On normed spaces with the Wigner Property

The aim of this paper is to generalize the Wigner Theorem to real normed spaces. A normed space is said to have the Wigner Property if the Wigner Theorem holds for it. We prove that every two-dimensional real normed space has the Wigner Property. We also study the Wigner Property of real normed spaces of dimension at least three. It is also shown that strictly convex real normed spaces possess the Wigner Property.

The Wigner Theorem describes those transformations of P 1 (H) (all rank-one projections on the Hilbert space H), which preserve the transition probability. Gehér and Šemrl [11] described the surjective isometries on the Grassmann space P n (H) of all rank n projections with respect to the gap metric (n ∈ N), which generalize the Wigner Theorem to the Grassmann space P n (H). We would also like to draw the readers' attention to the very recent paper [16] in which the author establishes a Wigner's type theorem for linear operators which map projections of a fixed rank to projections of other fixed rank.
Bargmann [3] proved the Wigner Theorem which is very close to Wigner's original statement.
Theorem 1 [3] Let (H, ·, · ) be an inner product space with dim(H) ≥ 2 and let T : H → H satisfy the equality for any x, y ∈ H. Then there exists an isometry or an anti-isometry A on H and a phase function ε : H → C with |ε(x)| = 1 such that T (

x) = ε(x)A(x) for any x ∈ H. Moreover, if T is surjective and H is a Hilbert space, then A is a unitary or an anti-unitary operator.
In connection with (1) let us draw the readers' attention to rich references concerning the Wigner equation. In particular, in [6] the authors proved the existence of a solution (satisfying some additional condition) to the equation where I : M → N is a mapping between inner product modules M and N over certain C * -algebras.
It is natural to ask whether the Wigner Theorem would still be true if H is a normed space. Unfortunately, there is no inner product in general normed spaces. However, since in an inner product space, the Eq. (1) is equivalent to the equality { T (x) + αT (y) : |α| = 1} = { x + α y : |α| = 1} (2) for any x, y ∈ H, one can raise the following problem. We call an operator T between two normed spaces a phase isometry if T satisfies the equality (2). If E and F are both normed spaces over the field K, two mappings T 1 : E → F and T 2 : E → F are said to be differ by a phase factor or to be phase equivalent if there exists ε : E → K with |ε(x)| = 1 such that T 1 (x) = ε(x)T 2 (x) for any x ∈ E.
Problem 1 Let E and F be normed spaces over the field K and let T : E → F be a surjective phase isometry. Is T phase equivalent to a linear isometry L from E to F?
Recall that in 1932, Mazur and Ulam [15] showed that any surjective isometry between two real normed spaces is an affine map, that is it is a translation of a linear isometry. Bourgain [2] gave an example which shows that the complex version of the Mazur-Ulam Theorem is not valid.
In this paper we are going to consider the above problem in the case when K = R. Notice that by the Wigner Theorem, this problem is solved when both E and F are real inner product spaces. Now, we are going to recall the following Definition 1 Let E be a real normed space. E is said to have the Wigner Property if for any real normed space F, and any surjective phase isometry T : E → F, T is phase equivalent to a linear isometry from E to F.
Recently, Tan and Huang [20] proved that smooth real normed spaces have the Wigner Property. They also proved that some classical real Banach spaces, such as L ∞ ( )-type space and an 1 ( )-space for some index set , have the Wigner Property.
Let us briefly summarize the contents of this paper. In Sect. 2, we will study the property of a surjective phase isometry between two real normed spaces and will give some properties of a surjective phase isometry operator, which will be used in the sequel. In Sect. 3, we will show that any two-dimensional real normed space has the Wigner Property. Section 4 deals with the Wigner Property in real finite dimensional normed spaces of dimension at least three. In particular, we will prove that any strictly convex real normed space has the Wigner Property.
In this paper, we will use the standard notations. E * denotes the dual space of the normed space E. S E and B E denote the unit sphere and the closed unit ball of the normed space E, respectively. w * − exp(B E * ) denotes the set of w * exposed points of the unit ball B E * while ext(B E * ) denotes the set of extremal points of that ball. sm(S E ) denotes the set of smooth points of the sphere S E . [x, y] := {λx +(1−λ)y : λ ∈ [0, 1]} for any x, y ∈ E. Finally, by |A| we will denote the cardinality of the set A and span{A} will denote the linear subspace generated by the set A.

