Phase-isometries on real normed spaces

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Abstract

We say that a map f from a normed space X to another normed space Y is a phase-isometry if the equality{f(x)+f(y),f(x)f(y)}={x+y,xy} holds for all x,yX. A normed space X is said to have the Wigner property if for any normed space Y and every surjective phase-isometry f:XY, there exists a phase function ε:X{1,1} such that εf is a linear isometry. We show that all smooth normed spaces, L(Γ)-type spaces and 1(Γ)-spaces enjoy this property.

Introduction

The study of isometries between two normed spaces dates back to 1930s. The classical Mazur-Ulam theorem ([19]) states that an isometry f of a real normed space X “onto” another real normed space Y with f(0)=0 is linear. For strictly convex Y, this result is trivial even for “into” maps (see [2]), but this is false in general. For example, if f:R2 is defined by f(t)=(t,sint) for all tR, then it is an “into” isometry with f(0)=0 which is not linear since the function sint on R is not linear. In 1968, T. Figiel proved in [8] the following substitute without the onto assumption for the Mazur-Ulam theorem: For every isometry f from a real normed space X to another real normed space Y with f(0)=0, there is a linear operator F of norm one from spanf(X) to X such that Ff is the identity on X. Recently, a statement [5, Lemma 2.4] equivalent to Figiel's theorem was obtained by Cheng, Dong and Zhang who showed that if f:XY is an isometry with f(0)=0, then for every xX there exists a linear functional φY with φ=x such that x(x)=φ(f(x)) for all xX. Figiel's theorem and its generalizations play an important role in the study of isometric embedding problems (see, for instance [6], [13]). More information and background on linear isometries of normed spaces can be found in the two volume set [9], [10].

Another important result related to linear isometries is the famous Wigner's theorem which plays a fundamental role in quantum mechanics. This result has been formulated in several ways (see [3], [4], [11], [14], [20], [22], [24], [25], [26] to list just some of them). One of them (see [24]) characterizes maps on a Hilbert space preserving the absolute value of the inner product of any pair of vectors. Namely, if H and K are complex Hilbert spaces and f:HK is a map with the property|f(x),f(y)|=|x,y|(x,yH), then there exists a phase function ε:HC with |ε(x)|=1 such that εf is a linear isometry or conjugate linear isometry. A real version of this result was presented in [22, Corollary 8(a)] by Rätz. It is worth mentioning that L. Molnár [20] described the form of all bijective maps on the set of all rank-one idempotents of a Banach space which preserve zero products in both directions. This not only extends Uhlhorn's version of Wigner's theorem to indefinite inner product spaces but also significantly generalizes a result of Van den Broek [26].

Wigner's theorem is so important that it worth studying from various points of view. A new general version of this theorem may certainly improve our understanding of it. We shall be trying to present a Wigner-type result for normed spaces over real fields. Let H and K be real inner product spaces. By the polar identity, we can easily see that a map f:HK satisfies (1.1) if and only if it satisfies{f(x)+f(y),f(x)f(y)}={x+y,xy}(x,yH).

This inspires us to introduce the following definition.

Definition 1

Let X and Y be normed spaces. A map f:XY is called a phase-isometry if it satisfies the functional equation{f(x)+f(y),f(x)f(y)}={x+y,xy}(x,yX).

Let X and Y be real normed spaces. Two maps f,g:XY are called phase equivalent if there exists a phase function ε:X{1,1} such that g=εf. We can easily see that a map that is phase equivalent to a linear isometry is definitely a phase-isometry. It is interesting in seeing whether the converse also holds for real normed spaces. Motivated by the Mazur-Ulam theorem, the following natural problem arises.

Problem 1

Let X and Y be real normed spaces, and let f:XY be a surjective phase-isometry. Is it true that f is phase equivalent to a linear isometry?

It is worth mentioning that in general surjectivity is needed for Problem 1. This can be seen from the preceding example f:R2 defined by f(t)=(t,sint) for all tR, where f is a phase-isometry not phase equivalent to any linear isometry.

By the real version of Wigner's theorem, Problem 1 is solved in the positive for real Hilbert spaces (see [18]). In our paper [15], we provide an affirmative answer to Problem 1 with X and Y being p(Γ) spaces (0<p<). The first author and Jia presented in [17] a similar result for L(Γ)-type spaces.

A normed space X is said to have the Wigner property if Problem 1 has an affirmative answer for an arbitrary target Y. It should be mentioned that the above results on real Hilbert spaces, p(Γ) spaces and L(Γ)-type spaces are not enough to allow these spaces to have the Wigner property since the target spaces are not arbitrary. We start in the present paper to attack Problem 1 for general normed spaces. Although we feel our paper is interesting in its own right, we hope that it will serve as a stepping stone to show that every surjective phase-isometry between two real normed spaces is phase equivalent to a linear isometry (i.e., all real normed spaces have the Wigner-property).

This paper is organized as follows. In Section 2, we first apply the fundamental theorem of projective geometry to present a sufficient condition such that Problem 1 can be solved in the positive for general normed spaces. Next, we give a Figiel-type result for phase-isometries. The technique of this result that we use draw on the famous Figiel theorem [8] and an idea of [5, lemma 2.4] for a special case ε=0. It plays a crucial role and is used frequently in every main theorem of the present paper.

Lemma 1 Main lemma

Let X and Y be real normed spaces, and let f:XY be a phase-isometry (not necessarily surjective). Then for every w-exposed point x of BX, there exists a linear functional φY of norm one such that x(x)=±φ(f(x)) for all xX.

As a result, our first result reads as follows.

Theorem 1

Let X be a real smooth normed space. Then X has the Wigner property.

The following result is an immediate consequence of Theorem 1.

Corollary 1

All real Lp(μ) spaces with 1<p< for an arbitrary measure μ have the Wigner property. In particular, real Hilbert spaces have the Wigner property.

In Section 3, by the properties of Birkhoff orthogonality, we continue to present another sufficient condition such that Problem 1 can be solved in the positive for some special normed spaces. As its application, we present one more result as follows.

Theorem 2

L(Γ)-type spaces and 1(Γ)-spaces have the Wigner property.

Section snippets

Main lemma and phase-isometries for smooth normed spaces

In this note, the letters X,Y are used to denote real normed spaces and X,Y are their dual spaces respectively. For a real normed space X, we denote by SX and BX the unit sphere and the closed unit ball of X, respectively. Throughout what follows, we shall freely use without explicit mention notation “±” to mean that either “+” or “−” holds.

We start this section with an interesting result showing that every surjective phase-isometry is an injective odd norm-preserving map.

Lemma 2

Let X and Y be real

Phase-isometries on L(Γ)-type and 1(Γ)-type spaces

In this section we consider phase-isometries from L(Γ)-type or 1-type spaces onto any normed spaces. We shall show that all such maps are phase equivalent to real linear isometries.

To show the following results of this section, we recall some notations and results about Birkhoff orthogonality. Let X be a real normed space. For all x,yX, let us denote by xy the Birkhoff orthogonality relation on X as:x+tyxfor alltR. This relation is clearly homogeneous, but neither symmetric nor

Acknowledgments

The authors wish to express their appreciation to Guanggui Ding for many very helpful comments regarding isometric theory in Banach spaces and the referees whose kind remarks helped them to make the presentation of the paper clearer.

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    The authors are supported by the Natural Science Foundation of China (Grant Nos. 11371201, 11201337, 11201338).

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