1. Introduction
Throughout this paper, we consider the spaces all over the real field. Let $X$
and $Y$
be Banach spaces. We say that a map $f: X \rightarrow Y$
is a phase-isometry if it satisfies
Two maps $f,\, g :X \rightarrow Y$
are called phase equivalent if there exists a phase function $\varepsilon : X \rightarrow \{-1,\, 1\}$
such that $g=\varepsilon \cdot f$
. In the present paper, it is the connection between phase-isometries and isometries that will concern us. Under the onto assumption, the well-known Mazur–Ulam theorem [Reference Mazur and Ulam16] states that an isometry $f$
of a real Banach space $X$
onto another Banach space $Y$
with $f(0)=0$
is linear. Thus, it is natural to ask
Question 1.1 Is it true that every surjective phase-isometry $f$
between two Banach spaces $X$
and $Y$
is phase equivalent to a linear isometry?
Another important result which is related to linear isometries is Wigner's theorem [Reference Wigner25] and its generalization (see [Reference Chevalier2, Reference Gehér4, Reference Györy5, Reference Molnár17, Reference Rätz19]). Rätz's result [Reference Rätz19, Corollary 8(a)] showed that every map $f$
between two real Hilbert spaces $H$
and $K$
with the property that
is phase equivalent to a linear isometry. Recently, Maksa and Páles [Reference Maksa and Páles12] revisited this real version of Wigner's theorem, and they used this to obtain that every phase-isometry between real Hilbert spaces is phase equivalent to a linear isometry. Namely, Question 1.1 has a positive answer when $X$
and $Y$
are real Hilbert spaces. In our paper [Reference Huang and Tan8], we provide an affirmative answer to Question 1.1 with $X$
and $Y$
being $l_p(\Gamma )$
spaces $(0< p<\infty )$
. The first author and Jia presented in [Reference Jia and Tan9] a similar result for $\mathcal {L}_{\infty }(\Gamma )$
-type spaces. Recall that a Banach space $X$
is said to have the Wigner property if Question 1.1 has an affirmative answer for an arbitrary target $Y$
. It should be mentioned that the previous results for real Hilbert spaces, $l_p(\Gamma )$
spaces or $\mathcal {L}_{\infty }(\Gamma )$
-type spaces are not enough to establish that these spaces have the Wigner property since the target spaces are not arbitrary. Very recently, Question 1.1 for general Banach space has started in our paper [Reference Tan and Huang21], in which we proved that smooth Banach spaces, $\mathcal {L}_{\infty }(\Gamma )$
-type spaces and $l_1(\Gamma )$
-spaces enjoy this property.
Although this manuscript is a continuation of our paper [Reference Tan and Huang21], the techniques we use in this note are totally different from those in [Reference Tan and Huang21]. We use a new method to construct the desired linear isometries. It is also a key step to solve Question 1.1. Actually, although our main result is about polyhedral spaces and CL-spaces, more properties of phase-isometries are shown to hold in general Banach spaces for the sake of possible applications. In §2, we introduce a notion of ‘star points’ to study the maximal convex subsets (i.e., facets) of the unit sphere. This notion enables us to show that every phase-isometry between two Banach spaces maps star points to star points in §3. By this, we can build a required isometry between two cones generated by the maximal convex subsets of corresponding unit spheres. We apply this result in §4 to establish that finite-dimensional polyhedral spaces (i.e., those spaces whose unit sphere is a polyhedron) and CL-spaces which include many classical Banach spaces have the Wigner property. It is known that CL-spaces are very famous and important in the development of the geometry of Banach spaces. Recall from [Reference Martín and Payá15] that
Definition 1.2 A Banach space $X$
is said to be a CL-space (an almost CL-space) if $B_X=\mbox {co}(M\cup -M)$
($B_X=\overline {\mbox {co}}(M\cup -M$
), respectively) for every maximal convex set $M$
of $S_X$
.
The notion of CL-spaces was first introduced in 1960 by R. Fullerton [Reference Fullerton3]. Examples of real CL-spaces are $L_1(\mu )$
spaces for any measure $\mu$
and its isometric preduals, in particular, $C(K)$
, where $K$
is a compact Hausdorff space (see [Reference Lima10, §3]). The $c_0$
and $l^{1}$
sums of CL-spaces are also proved to be CL-spaces (see [Reference Martín and Payá15]). For more information on CL-spaces, we refer to [Reference Cheng and Li1, Reference Hansen and Lima6, Reference López, Martín and Payá11, Reference Martín14, Reference Reisner20, Reference Tan, Huang and Liu22, Reference Tan and Liu23]. Let us mention that the key part in handling Question 1.1 is how to construct or find a proper linear isometry with the desired property. Although we can provide an affirmative answer to Question 1.1 for CL-spaces, we do not know whether this is true for almost-CL-spaces.
Throughout what follows, we shall freely use without explicit mention notation $``\pm "$
to mean that either $``+"$
or $``-"$
holds. For a Banach space $X$
, $B_X$
, $S_X$
and $X^{*}$
will stand for its unit ball, its unit sphere and its dual space, respectively. A subset $M$
of $S_X$
is said to be a maximal convex subset of $S_X$
if it is not properly contained in any other convex subset of $S_X$
.
2. Star points on the unit sphere of Banach spaces
In this section, we introduce the concept of star points in the unit sphere of Banach spaces. This concept plays an important role in showing main lemma and some propositions. Geometric characterizations of such points are also given for further results in the next section.