Preliminary results
In this section, we will study the general properties of surjective phase isometries between two real normed spaces. Lemmas 1 and 2 were given by Tan and Huang in the unpublished paper [20], so we include also their proofs for the readers' convenience.
Lemma 1 [20] Let E and F be real normed spaces and let T : Proof Putting y = x in the equality (2) we see that T preserves the norm. Next, putting y = −x in the equality (2), we get Using the equality (2) for x, y, z, we obtain This yields y, z ∈ {x, −x}. If z = x, then T (x) = −T (x) = 0, which contradicts to T (x) = 0, so we obtain z = −x. Now we will show that y = x. If not, we would get y = −x = z and This leads to the contradiction that T (x) = 0.
Lemma 2 [20] Let E and F be real normed spaces and let T : E → F be a phase isometry (that is not necessarily surjective). Then for every w * exposed point x * of B E * , there exists a linear functional ϕ ∈ F * of the norm one such that Proof First, we will prove that if E = R, then there exists a linear functional ϕ ∈ F * of the norm one such that ϕ(T (t)) ∈ {t, −t} for all t ∈ R. For every positive integer n, using the norm preserving property from Lemma 1, we have T (n) = n. The Hahn-Banach theorem guarantees the existence of a linear functional ϕ n ∈ S F * such that ϕ n = 1 and ϕ n (T (n)) = n. For every t ∈ [−n, n], we get or, alternatively, Then ϕ n (T (t)) ∈ {t, −t} for all t ∈ [−n, n]. It follows from Alaoglu's theorem that the sequence {ϕ n } has a cluster point ϕ in view of the w * topology. This entails that ϕ ≤ 1 and ϕ(T (t)) ∈ {t, −t} for every t ∈ R. Clearly, ϕ = 1 and ϕ is the desired mapping. Now suppose that dim(E) > 1 and u ∈ S E is a smooth point such that x * (u) = 1. Let G : R → F be defined by G(t) = T (tu) for t ∈ R . Then G is a phase isometry. By the proof above, there exists ϕ ∈ F * with ϕ = 1 such that ϕ(T (tu)) = ϕ(G(t)) ∈ {t, −t}.
Since u is a smooth point, it follows that x * is the only one supporting functional at u. Therefore, for every x ∈ X , From the Eq. (2), we get for all t > 0 and x ∈ E. For a fixed nonzero vector x ∈ E, the set (0, +∞) will be divided into four parts: Obviously, at least one of the sets {A i : i = 1, 2, 3, 4} is unbounded. We shall prove that if A i is unbounded, then for all i = 1, 2, 3, 4. Without loss of generality we can assume that A 1 is unbounded. Then, for every t ∈ A 1 , we get Leting t → +∞ in the two inequalities above, we get This completes the proof.

Lemma 3 Let E and F be real normed spaces, T : E → F be a surjective phase isometry. For all x, y ∈ E and a
This completes the proof.
Define the linear operator V : The following lemma is a simple case in dimension two of the fact that two linear functionals are linearly dependent if and only if they have the same kernel space. [13], the corollary on p.5). Because x * 1 , x * 2 ∈ S E * , we infer that k = 1 or k = −1.

Theorem 2 Let E and F be real normed spaces and let T : E → F be a surjective phase isometry. If for any two linear independent elements
Moreover, T is phase equivalent to a homogeneous surjective phase isometry.
Proof By Lemma 1, we only have to show that for all x ∈ S E and t > 0. If not, there would exist x 0 ∈ S E and t 0 > 0 such that neither Since By (4) and To prove the last part of the theorem, by the axiom of choice, there is a set L such that for every 0 = x ∈ E there exists a unique element y ∈ E such that x = sy for some s ∈ R. Define H : E → F by Then H is well defined, homogeneous and phase equivalent to T .
Next result shows that a surjective phase isometry preserves the strong convexity of its domain.

