Journal of Function Spaces
Volume 2018 (2018), Article ID 8085304, 14 pages
https://doi.org/10.1155/2018/8085304

Research Article

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

Osamu Hatori

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

Copyright © 2018 Osamu Hatori. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study isometries on algebras of the Lipschitz maps and the continuously differentiable maps with the values in a commutative unital -algebra. A precise proof of a theorem of Jarosz concerning isometries on spaces of continuous functions is exhibited.

1. Introduction

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 of Lipschitz functions on a compact metric space , equipped with the max norm , where denotes the Lipschitz constant of . Cambern [3] has considered isometries on spaces of scalar-valued continuously differentiable functions with norm given by for and determined a representation for the surjective isometries supported by such spaces. Rao and Roy [4] proved that surjective isometries on and with respect to the norm are of canonical forms in the sense that they are weighted composition operators. They asked whether a surjective isometry on with respect to the sum norm for is induced by an isometry on (note that for every . The reason is as follows. Let . Then is absolutely continuous. Hence the derivative exists almost everywhere on , and it is integrable by the theory of the absolutely continuous functions. Furthermore the equality

holds. As we see that is essentially bounded. In fact,

assures that . By (1) we have

It follows that . We conclude that . Thus .) 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 , , , and . 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 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 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.

Jiménez-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 ; see also [9]. Botelho and Jamison [10] studied isometries on with . 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 , , and , where is a compact metric space, , or ( denotes the unit circle on the complex plane), and is a compact Hausdorff space. We study isometries without assuming that they preserve unit. As corollaries of a general result we describe isometries on , , , and , respectively.

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 for an with on (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 is a surjective isometry fixing the unit. Then by applying a theorem of Jarosz [5] (Theorem 1 in this paper) we see that is also an isometry with respect to the supremum norm. By the Banach-Stone theorem is an algebra isomorphism. Then by applying Lumer’s method (cf. [30]) we see that 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 be a compact Hausdorff space. Let be a real or complex Banach space. The space of all -valued continuous maps on is denoted by . When (resp. ), is abbreviated by (resp. ). For a subset of , the supremum norm of on is for . When no confusion will result we omit the subscript and write only . Let be a compact metric space and . For , put

Then is called an -Lipschitz number of , or just a Lipschitz number of . When we omit the subscript and write only . The space of all such that is denoted by . When the subscript is omitted and it is written as .

When the closed subspace

of is called a little Lipschitz space. There are a variety of complete norms on and . In this paper we are mainly concerned with the norm of (resp., ) which is defined by

The norm of (resp., ) is defined by

Note that (resp., ) is a Banach space with respect to and , respectively. If is a Banach algebra, the norm is multiplicative. Hence (resp., ) is a (unital) Banach algebra with respect to the norm if is a (unital) Banach algebra. The norm fails to be submultiplicative even if is a Banach algebra. For a metric on , the Hölder metric is defined by for . is isometrically isomorphic to .

We are mainly concerned with in this paper. Then and are unital semisimple commutative Banach algebras with , when (resp., ) is abbreviated to (resp. ).

Let for or . We say that is continuously differentiable if there exists such that

for every . We denote . Put

Then with norm is a unital semisimple commutative Banach algebra. If is singleton we may suppose that is isometrically isomorphic to and we abbreviate by .

By identifying with we may assume that (resp., ) is a subalgebra of by the correspondence

Throughout the paper we may suppose that

We say that a subset of is point separating if separates the points of . The unit of commutative Banach algebra is denoted by . The maximal ideal space of is denoted by . Suppose that is a unital point separating subalgebra of equipped with a Banach algebra norm. Then is semisimple because is a maximal ideal of for every and the Jacobson radical of 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 with is denoted by . For we put

Recently Tanabe pointed out by a private communication that exists and it is finite for every . (In fact, it is easy to see that is increasing since is convex. We also see that .) Let be a compact Hausforff space and a liner subspace of which contains constant functions. A seminorm on is called one-invariant (in the sense of Jarosz) if for all . Let . A norm on is called a -norm if there is a one-invariant seminorm on such that . A natural norm is a -norm for some .

Theorem 1 (Jarosz [5]). Let and be compact Hausdorff spaces, let and be complex-linear subspaces of and , respectively, and let . 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 . Then if , or if and are regular subspaces of and , respectively, then is an isometry from onto .