Before our conclusions, we need some notation and concepts most of which come from [Reference Holmes7]. By the Hahn–Banach and Krein–Milman theorems, we can see that every maximal convex subset $M$
of $S_X$
has the form
for some extreme point $x^{*}$
of $B_{X^{*}}$
. We denote by $\mathfrak {M}(X)$
the set made up of all maximal convex subsets of $S_X$
, i.e.,
Given $x\in S_X$
, the star of $x$
with respect to $S_X$
is defined by
It is easy to check that $\|x+y\|=2$
if and only if $[x,\,y]\subset S_X$
for every $x,\, y\in S_X$
. Then we can rewrite
Clearly, for every $x\in S_X$
, there is a maximal convex subset of $S_X$
containing $x$
. Thus $St(x)$
is precisely the union of all maximal convex subsets of $S_X$
containing $x$
, i.e.,
It is interesting to give a result demonstrating the relationship between the maximal convex subsets and the star sets of unit sphere. The proof here is based on an idea of [Reference Mori18, Proposition 2.3], and given only for the sake of completeness.
Proposition 2.1 Let $X$
be a Banach space, and let $M\in \mathfrak {M}(X)$
. Then $M$
is precisely the intersection of all $St(x)$
with $x\in M,$
that is, $M=\bigcap_{x\in M} St(x)$
.
Proof. Note first that $M\subset St(x)$
for every $x\in M$
implies that
To prove the converse, choose $z\in \bigcap_{x\in M} St(x)$
. Then $\|z+x\|=2$
for all $x\in M$
. Set two open convex sets
where $\mbox {int}B_X$
denotes the interior points of $B_X$
. Obviously, $S_1\cap S_2=\emptyset$
. By the separation theorem, there is a linear functional $x^{*}\in S_{X^{*}}$
and a real number $c$
such that $x^{*}(z_1)>c>x^{*}(z_2)$
for every $z_1\in S_1$
and every $z_2\in S_2$
. It follows that
for every $x\in M$
. Consequently,
for every $x\in M$
. This shows that $c=0$
and $x^{*}(z)=x^{*}(x)=1$
for every $x\in M$
. By the maximality of $M$
, we have $M=\{x\in S_X: x^{*}(x)=1\}$
, and hence $z\in M$
. The proof is complete.
We will provide a concept which turns out to be a useful tool for analysing maximal convex subsets of unit sphere of Banach spaces.
Definition 2.2 Let $X$
be a Banach space. A point $x\in S_X$
is called a star point of $S_X$
if $St(x)$
is a maximal convex subset of $S_X$
.
We can see that an element $x\in S_X$
is a star point if and only if $St(x)$
is convex since that $St(x)$
is convex is sufficient to show that $St(x)$
is a maximal subset of $S_X$
. For better understanding of star points, we now give two characterizations of star points on the unit sphere which will be of use later.
Proposition 2.3 Let $X$
be a Banach space and $x\in S_X$
. Then the following statements are equivalent:
(1) $x$
is a star point of $S_X$.(2) For all $y,\, z\in St(x),$
we have $\|y+z\|=2$.(3) There is a unique maximal convex set of $S_X$
containing $x$.
Proof. That (1) implies (2) is obvious. To prove that (2) implies (3), suppose that there are two sets $M_1,\, M_2\in \mathfrak {M}(X)$
containing $x$
. Clearly, $M_1\cup M_2 \subset St(x)$
. Now, take an arbitrary $z\in M_1$
. Since (2) holds, it follows that $[z,\, y]\subset S_X$
for every $y\in M_2$
. Hence we have $M_2 \subset \mbox {co}(M_2,\, z) \subset S_X$
. Then the maximality of $M_2$
ensures us that $\mbox {co}(M_2,\,z)=M_2$
, and thus $z\in M_2$
. It follows that $M_1\subset M_2$
, and so $M_1=M_2$
. Finally, (3) implies (1) following from the identity (2.1).
By the previous proposition, we can present an example here that draws upon [Reference Holmes7, Exercise 2.18], which shows that every maximal convex set of the unit sphere in a separable Banach space has star points.
Example 2.4 If $X$
is a separable Banach space, then $S_X$
has star points. Indeed, let $M\in \mathfrak {M}(X)$
. Choose $\{x_n\}$
to be a dense subset of $M$
. Then the point $x_0=\sum \nolimits 2^{-n}x_n$
is a star point of $S_X$
. Actually, for every $M_1\in \mathfrak {M}(X)$
containing $x_0$
, there exists a functional $x^{*}\in S_{X^{*}}$
such that $M_1=\{x\in S_X: x^{*}(x)=1\}$
. In particular, $x^{*}(x_0)=1$
implies that $x^{*}(x_n)=1$
for all $n\in \mathbb {N}$
. This and the maximality of $M$
show that $M=M_1$
. So there is a unique maximal convex set of $S_X$
containing $x_0$
. This completes the proof by Proposition 2.3.
The notion of star points relates to the familiar notion of smooth points in Banach spaces.
Remark 2.5 Recall that $x\in S_X$
is called a smooth point if there exists only one support functional $x^{*}\in S_{X^{*}}$
at $x$
. Proposition 2.3 and the Hahn–Banach theorem establish that if $x$
is a smooth point of $S_X$
, then it is a star point of $S_X$
. It should be remarked that the smooth points of $S_X$
are dense in $S_X$
, whenever $X$
is a separable Banach space [Reference Holmes7, p. 171]. Therefore, for any separable Banach space $X$
, the set consisting of star points is dense in $S_X$
.
By the above results, one may wonder whether every Banach space possesses a star point. The following easy example shows that it is not true.