Theorem 3 Let E and F be two real normed spaces, and let E be strictly convex. If there exists a surjective phase isometry T : E → F, then F is also a strictly convex real normed space.
Proof If F is not strictly convex, then there exist x, y ∈ S F such that [x, y] ⊂ S F . Since T is surjective, there exist x , y ∈ S E such that T (x ) = x and T (y ) = y.
Because T is a phase isometry, we have Therefore [x , y ] ⊂ S E or [−x , y ] ⊂ S E , which contradicts to the strict convexity of E.

Two-dimensional normed spaces with the Wigner Property
In this section, we will show that any two-dimensional real normed spaces have the Wigner Property. First, we recall some definitions and notations.
Definition 2 [8] Let E be a real normed space. For any x, y ∈ S E , x = −y, we define the arc of x and y to be the set Freese et al. [8, Theorem 2.1, Theorem 2.2] gave the following important theorem describing a certain property of a unit sphere of a two-dimensional real normed space.

Theorem 4 [8] Let E be a two-dimensional real normed space and let x, y be linearly independent elements of S E . If z ∈
Let E be a normed space. Suppose that x, y ∈ E. Then x is said to be isosceles or thogonal to y, denoted by x⊥ I y, if x + y = x − y . Alonso [1] proved the existence and uniqueness of an isosceles orthogonal element of a unit sphere to any element of the unit sphere under consideration.

Theorem 5 Let E and F be real normed spaces, dim(E) = 2 and let T : E → F be a surjective phase isometry. Then T is phase equivalent to a homogeneous surjective phase isometry.
Proof By Lemma 7, we infer that dim(F) = 2. Using Theorem 2 we deduce that T is phase equivalent to a homogeneous surjective phase isometry.

Theorem 6 Let E and F be real normed spaces, dim(E) = 2 and let T : E → F be a surjective phase isometry.
If there exist two linearly independent vectors x 0 , y 0 ∈ S E such that the following conditions holds for any a, b ∈ R: are two real numbers α(a, b), β(a, b) with |α(a, b) β(a, b)T (by 0 ); (ii) ax 0 + by 0 ≥ max{a, b} for a ≥ 0 and b ≥ 0.

Then T is phase equivalent to a linear isometry.
Proof Since phase equivalence is an equivalence relationship between all surjective phase isometries, by Theorem 5, we may assume that T is a homogeneous surjective phase isometry.
The equation α(a, 1)β(a, 1) = α(1, 1)β(1, 1) for all a = 0, and the definition of L show that This means that L is a linear isometry from E onto F. The proof is complete.
Theorem 6 is important in the study of surjective phase isometry operators between two dimensional real normed spaces. We will use it to show first that any two dimensional strictly convex real normed space and next to show that any two dimensional non-strictly convex real normed space has the Wigner Property.

Theorem 7 Let E be a two dimensional strictly convex real normed space and F be a real normed space, and let T : E → F be a surjective phase isometry. Then T is phase equivalent to a linear isometry.
Proof Since the set of smooth points of S E is dense in S E , we can choose x 0 , y 0 ∈ S E and 0 < λ 0 < 1 such that x 0 − y 0 < 1 and λ 0 We have Similarly, we obtain y * 0 (x 0 ) > 0, z * 0 (x 0 ) > 0 and z * 0 (y 0 ) > 0. For any a > 0, b > 0, we have Let 0 < λ < 1. Since without loss of generality, we may assume that T (x 0 ) − T (y 0 ) = x 0 − y 0 . We have Other cases can be discussed similarly. If By Theorem 3, the normed space F is strictly convex, so we obtain By Theorem 6, we infer that T is phase equivalent to a linear isometry.