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 . 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 from the uniform closure of onto the uniform closure of . Then a theorem of Nagasawa asserts that there exists a homeomorphism such that (, ). As we have the conclusion.

Corollary 3. Let be a compact metric space for . Suppose that is a surjective complex-linear isometry with respect to the norm . Assume . Then there exists a surjective isometry such that

Conversely if is of the form as (14), then is a surjective isometry with respect to both of and such that .

Proof. As is a unital semisimple commutative Banach algebra with maximal ideal space , Corollary 2 asserts that there is a homeomorphism such that

Then by a routine argument we see that is an isometry.
Converse statement is trivial.

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

need not be a Banach algebra since need not be submultiplicative. On the other hand, is a natural norm in the sense of Jarosz (see [5]) such that . Then by Theorem 1 we have the following.

Corollary 4. Let be a compact metric space for . Suppose that is a surjective complex-linear isometry with respect to the norm . Assume . Then there exists a surjective isometry such that

Conversely if is of a similar form as (16), then is a surjective isometry with respect to both of and such that .

Proof. As is a natural norm, we have by Corollary 2 that there is a homeomorphism such that

Then by a routine argument we see that is an isometry.
Converse statement is trivial.

Without the assumption that 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].

Lemma 5 (Jarosz [5]). Assume is a regular subspace of with and let . Then for any and any open neighborhood of , there is an such that

and for all .

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 ’s between 11 lines and 5 lines from the bottom of page 69 read as .(ii)Next reads as on the bottom of page 69.(iii)We point out that the term which appears on the first line of the first displayed inequalities on page 70 reads if .(iv)The term on the right hand side of the second line of the same inequalities reads as .(v)Two ’s on the same line read as .(vi)On the next line reads as .(vii)For any we infer that

 Hence we have if by the first displayed inequalities of page 70.(viii)The inequality on the fifth line on page 70 reads as .

Let be a nonempty convex subset of the complex plane and . Put

Note that we may write

Let be a subspace of for a compact Hausdorff space. For we put and , where denotes the closed convex-hull. We define the functions

Proof of Theorem 1. Let . First we note that

since is the closed convex-hull of a compact set . We prove the inequalities

which appear on p. 68 in [5]. Put . As is compact, there exists such that . Hence

As

we have

Let . By the definition of , we infer that , hence we have for every . Then

Letting , we have

As is arbitrary we have

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

for every . By (24) and (31) we infer that

As is -invariant we have

As is an isometry, , and is -invariant, we have

Thus

It follows that

Recall that and . It follows that

Suppose that . Then we have by (37) that for every and . By Lemma 1 in [5] we infer that . Thus we have . We have proved that is an isometry from onto if .
Suppose that and are regular subspaces of and , respectively. Let . Put

Suppose that . For any and any nonempty compact convex subset , we have that

for all , where . Then by (37) we have

for all . It follows by Lemma 1 in [5] that

and therefore

If , then a similar calculation shows that

and

It follows that in any case (, ) we obtain

We will prove that

for all . Once it is proved, applying the same argument for instead of , we see that for every . As is a bijection, it follows that for every . It will follow that for every . A proof of (46) is the following. For every , denote

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

(1) is a continuous mapping from onto .(2)For each , the set is dense in .(3)For each and each , it holds that .

Suppose that these assertions are proved. Let . By (2), for any , there is a sequence of functions in such that as . By (3) we have

for every . Letting we have

by (1). As is arbitrary, we have that

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 and which appear in the formulae and in [5] seem inappropriate; they read, for example, as and , respectively.
We now proceed to prove the first statement. Aiming for a contradiction, suppose that is not continuous from to . Let be a positive real number less than . Then there is a function such that and . Then there exist such that by [29, Proposition 6.3]. Since is complex-linear we may suppose that .
By (41) and (45), we deduce that and . As and , we infer that and . Let . Let . Then asserts that . Thus

Hence . As we have

Consider the open neighborhood of in given by

We infer that is a proper subset of by (52). Then, by [5, Lemma 2], there exists such that , , for every and for all . If denotes the closed rectangle whose vertices are the four points , we have

Consider now the set

We claim that . Suppose that . As is compact, there exists a positive integer such that , where . Then (52) gives . As is the closed convex-hull of , it is contained in the closed convex set . On the other hand, by (52). As , this contradicts , and this proves our claim. Hence there is with . As , it follows that and so . Hence . Thus is in . Thus we have