Example 2.6 Let $\Gamma$
be an uncountable set. Note that for every $x\in l_1(\Gamma )$
, $St(x)$
is not a convex subset of $S_{l_1(\Gamma )}$
. It follows that the unit sphere of $l_1(\Gamma )$
does not have star points.
Let $X$
be a Banach space and $M\in \mathfrak {M}(X)$
. Recall that a point $x\in M$
is called a non-proper support point of $M$
if any $x^{*}\in S_{X^{*}}$
with $x^{*}(x)=1$
, we have $x^{*}(z)=1$
for all $z\in M$
. It is routine to verify that there is only a unique maximal convex set $M$
of $S_X$
containing $x$
. So a non-proper support point of $M$
is a star point. Now we may conclude a direct relation from the above arguments that for every $x\in S_X$
,
However, none of the converse is true, even for the finite-dimensional spaces.
Example 2.7
(1) For each $0< t<1$
, let us write $X_t=(\mathbb {R}^{2},\,\|\cdot \|)$, where the unit sphere is given by
Then the point $(t,\, \sqrt {1-t^{2}})$
is a star point but not a non-proper support point.
(1) Define an equivalent norm $\|\cdot \|$
on $l_1$ by
where $\|\cdot \|_{l_1}$
and $\|\cdot \|_{l_2}$
denote the usual $l_1$
-norm and $l_2$
-norm, respectively. Then the unit vectors $\{e_n\}_{n\geq 1}\subset X=(l_1,\, \|\cdot \|)$
are all non-proper support points but not smooth points. Indeed, $X$
is a Banach space with a strict convex norm. Therefore, every maximal convex subset of $S_X$
is a singleton. It follows that all points of the unit sphere are non-proper support points, whereas there are more than one support functional in $S_{X^{*}}$
for each $e_n$
with $n\geq 1$
. For example, $e_n^{*}=(a_i)_{i\geq 1}$
where $a_i\in \mathbb {R}$
with $a_n=1$
and $a_i=0$
for $i\neq n$
. Then for each $j\neq n$
, $x_n^{*}=e_n^{*}+\tfrac {1}{2}e_j^{*}\in S_{X^{*}}$
is a support functional at $e_n$
. Thus, $e_n$
is not a smooth point. One can easily check that this conclusion also holds for the $l _1^{(m)}$
which is $l_1$
-spaces in dimension $m$
with $m\geq 2$
.
Remark 2.8 Although three conceptions are quite different in general Banach spaces, they are the same in some classical Banach spaces, such as $L_p(\mu )$
($\mu$
is a measure and $1\leq p\leq \infty$
) and $C(K)$
($K$
is a compact Hausdorff space).
3. The properties of phase-isometries on Banach spaces
In this section, we start the study of surjective phase-isometries between two Banach spaces $X$
and $Y$
. We start this section with a basic property of surjective phase-isometry which was verified in [Reference Tan and Huang21, lemma 2.1].
Lemma 3.1 (Tan and Huang [Reference Tan and Huang21, lemma 2.1])
Let $X$
and $Y$
be Banach spaces, and let $f: X \rightarrow Y$
be a surjective phase-isometry. Then $f$
is an injective norm-preserving map and $f(-x)=-f(x)$
for all $x\in X$
.
An elementary result is presented to show that every phase-isometry carries star points to star points and we do not know whether this is true for smooth points and non-proper support points.
Lemma 3.2 Let $X$
and $Y$
be Banach spaces, and let $f:X\rightarrow Y$
be a surjective phase-isometry. Then for every $x\in S_X,$
$St(x)\in \mathfrak {M}(X)$
if and only if $St(f(x))\in \mathfrak {M}(Y)$
.
Proof. Since the inverse of $f$
denoted by $f^{-1}$
is also a surjective phase-isometry, we only need to prove that if $St(x)\in \mathfrak {M}(X)$
then $St(f(x))\in \mathfrak {M}(Y)$
. Suppose that it is not true. Then Proposition 2.3 allows the existence of two distinct elements $y_1,\, y_2\in St(f(x))$
such that $\|y_1+y_2\|<2$
. We set $z_1:=\tfrac {1}{2}(y_1+f(x))$
and $z_2:=\tfrac {1}{2}(y_2+f(x))$
. It is easy to see that $z_1,\, z_2\in St(f(x))$
satisfy
Since $f$
is surjective, there are $x_1,\,x_2\in S_X$
such that $f(x_1)=z_1$
and $f(x_2)=z_2$
. Observe from the fact that $f$
is a phase-isometry that
Thus, $\|x_1+x_2\|=2$
or $\|x_1-x_2\|=2$
, which leads to a contradiction that $\|z_1+z_2\|=2$
or $\|z_1-z_2\|=2$
. The proof is complete.
We shall take up to construct maps concerning the maximal convex sets of the unit sphere, which is also a key step to obtain a desired linear isometry from $X$
onto $Y$
.
Definition 3.3 Let $X$
be a Banach space and $M \in \mathfrak {M}(X)$
. Denote by $C_M^{X}$
the cone generated by $M,$
i.e., $C_M^{X}=\cup _{\lambda \geq 0}\lambda M$
. We see that $\|x+y\|=\|x\|+\|y\|$
for all $x,\, y\in C_M^{X}$
. We shall freely use this observation without explicit mention throughout what follows.
