawa asserts that there exists a homeomorphism  :   →   such that T()() =  ∘ () ( ∈ ,  ∈   ).As T| =  we have the conclusion.
Then there exists a surjective isometry  :  2 →  1 such that Conversely if  : Lip( 1 ) → Lip( 2 ) is of the form as (14), then  is a surjective isometry with respect to both of ‖ ⋅ ‖  and ‖ ⋅ ‖  such that 1 = 1.
Proof.As (Lip(  ), ‖ ⋅ ‖  ) is a unital semisimple commutative Banach algebra with maximal ideal space   , Corollary 2 asserts that there is a homeomorphism  :  2 →  1 such that Then by a routine argument we see that  is an isometry.Converse statement is trivial.
Without assuming 1 = 1, we have that  is a weighted composition operator.We exhibit a general result as Theorem 14 (see also [30]).
Proof.As ‖ ⋅ ‖  is a natural norm, we have by Corollary 2 that there is a homeomorphism  :  2 →  1 such that Then by a routine argument we see that  is an isometry.Converse statement is trivial.
Without the assumption that 1 = 1 in Corollary 4, one may expect that  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  and  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 of Jarosz.The following is Lemma 2 in [5].
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 /2's between 11 lines and 5 lines from the bottom of page 69 read as /3.
(ii) Next  ∈ \ 1 reads as  ∈  1 on the bottom of page 69.
(iii) We point out that the term ∑  0 −1 =1 (  () − 1) which appears on the first line of the first displayed inequalities on page 70 reads 0 if  0 = 1.
(iv) The term 1 +  on the right hand side of the second line of the same inequalities reads as 1 + /3.
(v) Two /2's on the same line read as /3.
Let  be a nonempty convex subset of the complex plane and  ∈ [0, 2).Put Note that we may write Let  be a subspace of () for a compact Hausdorff space.
(3) For each  > 0 and each  ∈ A  , it holds that Suppose that these assertions are proved.Let  ∈ .By (2), for any  > 0, there is a sequence 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 −/2 and /2 which appear in the formulae (7) and (8) in [5] seem inappropriate; they read, for example, as 3/4 and /4, respectively.
This completes the proof of the theorem.

Hermitian Operators on a Banach Algebras of Continuous Maps Whose Values Are in a Uniform Algebras
Let  and  be compact Hausdorff spaces.Let  be a unital subalgebra of () which separates the points of .
Throughout this section we assume  is a Banach algebra with the norm ‖ ⋅ ‖  and  is a uniform algebra on .Recall that a uniform algebra on  is a uniformly closed subalgebra of () which contains constants and separates the points of .
Note that B separates the points of × since  separates the points of  and  separates the points of .We assume that there exists a compact Hausdorff space M and a complexlinear map  : B → (M) such that ker  = 1 ⊗ .We assume that ‖‖ B = ‖‖ ∞(×) +‖()‖ ∞(M) for every  ∈ B.
Hence  is continuous.Defining ‹ ⋅ ‹ is a one-invariant seminorm in the sense of Jarosz; ‹ ⋅ ‹ is a seminorm on B such that ‹ + 1‹ = ‹‹ for every  ∈ B.
Hence the norm ‖ ⋅ ‖ B is a natural norm (see [5, p.67]) Note that B is a regular subspace of ( × ) 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  −1 must be Hermitian if  is Hermitian and  is a surjective isometry.Definition 6.Let A be a unital Banach algebra.We say that for every  ∈ R. The set of all Hermitian elements of A is denoted by (A).
If A is a unital  * -algebra, then (A) is the set of all self-adjoint elements of A. Hence (  (C)) is the set of all Hermitian matrices, and (()) =  R ().Definition 7. Let  be a complex Banach space.The Banach algebra of all bounded operators on  is denoted by ().We say that  ∈ () is a Hermitian operator if  ∈ (()).
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   be a complex Banach space for  = 1, 2. Suppose that  :  1 →  2 is a surjective isometry and  :  1 →  1 is a Hermitian operator.Then  −1 :  2 →  2 is a Hermitian operator.(107) In any case we have there exists  ∈ R such that     exp ()     B ≥     exp ()    ∞(×) > 1, which contradicts our assumption.We have that Thus for every (, ) ∈  ×  and  ∈ R, |exp((, ))| = 1.
Note that  ∈  is Hermitian if and only if  ∈ ∩ R () by [37,Proposition 5].Hence Proposition 9 asserts that  is a Hermitian element in B if and only if  = 1⊗ for a Hermitian element  in .
Proposition 10.Suppose that  : B → B is a surjective unital isometry.Then  is an algebra isomorphism.
Proof.As we have already mentioned, B is a regular subspace (in the sense of Jarosz) with a natural norm.Then by Theorem 1  is also an isometry with respect to the supremum norm on  × .Then  is uniquely extended to a surjective isometry, with respect to the supremum norm, Ũ, from the uniform closure B onto itself.Since B is a uniform algebra, a theorem of Nagasawa [32] asserts that Ũ is an algebra isomorphism since Ũ(1) = 1.Thus  is an algebra isomorphism from B onto itself.Proof.By Proposition 10, every surjective unital isometry on B is multiplicative.Then by [37,Theorem 4], we have the conclusion.