We claim that

where . Let . Suppose first that . Since by (54), we have

Suppose next that . Then and so . Moreover, . Therefore we have

It follows from (58) and (59) that

and hence

as is claimed. Therefore we have

Put . We claim that . If , there is nothing to prove. Suppose that . Then, by (41), we have

Since by (54), we have

As does not include a closed disk with the radius greater than , we conclude that .
In the following we will consider two cases: and . Suppose first that . Then (64) yields . From we deduce that . Hence we have

Since is convex we have

From (39) we infer that

Since , from (56) and (67) we obtain that

By (63) and , we deduce that . Thus there is such that . It follows that ; hence we have

as and . We get by (62) and (69) that

On the other hand, is invariant for any by (37). From (68) and (70) we deduce that and this contradicts that .
For the second case, suppose next that . Then, by (43), we have

and, by (54), it follows that . Moreover, since . Then

Hence, . Using (39), we infer that

By (71), we obtain that , and, as , we infer that . Hence , so that

as . Since , we obtain by (56) and (73) that

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

Since is invariant for any by (37), from (75) and (76) we deduce that and this is impossible since .
Next we show a proof of the second assertion (2). Let . We prove that there exists a sequence which uniformly converges to such that as . Without loss of generality we may assume that . Then there exists such that by [29, Proposition 6.3]. We may assume that . Suppose that . Put

and

(In the following we identify and ; that is, we identify and for every .) Since we assume that we infer by a simple calculation that

for with . We assume that . By [5, Lemma 2] there exists such that , , on , and on . Put . Then and

Let . Then we have

and

Hence

It follows that we have

for . Suppose that . Then

and hence

for . Since , we have by combining (84) and (86) that

As is convex we obtain

Recall that for and a complex number

denotes the closed disk with center and radius . We observe that . Recall that

Let be the line defined by the equation

part of which is a part of the boundary of . Let be the line defined by the equation

By some calculation we have that the distance between and is and it coincides with the distance between the point and the line . Hence we see that

Thus .
Next we prove that for every with . It will follow that and the equality will hold. Let with . We prove the case where . A proof for the case where is the same and we omit it. We divide into two parts:

and

Suppose that and . The distance between and is . Hence . Suppose that and ; that is, . Let be the line passing through which is parallel to . Let be the unique point in the intersection of and the -axis. Then for some . Then the distance between and is , which is equal to the distance between the point and the line . On the other hand

It follows that . We conclude that if satisfies , then . Thus we have

Since , we have

On the other hand, as ensures that as . It follows that, for every , is dense in .
Finally we show a proof of the third assertion (3). As is pointed out in the proof of [5, Theorem], for any and any . Let and . Suppose that . Then by (43) we have . Hence we have

As , we conclude by (45) that

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 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 . For functions and , let be the function defined by for , and for a subspace of and a subspace of , put

and

Throughout the section is a unital subalgebra of with a Banach algebra norm . We assume that . Note that separates the points of since separates the points of and separates the points of . We assume that there exists a compact Hausdorff space and a complex-linear map such that . We assume that for every . Hence is continuous. Defining

is a one-invariant seminorm in the sense of Jarosz; is a seminorm on such that for every . Hence the norm is a natural norm (see [5, p.67]) Note that 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 must be Hermitian if is Hermitian and is a surjective isometry.

Definition 6. Let be a unital Banach algebra. We say that is a Hermitian element if

for every . The set of all Hermitian elements of is denoted by .

If is a unital -algebra, then is the set of all self-adjoint elements of . Hence is the set of all Hermitian matrices, and .

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 . Suppose that is a surjective isometry and is a Hermitian operator. Then is a Hermitian operator.

Proposition 9. An element is Hermitian if and only if there exists such that .

Proof. Suppose that is a Hermitian element. Then

for every . Suppose that there exists a point with , where denotes the imaginary part of a complex number. Suppose that . Then

Suppose that . Then

In any case we have there exists such that

which contradicts our assumption. We have that

Thus for every and , . Hence for every . By (105) we have , which ensures that for every . Thus for every . We have

and hence for every with we have

It follows that

as . Since , for each there exists such that

By (112) we have

as . We have that is a Cauchy sequence in ; thus we infer that is a Cauchy sequence in . Since is uniformly closed as it is a uniform algebra, there exists such that

and hence

as . It follows by (114) that ; thus

By (109) we see that ; thus we have and .
Suppose conversely that for . We infer that and for every and . Hence for every . Since

we have . It follows that

for every . We conclude that is a Hermitian element in .