Let $X$
and $Y$
be Banach spaces, and let $f:X\rightarrow Y$
be a surjective phase-isometry. Suppose that $M \in \mathfrak {M}(X)$
and $x_0 \in M$
. We define a map $f_{M}: C_M^{X} \rightarrow Y$
with respect to $x_0$
given by
We can also define a phase-isometry $F_M:X\rightarrow Y$
given by
Remark 3.4 It is easy to see that for every $x\in C_M^{X}$
, we have
Moreover, we can see that the map $F_M:X\rightarrow Y$
is an odd mapping, since $f(-x)=-f(x)$
for every $x\in X$
by Lemma 3.1. It follows that $F_M:X\rightarrow Y$
is surjective phase-isometry. Then, Lemma 3.1 again establishes that $F_M$
is a bijection and $f_{M}: C_M^{X} \rightarrow Y$
is injective.
We now deal with the case that maximal convex set $M$
has a star point. An interesting property of $f_{M}$
is presented below.
Proposition 3.5 Let $X$
and $Y$
be Banach spaces, and let $f:X\rightarrow Y$
be a surjective phase-isometry. Suppose that $M \in \mathfrak {M}(X)$
contains a star point $x_0$
. Let $f_{M}: C_M^{X} \rightarrow Y$
be a map with respect to $x_0$
. Then $M_1=f_M(M)\in \mathfrak {M}(Y)$
and $f_{M}(C_M^{X})$
is a cone in $Y$
generated by $M_1$
. Furthermore, $f_{M}: C_M^{X} \rightarrow C_{M_1}^{Y}$
is a surjective isometry.
Proof. By Lemma 3.2 and Remark 3.4, it is apparent that $St(f(x_0))\in \mathfrak {M}(Y)$
and $f_M(M)\subset St(f(x_0))$
. For every $y\in St(f(x_0))$
with $f(x)=y$
for some $x\in S_X$
. Since $f$
is a phase-isometry, we see that
It follows that $x$
or $-x \in St(x_0)=M$
by Proposition 2.3. As a consequence, we get $y=f_M(x)$
with $x\in M$
or $y=f_M(-x)$
with $-x\in M$
, which implies that $f_M(M)=St(f(x_0))$
.
Set $M_1:=f_{M}(M)$
. It remains to verify that $f_M(C_M^{X})=C_{M_1}^{Y}$
. By the definition of $f_{M}$
, for every $0\neq x\in C_M^{X}$
, we have
This allows us to conclude that
An immediate conclusion from this is that $f_M(x)/\|f_M(x)\| \in St(f_M(x_0))=M_1$
, and so $f_M(x)\in C_{M_1}^{Y}$
. For every $0\neq y\in C_{M_1}^{Y}$
with $f(x)=y$
for some $x\in X$
. Note that $\|f(x)+f(x_0)\|=\|f(x)\|+\|f(x_0)\|$
, and since $f$
is a phase-isometry, we have either
or
It follows that $\pm x/\|x\|\in St(x_0)=M$
. As a consequence, $f(x)=f_{M}(x)$
with $x\in C_M^{X}$
or $f(x)=f_{M}(-x)$
with $-x\in C_M^{X}$
. Both cases show that $y=f(x)\in f_{M}(C_M^{X})$
. Thus, we finish the proof of $f_{M}(C_M^{X})=C_{M_1}^{Y}$
.
Finally, as seen from the above for all $x,\,y\in C_M^{X}$
, we have
That $f_{M}$
is a phase-isometry leads to
as required, which finishes the proof.
One may have a question whether the definition of $f_M$
depends on the choice of $x_0$
. It may have something to do with the choice of $x_0$
in some degree.
Lemma 3.6 Let $X$
and $Y$
be Banach spaces, and let $f:X\rightarrow Y$
be a surjective phase-isometry. Suppose that $M\in \mathfrak {M}$
has a star point. Then $f_{M}=\pm f'_{M},$
where $f_{M}$
and $f'_{M}$
are defined with respect to two distinct points $x_0$
and $x_0'$
of $M$
respectively.
Proof. We can assume that $x_0\in M$
is a star point. Then $f(x_0)$
is also a star point of $S_Y$
by Lemma 3.2. Since $f$
is a phase-isometry, we have
If $\|f(x_0)+f(x'_0)\|=\|x_0+x'_0\|=2$
, then $f(x'_0)\in St(f(x_0))$
. We conclude from this and $f_M$
's definition that for every $0\neq x\in C_M^{X}$
and $\theta \in \{-1,\,1\}$
, $f_M(x)=\theta f(x)$
implies that $f_M'(x)=\theta f(x)$
. Namely, $f_M=f_M'$
. Another possibility is that
Applying a similar argument as above shows that $f_M'=-f_M$
.
We still need more lemmas to investigate the property of these $f_{M}$
's. The first one bases completely on the idea of [Reference Tanaka24, Lemma 3.4] (we apply almost the same proof). For our use and convenience, we present here a generalized version which also establishes that all surjective phase-isometries are separably determined in some sense. For every subset $A$
of a Banach space, $[A]$
stands for its closed linear span.
Lemma 3.7 Let $X$
and $Y$
be two Banach spaces, and let $f: X \rightarrow Y$
be a surjective phase-isometry. Then for every separable subset $A\subset X,$
there are separable subspaces $X_0\subset X$
and $Y_0\subset Y$
such that $A\subset X_0$
and $f(X_0)=Y_0$
.