Theorem 8 Let E be a two-dimensional non-strictly convex real normed space and F be a real normed space, let T : E → F be a surjective phase isometry. Then T is a phase equivalent to a linear isometry.
Proof We will divide the proof into two cases. Case one, if E is isometric to l (2) 1 . Let x 0 = e 1 = (1, 0) and y 0 = e 2 = (0, 1). Then, by Lemma 8 and Theorem 6, we infer that T is a phase equivalent to a linear isometry.
Case two, if E is not isometric to l Without loss of generality, we assume that }, for i = 1, 2, 3. By Lemma 5, we obtain that λx 0 −(1−λ)y 0 ∈ {±x 2 }. Then, by Lemma 8 and Theorem 6, we infer that T is a phase equivalent to a linear isometry. This completes the proof. Now, we can obtain the main result of this section using Theorems 7 and 8.

Theorem 9
If E is a two-dimensional real normed space, then E has the Wigner Property.

Wigner Property on real Banach spaces of dimension at least three
In the proof of the Wigner Theorem from [22], Uhlhorn highlights the connection between the Wigner Theorem and the First Fundamental Theorem of projective geometry. The First Fundamental Theorem of projective geometry says that an abstract automorphism of the set of lines in vector spaces which preserves "incidence relations" must have a simple algebraic form (see [7]). In this section, we will show that the First Fundamental Theorem of projective geometry also plays an important role in the study of real normed spaces with the Wigner Property.
Let f be a mapping from a set X into a set Y and let D be a subset of 2 X -the power set of X . A mapping F : D → 2 Y is said to be induced or generated by f if for every M ∈ D, F(M) = { f (m) : m ∈ M}. As usual, this last set is also noted f (M). If X is a real vector space, we denote the projectivised space (that is the set of all one-deminsional subspaces) by P(X ). The element of P(X ) generated by 0 = x ∈ X will be denoted by [x] := R · x.
Gehér [9, Theorem 3] proved the following special case of the First Fundamental Theorem of projective geometry for real vector spaces, which will be used later.

Proposition 1 [9]
Let E 1 and E 2 be two real vector spaces of dimensions at least three. If g : P(E 1 ) → P(E 2 ) satisfies the following conditions:

By Proposition 1, there exists an injective linear map
for every x ∈ E. Consequently, there exists a function λ : E → R such that T (x) = λ(x)A(x) for every x ∈ X . Since T is homogeneous, λ(t x) = λ(x) for every x ∈ E and 0 = t ∈ R. Moreover, suppose that x, y ∈ E are two linearly independent vectors. Let us write T (x +y) = αT (x)+βT (y) for some real numbers α and β with |α| = |β| = 1. We immediately obtain so λ(x + y) = αλ(x) = βλ(y). As a consequence, |λ(x)| is a constant for any x ∈ E, which we denote by λ. Hence we can define a desired phase function ε : E → {−1, 1} such that T = ελA. Thus T is phase equivalent to the linear isometry λA.
Using the above result we can prove the following Theorem 11 Let E be a real strictly convex normed space. Then E has the Wigner Property.
Proof If dim(E) = 2, then E has the Wigner Property by Theorem 9. Let us assume that dim(E) ≥ 3, F is any real normed space and T : E → F is a surjective phase isometry. We will show that T is a phase equivalent to a linear isometry. By Theorem 10, we need only to prove that T (span{x, y}) = span{T (x), T (y)} for any two linear independent elements x, y ∈ E.
Since every inner product space is a strictly convex normed space, by Theorem 11, one can easily get the following Corollary 1 Let E be a real inner product space and F be a real normed space. Let T : E → F be a surjective phase isometry. Then F is a real inner product space.

Theorem 12
Let E be a real normed space. If for any three different points x, y, z ∈ E with x ∈ { y , z } there exists x * ∈ w * − ex p(B E * 0 ) such that x * (x) / ∈ {±x * (y), ±x * (z)}, where E 0 = span{x, y, z}, then E has the Wigner Property.
It is obvious that the dimension of the subspace E 0 in Theorem 12 is less or equal to three. Thus a three-dimensional subspace plays a very important role in the study of real normed spaces with the Wigner Property. Hence it seems to be natural to rise the following two problems. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.