Note that is Hermitian if and only if by [37, Proposition 5]. Hence Proposition 9 asserts that is a Hermitian element in if and only if for a Hermitian element in .

Proposition 10. Suppose that is a surjective unital isometry. Then 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 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, , from the uniform closure onto itself. Since is a uniform algebra, a theorem of Nagasawa [32] asserts that is an algebra isomorphism since . Thus is an algebra isomorphism from onto itself.

Theorem 11. A bounded operator is a Hermitian operator if and only if is a Hermitian element in and , the multiplication operator by .

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

5. Banach Algebras of -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 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 is a unital point separating subalgebra of such that equipped with a Banach algebra norm . Suppose that is self-adjoint. We say that is a natural -valuezation of if there exists a compact Hausdorff space and a complex-linear map such that and which satisfies

The term “a natural -valuezation of ” comes from the natural norm defined by Jarosz [5]. In fact the norm is a natural norm in the sense of Jarosz [5].

Note that need not be an admissible quadruple defined by Nikou and O’Farrell [38] (cf. [31]) since we do not assume that , which is a requirement for the admissible quadruple. On the other hand if is an admissible quadruple of type L defined in [30], then is a natural -valuezation of due to Definition 12.

Example 13. Let and for , a singleton. Then is algebraically isomorphic to . Suppose that is the maximal ideal space of and is defined by , where denotes the Gelfand transform in . Then is a natural -valuezation of . The Banach algebra with the norm is isometrically isomorphic to .

Let be a compact Hausdorff space. Note that a closed subalgebra of which appears in Example 12 in [30] is an example of a natural -valuezation of . The Banach algebras and which appear in Examples 16 and 17 in [30], respectively, are also examples of natural -valuezations of .

6. Isometries on Natural -Valuezations

The main theorem in this paper is the following.

Theorem 14. Suppose that is a natural -valuezation of for . We assume that

for every and with on for . Suppose that is a surjective complex-linear isometry. Then there exists such that on , a continuous map such that is a homeomorphism for each , and a homeomorphism which satisfies

for every .

In short a surjective isometry between -valuezations is a weighted composition operator of a specific form: the homeomorphism has the second coordinate that depends only on the second variable . 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 -valuezation of for . We assume that

for every and with on for . Suppose that is a surjective complex-linear isometry. A crucial part of a proof of Theorem 14 is to prove Proposition 15.

Proposition 15. Suppose that is not a singleton. There exists with on such that .

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

then were an admissible quadruple of type L. Although 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 and are singletons is the same as the proof of Theorem 8 in [30].
Suppose that is not a singleton. By Proposition 15 there exists with on such that . Letting by , , we see by the hypothesis for every that is a surjective unital isometry from onto . Then Corollary 2 asserts that is an algebra isomorphism. Let . By Proposition 9, is a Hermitian element in . Then by Theorem 11, is a Hermitian operator on . By Proposition 8   is a Hermitian operator on . Then by Theorem 11 there exists such that . Hence an operator is defined. Since is an algebra isomorphism, it is easy to see that is a real algebra isomorphism from onto . Then defined by for gives a complex algebra isomorphism. Gelfand theory asserts that there is a homeomorphism such that , . It follows that

Since we have

Define by , . Since is an algebra isomorphism, the map is a unital homomorphism. Since the maximal ideal space of is and the maximal ideal space of is , there is a continuous map such that

It follows by (126) and (127) that

for every and . Thus

for every . By the Stone-Weierstrass theorem is uniformly dense in ; hence any element in is uniformly approximated by . As is also an isometry with respect to the uniform norm, we see that

for every and

As is an algebra isomorphism, the map defined by gives a homeomorphism. Therefore, for every , the map

is a homeomorphism.
Suppose that is not a singleton. By the same way as in the last part of the proof of Theorem 8 in [30] we have that 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 is a surjective isometry with respect to the norm defined by for . Then is a constant function of unit modulus such that

or

The converse statement also holds.

Proof. By Example 13 we may suppose that is a Banach algebra of -valuezation. Applying Theorem 14 we have that for with . Since our is a singleton, is a constant function of unit modulus. We also see that the corresponding continuous map can be considered as a homeomorphism from onto ; therefore we have that

The rest is a routine argument to prove that is an isometry; hence , or , .
The converse statement is trivial.

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

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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