Proof. Let $X_1=[A]$
and $Y_1=[f(X_1)]$
. Choose $\{x_n\}\subset X_1$
which is dense in $X_1$
. To see that $Y_1$
is separable, it suffices to show that the set $\{f(x_n)\}\cup \{-f(x_n)\}$
is dense in $f(X_1)$
. By Lemma 3.1, it is clear that $\{f(x_n)\}\cup \{-f(x_n)\} \subset f(X_1)$
. Now for every $y\in f(X_1)$
with $f(x)=y$
for some $x\in X_1$
, there is a subsequence $\{x_{n_k}\}$
of $\{x_n\}$
such that $x_{n_k}$
converges to $x$
in norm. For every $k\geq 1$
, choose $y_k\in Y$
satisfying
Thus, it is obvious that $\|y_k-y\|=\|y_k-f(x)\|=\|x_{n_k}-x\|.$
This means that $\{f(x_n)\}\cup \{-f(x_n)\}$
is dense in $f(X_1)$
as desired. Define inductively separable subspaces $X_n\subset X$
and $Y_n\subset Y$
such that for all $n\geq 2$
,
Then it is easy to verify that $X_0=\bigcup_{n=1}^{\infty } X_n$
and $Y_0=\bigcup_{n=1}^{\infty } Y_n$
are just the subspaces that we need.
We will follow the line of the proof of [Reference Tanaka24, Lemma 3.5] to obtain a similar conclusion for phase-isometries instead of isometries. By Lemmas 3.6 and 3.7, we are now able to generalize Proposition 3.5 to the general case where the assumption on star points can be removed.
Proposition 3.8 Let $X$
and $Y$
be Banach spaces, and let $f:X\rightarrow Y$
be a surjective phase-isometry. Suppose that $M \in \mathfrak {M}(X)$
and $f_{M}: C_M^{X} \rightarrow Y$
is the map with respect to some $x_0 \in M$
. Then $f_M(M)\in \mathfrak {M}(Y)$
and $f_{M}(C_M^{X})$
is a cone in $Y$
generated by $f_{M}(M)$
. Furthermore, $f_{M}$
is an isometry on $C_M^{X}$
.
Proof. We first conclude that this conclusion is true under the assumption that $X$
is a separable Banach space. By Example 2.4, $M$
has a star point. In other words, if the chosen $x_0\in M$
is not a star point, then by Lemma 3.6 we can replace it by a star point $x'_0\in M$
. Proposition 3.5 yields the desired conclusion.
Now we handle the general case. For every finite subset $A\subset M$
with $x_0\in A$
, we apply Lemma 3.7 to $F_M$
to obtain two separable subspaces $X_A\subset X$
and $Y_A\subset Y$
such that $A\subset X_A$
and $F_M(X_A)=Y_A$
. Let $M_A\in \mathfrak {M}(X_A)$
such that $M\cap X_A\subset M_A$
. The previous argument ensures us that
where $f_{M_A}: C_{M_A}^{X_A} \rightarrow Y_A$
is the induced map with respect to $x_0$
and $M_A$
. As $A\subset M\cap M_A$
, we have $f_{M}(x)=f_{M_A}(x)$
for every $x\in A$
. It follows that
So we can choose $M_1\in \mathfrak { M}(Y)$
such that $f_M(M)\subset M_1$
. For the converse, since the inverse $f^{-1}$
of $f$
is also a surjective phase-isometry, there is a map $h_{M_1}:C_{M_1}^{Y}\rightarrow X$
with respect to $f(x_0)\in M_1$
and induced from $f^{-1}$
. We can see that
for all $x\in C_M^{X}$
. Using the above argument for $h_{M_1}$
and $M_1$
, we also have
This and the maximality of $M$
entail that $M=h_{M_1}(M_1)$
, and so $f_M(M)=M_1$
.
We also need to prove that $f_M(C_M^{X})=C_{M_1}^{Y}$
, where $M_1=f_M(M)$
. For all $0\neq z\in C_M^{X}$
and $x\in M$
, set $B:=\{z,\, x,\, x_0\}$
. Following a similar argument as above, we get two separable subspaces $X_B\subset X$
and $Y_B\subset Y$
such that $B\subset X_B$
and $F_M(X_B)=Y_B$
. Let $f_{M_B}: C_{M_B}^{X_B} \rightarrow Y_B$
be the induced map with respect to $x_0$
and $M_B$
. Then $f_{M_B}(M_B)\in \mathfrak {M}(Y_B)$
and $f_{M_B}(C_{M_B}^{X_B})\subset Y_B$
is a cone generated by $f_{M_B}(M_B)$
. It follows that
Consequently, we obtain $f_{M}(z)/\|f_{M}(z)\|\in St (f_{M}(x))$
for every $x\in M$
. Applying Proposition 2.1, we get $f_{M}(z)/\|f_{M}(z)\|\in M_1$
, and so $f_M(C_M^{X})\subset C_{M_1}^{Y}$
. Conversely, repeating the same argument to $h_{M_1}: C_{M_1}^{Y} \rightarrow X$
, we obtain $h_{M_1}(C_{M_1}^{Y})\subset C_M^{X}$
. Observe that the map $h_{M_1}$
is injective and
for all $x\in C_M^{X}$
. We get $f_M(C_M^{X})=C_{M_1}^{Y}$
as desired. It is simply verified that $f_M: C_M^{X} \rightarrow Y$
is an isometry by noticing that
for all $x,\,y\in C_M^{X}$
. The proof is complete.
Remark 3.9 It should be noted that when $E$
is a subspace of a Banach space $X$
and $M$
is a maximal convex set of $S_X$
, $M_E=M\cap E$
may not be a maximal convex set of $S_E$
by [Reference Tan and Liu23, Remark 2.3]. So we can not replace $M_A$
by $M\cap X_A$
to simplify the proof of Proposition 3.8.
Proposition 3.8 also allows us to conclude that Lemma 3.6 still works in the general case. For this reason, in what follows we shall briefly use the notation $f_M$
referring to the map defined in Definition 3.3 without explicit mention $x_0$
unless it is necessary. We sum up now the result as follows.