Banach Algebras of 𝐶(𝑌)-Valued Maps
Suppose that  is a compact Hausdorff space.Suppose that  is a unital point separating subalgebra of () equipped with a Banach algebra norm.Then  is semisimple because { ∈  : () = 0} is a maximal ideal of  for every  ∈  and the Jacobson radical of  vanishes.The inequality ‖‖ ∞ ≤ ‖‖  for every  ∈  is well known.We say that  is natural if the map  :  →   defined by   →   , where   () = () for every  ∈ , is bijective.We say that  is self-adjoint if  is natural and conjugate-closed in the sense that  ∈  implies that  ∈  for every  ∈ , where ⋅ denotes the complex conjugation on .Definition 12. Let  and  be compact Hausdorff spaces.Suppose that  is a unital point separating subalgebra of () equipped with a Banach algebra norm ‖ ⋅ ‖  .Suppose that  is self-adjoint.Suppose that B is a unital point separating subalgebra of ( × ) such that  ⊗ () ⊂ B equipped with a Banach algebra norm ‖ ⋅ ‖ B. Suppose that B is selfadjoint.We say that B is a natural ()-valuezation of  if there exists a compact Hausdorff space M and a complexlinear map  : B → (M) such that ker  = 1 ⊗ () and ( R ( × ) ∩ B) ⊂  R (M) which satisfies The term "a natural ()-valuezation of " comes from the natural norm defined by Jarosz [5].In fact the norm ‖ ⋅ ‖ B is a natural norm in the sense of Jarosz [5].
Note that (, (), , B) need not be an admissible quadruple defined by Nikou and O'Farrell [38] (cf.[31]) since we do not assume that {(⋅, ) :  ∈ B,  ∈ } ⊂ , which is a requirement for the admissible quadruple.On the other hand if (, (), , B) is an admissible quadruple of type L defined in [30], then B is a natural ()-valuezation of  due to Definition 12.

Isometries on Natural 𝐶(𝑌)-Valuezations
The main theorem in this paper is the following.Theorem 14. Suppose that B is a natural (  )-valuezation of   ⊂ (  ) for  = 1, 2. We assume that for every  ∈ B1 .
In short a surjective isometry between ()-valuezations is a weighted composition operator of a specific form: the homeomorphism  2 ×  2 →  1 ×  1 , (, )  → ((, ), ()) has the second coordinate that depends only on the second variable  ∈  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 B is a natural (  )-valuezation of  ⊂ (  ) for  = 1,2.We assume that for every  ∈ B and ℎ ∈ (  ) with |ℎ| = 1 on   for  = 1, 2. Suppose that  : B1 → B2 is a surjective complex-linear isometry.A crucial part of a proof of Theorem 14 is to prove Proposition 15.
A similar result for admissible quadruples of type L is proved in [30,Proposition 9].If we assumed that then B were an admissible quadruple of type L.Although B 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].

Application of Theorem 14
We exhibit applications of Theorem 14.

Theorem 11 .
A bounded operator  : B → B is a Hermitian operator if and only if (1) is a Hermitian element in B and  =  (1) , the multiplication operator by (1).
Let  and  be compact Hausdorff spaces, let  and  be complex-linear subspaces of () and (), respectively, and let ,  ∈ P. Assume  and  contain constant functions, and let ‖⋅‖  , ‖⋅‖  be a -norm and -norm on  and , respectively.Assume next that there is a linear isometry  from (, ‖ ⋅ ‖  ) onto (, ‖ ⋅ ‖  ) with 1 = 1.