Corollary 3.10 Let $X$
and $Y$
be Banach spaces, and let $f:X\rightarrow Y$
be a surjective phase-isometry. Suppose that $M\in \mathfrak {M}(X)$
and $x_0,\, x_0'\in M$
. Then $f_{M}=\pm f'_{M}$
, where $f_{M}$
and $f'_{M}$
are defined with respect to $x_0$
and $x_0'$
of $M$
respectively.
For every Banach space $X$
, since $S_X$
consists of all its maximal convex subsets $M$
, it seems that a demanded isometry $F$
which is phase-equivalent to $f$
on the entire space $X$
can be induced from these $f_{M}$
's. It can not be simply got by defining $F:X\rightarrow Y$
by $F(x)=f_{M}(x)$
for every $M\in \mathfrak {M}(X)$
and $x\in M$
. A difficulty should be noted here that for every $x$
in the intersection of the two cones $C_{M_1}^{X}$
and $C_{M_2}^{X}$
of distinct sets $M_1,\, M_2\in \mathfrak {M}(X)$
. It may happen that $f_{{M_1}}(x)\neq f_{{M_2}}(x)$
, for example, $f_{{M_1}}(x)=f(x)$
, while $f_{{M_2}}(x)=-f(x)$
. This means that the definition of $F$
may be actually not valid. We shall present a lemma below as a first step to overcome this difficulty.
Lemma 3.11 Let $X$
and $Y$
be Banach spaces, and let $f:X\rightarrow Y$
be a surjective phase-isometry. Suppose that $M_1,\, M_2$
are in $\mathfrak {M}(X)$
with $M_1 \cap M_2 \neq \emptyset$
. If $f_{M_1}(x_0)=\theta f_{M_2}(x_0)$
for some $x_0\in M_1 \cap M_2$
and some $\theta \in \{-1,\,1\},$
then $f_{M_1}(z)=\theta f_{M_2}(z)$
for all $z\in C_{M_1}^{X} \cap C_{M_2}^{X}.$
Proof. We may assume that $\theta =1$
to simplify the notation, that is $f_{M_1}(x_0)=f_{M_2}(x_0)$
. Suppose, on the contrary, that there is a nonzero element $z_0\in C_{M_1}^{X} \cap C_{M_2}^{X}$
such that $f_{{M_1}}(z_0)=-f_{{M_2}}(z_0).$
We observe first from Proposition 3.8 that
Let $u_0$
be the mid-point of $x_0$
and $z_0$
, i.e., $u_0=(x_0+z_0)/2$
. Obviously, $u_0\in C_{M_1}^{X}\cap C_{M_2}^{X}$
. If $f_{M_1}(u_0)=-f_{M_2}(u_0)$
, then
If $f_{M_1}(u_0)=f_{M_2}(u_0)$
, then
Either case leads to a contradiction. Thus, the proof is complete.
4. Phase-isometries on finite-dimensional polyhedral Banach spaces and CL-spaces
In this section, we shall apply the preceding results about surjective phase-isometries to the study of finite-dimensional polyhedral Banach spaces ( i.e., for those whose unit sphere is a polyhedron) and CL-spaces.
For our result on finite-dimensional polyhedral Banach spaces, we need a result [Reference Huang and Tan8, Lemma 2.5] which builds a connection between functionals of dual of respective spaces in terms of a phase-isometry. For a Banach space $X$
, a functional $x^{*}\in S_{X^{*}}$
is called a $w^{*}$
-exposed point of $B_{X^{*}}$
if $x^{*}$
is the unique support functional at some smooth point $x\in S_X$
.
Lemma 4.1 Let $X$
and $Y$
be real Banach spaces, and let $f: X \rightarrow Y$
be a phase-isometry (not necessarily surjective). Then for every $w^{*}$
-exposed point $x^{*}$
of $B_{X^{*}},$
there exists a linear functional $\varphi \in Y^{*}$
of norm one such that $x^{*}(x)=\pm \varphi (f(x))$
for all $x\in X$
.
The previous result enables us to show that all finite-dimensional polyhedral Banach spaces have the Wigner property.
Theorem 4.2 Let $X$
be an $n$
-dimensional polyhedral Banach space. Then $X$
has the Wigner property.
Proof. Let $Y$
be a Banach space, and let $f: X\rightarrow Y$
be a surjective isometry. Given $M_0\in \mathfrak {M}(X)$
, let $f_{M_0}$
be defined as in Definition 3.3. Since $X$
is a polyhedral Banach space of finite dimension, the cone $C_{M_0}^{X}$
generated by $M_0$
is a convex body of $X$
. Proposition 3.8 guarantees that $f_{M_0}(C_{M_0}^{X})$
is also a convex body in $Y$
. P. Mankiewicz [Reference Mankiewicz13] has proved that every isometry between convex bodies is the restriction of an affine onto isometry between the corresponding spaces. By this result, there is a linear isometry $g_{M_0}: X\rightarrow Y$
such that its restriction on the cone $C_{M_0}^{X}$
is $f_{M_0}$
.
We now apply Lemma 4.1 to obtain that for every $w^{*}$
-exposed point $x^{*}$
of $B_{X^{*}}$
, there exists a linear functional $\varphi \in S_{Y^{*}}$
such that
We claim that for every $M\in \mathfrak {M}(X)$
, either $x^{*}(x)=\varphi (f_M(x))$
for all $x\in C_M^{X}$
or $x^{*}(x)=-\varphi (f_M(x))$
for all $x\in C_M^{X}$
holds. It remains to show that the third possibility, that is $x^{*}(x)=\varphi (f_M(x))\neq 0$
and $x^{*}(y)=-\varphi (f_M(y))\neq 0$
for some $x,\, y \in C_M^{X}$
, leads to a contradiction. Indeed, it follows from (4.1) that
If $x^{*}(x+y)=\varphi (f_M(x)+f_M(y))$
, this shows that $x^{*}(y)=-\varphi (f_M(y))=0$
, a contradiction. If $x^{*}(x+y)=-\varphi (f_M(x)+f_M(y))$
, then obviously $x^{*}(x)=-\varphi (f_M(x))=0$
. This is a contradiction which proves the claim. For every $w^{*}$
-exposed point $z^{*}$
of $B_{X^{*}}$
, we can choose $\varphi _{z^{*}}\in S_{Y^{*}}$
such that
To see our conclusion, we shall prove that $f$
is phase equivalent to $g_{M_0}$
. Now for every $M\in \mathfrak {M}(X)$
, although it may occur that $M\cap M_0=\emptyset$
, there are $M_1,\,\cdots ,\, M_m\subset \mathfrak {M}(X)$
such that $M_i\cap M_{i+1}\neq \emptyset$
for $i=0,\,1,\,\cdots ,\,m$
, where we use the notation $M_{m+1}=M$
. We deduce from Lemma 3.11 and $M_1\cap M_0\neq \emptyset$
that there is $\theta _1\in \{-1,\,1\}$
such that $g_{M_0}(x)=\theta _1 f_{M_1}(x)$
for all $x\in C_{M_0}^{X}\cap C_{M_1}^{X}$
. This and the claim show that
for all $y\in C_{M_1}^{X}$
. Inductively, there are $\{\theta _i\}_{i=2}^{m+1}\subset \{-1,\,1\}$
such that
for all $y\in C_{M}^{X}$
. Set $\theta _M=\theta _1\cdots \theta _{m+1}$
. Then
for all $y\in C_M^{X}$
. Let $W$
be the set consisting of all $w^{*}$
-exposed points of $B_{X^{*}}$
. By Proposition 3.8, the set $\{\varphi _{z^{*}}\}_{z^{*}\in W}$
is a norming set of $Y$
. So (4.2) ensures us that $g_{M_0}=\theta _M f_M$
on $C_{M}^{X}$
. This completes the proof.
The second conclusion of this section is that CL-spaces have the Wigner property. In what follows, to simplify the notation, let us make some explanation. $X$
and $Y$
are always used to represent a CL-space and a Banach space, respectively. For every surjective phase-isometry $f :X\rightarrow Y$
and every $M\in \mathfrak {M}(X)$
, let $f_M$
be defined as in Definition 3.3. The unique linear functional $x^{*}\in S_{X^{*}}$
such that $M=\{x\in S_X: x^{*}(x)=1\}$
will be denoted by $x_M^{*}$
. By Proposition 3.8, $f_M(M)\in \mathfrak {M}(Y)$
. Thus, there is a unique $y^{*}\in S_{Y^{*}}$
such that $f_{M}(M)=\{y\in S_Y: y^{*}(y)=1\}$
and we denote it by $y_M^{*}$
.
For our second result, we present a basic property for CL-spaces, which may be previously known. Its brief proof is given here since we can not find it in literature.
Lemma 4.3 Let $X$
be a CL-space. For all $M_1,\, M_2\in \mathfrak {M}(X)$
. If $M_1\neq -M_2,$
then $M_1\cap M_2\neq \emptyset$
.
Proof. If $M_1=M_2$
, then the desired conclusion is obviously true. Now we assume that $M_1\neq M_2$
. Choose an element $z\in M_1$
which is not in $M_2\cup -M_2$
. Since $B_X=\mbox {co}(M_2\cup -M_2)$
, there are $x_1\in M_2,\, x_2\in -M_2$
and $\lambda \in (0,\,1)$
such that
The identity $x_{M_1}^{*}(z)=1$
implies that $x_1,\, x_2\in M_1$
. We conclude from this that $x_1\in M_1\cap M_2\neq \emptyset$
.
Lemma 4.4 Let $X$
be a CL-space and $Y$
a Banach space, and let $f:X\rightarrow Y$
be a surjective phase-isometry. If $M_1,\, M_2\in \mathfrak {M}(X)$
with $M_1 \cap M_2 \neq \emptyset$
and $f_{M_1}(x_0)=\theta f_{M_2} (x_0)$
with $\theta \in \{-1,\, 1\}$
for some $x_0\in M_1 \cap {M_2},$
then $y^{*}_{M_1}\circ f_{{M_2}}=\theta x^{*}_{M_1}$
on the cone $C_{M_2}^{X}$
.
Proof. We may assume that $f_{M_1}(x_0)=f_{{M_2}}(x_0)$
for some $x_0\in M_1\cap {M_2}$
. Namely, the value of $\theta$
is chosen to be $1$
. Moreover, by Corollary 3.10, we can assume that $f_{M_1}$
and $f_{M_2}$
are defined with respect to $x_0$
. If $z\in C_{M_2}^{X}\cap C_{M_1}^{X}$
, then by Lemma 3.11, we have $f_{M_2}(z)=f_{M_1}(z)$
. This gives the desired conclusion in this case. Note that $f(-x)=-f(x)$
for every $x\in X$
. Then in the case of $z\in C_{M_2}^{X}\cap -C_{M_1}^{X}$
, we have $f_{M_2}(z)=-f_{M_1}(-z)$
following from $\|f(z)+f(x_0)\|=\|z+x_0\|$
or $\|f(z)-f(x_0)\|=\|z+x_0\|$
. This entails the desired result.
Now we handle the remaining case of $z\in C_{M_2}^{X}$
and $z \notin C_{M_1}^{X}\cup -C_{M_1}^{X}$
. Since $X$
is a CL-space, there are $x_1\in M_1,\, x_2\in -M_1$
and $\lambda \in (0,\,1)$
such that
It follows that $x_1,\, x_2\in {M_2}$
, and therefore
This together with the choice of $y^{*}_{M_1}$
yields
Thus, equality holds in the above, i.e., we have
On the other hand, we get
This allows us to assert that
Since we have $y^{*}_{M_1}(f_{{M_2}}(\|z\|x_1))=x^{*}_{M_1}(\|z\|x_1)=\|z\|$
, it follows that $y^{*}_{M_1} (f_{{M_2}}(z))=x^{*}_{M_1}(z)$
as expected. The proof is complete.
The previous lemma reveals the connections between distinct maximal convex sets of unit sphere of CL-spaces. We will prove the main result mentioned in the abstract: Every CL-space has the Wigner property.
Theorem 4.5 Let $X$
be a CL-space. Then $X$
has the Wigner property.
Proof. Note that $\mathfrak {M}(X)$
is the set consisting of all maximal convex subsets of $S_X$
. One can easily see that
By the axiom of choice, there is a set $\mathfrak {M}(X)^{+}\subset \mathfrak {M}(X)$
such that for every $M\in \mathfrak {M}(X)$
either $M$
or $-M$
belongs to $\mathfrak {M}(X)^{+}$
and $M_1\cap M_2\neq \emptyset$
for every $M_1,\, M_2\in \mathfrak {M}(X)^{+}$
by Lemma 4.3. Fix $M_0\in \mathfrak {M}(X)^{+}$
. Lemma 3.11 guarantees that there is a set $\{\theta _M\}_{M\in \mathfrak {M}(X)^{+}}$
of signs such that $f_{{M_0}}(x_0)=\theta _M f_{M}(x_0)$
for some $x_0\in M\cap M_0$
and each $\theta _M$
does not depend on the choices $x_0$
in $M\cap M_0$
. So we can define a phase-isometry $F: X\rightarrow Y$
given by
for all $M\in \mathfrak {M}(X)^{+}$
. To see that $F$
is well defined, it remains to check that
for all $x\in C_{M_1}^{X}\cap C_{M_2}^{X}$
, where $M_1,\, M_2\in \mathfrak {M}(X)^{+}$
. By Lemma 3.11, it suffices to show that there exists some $z_0\in M_1\cap M_2$
such that
Lemma 4.4 ensures us the existence of a corresponding $y_{M_0}^{*}\in S_{Y^{*}}$
such that
for all $x\in M_1\cap M_2$
. Consequently,
for all $x\in M_1\cap M_2$
. Choose $z_0\in M_1 \cap M_2$
. If $x_{M_0}^{*}(z_0)\neq 0$
, then (4.3) and $f_{M_1}(z_0)=\pm f_{{M_2}}(z_0)$
entail that $\theta _{M_1}f_{M_1}(z_0)= \theta _{M_2}f_{{M_2}}(z_0)$
. If $x_{M_0}^{*}(z_0)=0$
, then there are $z_1\in M_0,\, z_2\in -M_0$
and $\lambda \in (0,\,1)$
such that
We observe that $z_1\in M_1\cap M_2 \cap M_0$
is the desired element. Clearly, $F$
is a surjective phase-isometry, and it is phase-equivalent to $f$
. By Lemma 3.1 and Proposition 3.8, we see that $F$
is bijective, $F(M)\in \mathfrak {M}(Y)$
and $F(C_M^{X})$
is a cone in $Y$
generated by $F(M)$
.
We claim that for every $M\in \mathfrak {M}(X)$
, the relation $y^{*}_M \circ F=\theta _{M} x^{*}_M$
holds on $X$
. We can assume that $M\in \mathfrak {M}(X)^{+}$
. By the above observation, we only need to check that
for every $z\in C_{M'}^{X}$
with $M'\in \mathfrak {M}(X)^{+}$
. Note that $M\cap M'\neq \emptyset$
and $\theta _M f_M(v)=\theta _{M'}f_{M'}(v)$
for all $v\in M\cap M'$
. Lemma 4.4 guarantees that
for all $z\in C_{M'}^{X}$
. This finishes the claim.
We now prove that $F:X\rightarrow Y$
is a surjective isometry, and so $F$
is just the desired linear isometry by Mazur–Ulam theorem. Indeed, for all $u,\, v\in X$
, it is easily seen that there is an $M\in \mathfrak {M}(X)$
such that $x^{*}_M(u-v)=\|u-v\|$
. The claim implies that
Similarly, we have
Since $F$
is a phase-isometry, the previous two inequalities should be equalities. Namely $F:X\rightarrow Y$
is an isometry. The proof is complete.
An immediate conclusion of Theorem 4.5 is the following
Corollary 4.6 $L_1(\mu )$
-spaces and $C(K)$
have the Wigner property, where $\mu$
is a positive measure and $K$
is a compact Hausdorff space.
Acknowledgements
The authors wish to express their appreciation to Guanggui Ding for many very helpful comments regarding isometric theory in Banach spaces and the anonymous referee's helpful suggestions that improve the original manuscript greatly.
The authors are supported by the Natural Science Foundation of China (Grant Nos. 11371201, 11201337, 11201338, 11301384).
