数理解析研究所講究録
第 1996 巻 2016 年 107-112

107

POSITIVE CONES AND GGVOSAMU HATORIABSTRACT. We show that a gyrometric preserving transformations between connected open subsets of the positive cones are
uniquely extended to a gyrometric preserving surjection between
the underlying cones.

1. $IN^{r}fRODUC’$rION
A celebrated Mazur-Ulam theorem states that a surjective isometry from a normed real-linear space onto a normed real-linear space isaﬃne in the sense that it is a real-linear isometry followed by a translation (cf.[9]).. Mankiewicz [6] generalized in the way that any surjective
isometry between connected open subsets of normed real-linear spaces
is extended to a surjective isometry between underlying normed reallinear spaces. As the main motivation to our present investigation wemention that in the paper [1] we prove that a gyrometric preserving
surjection from a GGV onto a GGV is a gyrometric preserving isomorphism followed by the translation. The idea of the proof in [1]emploies the one in the proof of the celebrated Mazur-Ulam theorem
due to V\"ais\"al\"a [9]. The following is expected to be true: a gyrometric preserving surjection between connected open subsets ofareextended to a gyrometric preserving surjection between the underlying
GGV’s. We have not yet prove the above conjecture. There is an obstruction to prove the conjecture by applying simply the way similar
to the proof of Mankiewicz. We explain it; we show how to extend thegiven isometry between connected open subsets to a global isometry.
The essential part of the proof of Mankiewicz [6] is as follows. Suppose thatis a normed real-linear space for $j=1$ , 2. Suppose that
$B_{j}=\{x\in X_{j} : \Vert x\Vert<\epsilon\}$ for:is
O. Suppose thata surjective isometry. We give :byfor a$z\in X_{1}$ , where $r>0$ is a large enough such that. To show
an idea we omit preciseness: we omit to prove is well defined in the$GGV^{\rangle}s$$X_{j}$$T$$\epsilon>$$\tilde{T}$$B_{1}arrow B_{2}$$\tilde{T}(z)=rT(\frac{1}{r}z)$$X_{1}arrow X_{2}$$\frac{1}{r}z\in B_{1}$$\tilde{T}$2010 Mathematics Subject Classification. $47B49,$ $46J10.$
Key words and phrases. Isometry, Jordan -isomorphism, positive element,$*$

Thompson metric,$C^{*}$-algebra.

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OSAMU HATORI

sense that it is independent of the choice of . We infer that is a
. We show that it is an isometryonto
surjective isometry from$\tilde{T}$$r$$X_{1}$$X_{2}$

by applying the distributive law:$\Vert z-w\Vert=\Vert r(\frac{1}{r}z-\frac{1}{r}w)\Vert=r\Vert\frac{1}{r}z-\frac{1}{r}w\Vert$$=r \Vert T(\frac{1}{r}z)-T(\frac{1}{r}w)\Vert=\Vert r(T(\frac{1}{r}z)-T(\frac{1}{r}w)\Vert$$= \Vert rT(\frac{1}{r}z)-rT(\frac{1}{r}w)\Vert=\Vert\tilde{T}(z)-\tilde{T}(w) z, w\in X_{1}$A distributive law such as $r\otimes(a\oplus b)=(r\otimes a)\oplus(r\otimes b)$ is not assumed
for GGV’s. Hence it is an obstruction for extending a gyrometric preserving map between connected open sets to a global one.

2. GENERALIZED GYROVECTOR SPACES
The definition of a generalized gyrovector spaces (GGV) is the following.
be aDefinition 1 (A generalized gyrovector space [1]). Let$map\otimes:\mathbb{R}\cross
GarrowG$. Let begyrocommutative gyrogroup with theWe say thatan injection from into a real normed space$(or (G, \oplus, \otimes)$ just for a simple notation) is a generalized
gyrovector space or a GGV in short if the following conditions (GGVO)
to (GGV8) are fulfilled:(GGVO) $\Vert\phi(gyr[u, v]a)\Vert=\Vert\phi(a)\Vert$ for any $u,$ $v,$ $a\in G$ ;(GGVI) $1\otimes a=a$ for every $a\in G$ ;;(GGV2) $(r_{1}+r_{2})\otimes a=(r_{1}\otimes a)\oplus(r_{2}\otimes a)$ for any $a\in G,$;
for any $a\in G,$(GGV3)$r\in$$(\phi(|r|\otimesa))/\Vert\phi(r\otimesa)\Vert=\phi(a)/\Vert\phi(a)$for any(GGV4), where denotes the identity element of the gyrogroup;;(GGV5) $gyr[u, v](r\otimes a)=r\otimes gyr[u, v]a$ for any $u,$ $v,$ $a\in G,$$[r_{1}\otimesv,r_{2}\otimesv]=id_{G}$
$v\in
G,$
;
for any(GGV6) gyr$\Vert\phi(G)\Vert=\{\pm\Vert\phi(a)\Vert\in\mathbb{R}
:
a\in
G\}$
is a real one-dimensional(GGVV)and scalar multiplication
vector space with vector addition;;
for any $a\in G,$(GGV7)$b\inG.$
for any $a,$
(GGV8)$(G, \oplus)$$\phi$$G$$(\mathbb{V},$$\Vert$$(G, \oplus, \otimes, \phi)$$r_{1},$$(r_{1}r_{2})\otimes a=r_{1}\otimes(r_{2}\otimes a)$$r_{1},$$r_{2}\in \mathbb{R}$$r_{2}\in \mathbb{R}$$a\in G\backslash \{e\},$$\mathbb{R}\backslash \{0\}$$e$$(G, \oplus)$$r\in \mathbb{R}$$r_{1},$$r_{2}\in \mathbb{R}$$\oplus’$$\otimes’$$r\in \mathbb{R}$$\Vert\phi(r\otimes a)\Vert=|r|\otimes’\Vert\phi(a)\Vert$$\Vert\phi(a\oplus b)\Vert\leq\Vert\phi(a)\Vert\oplus’\Vert\phi(b)\Vert$

be a GGV. Let $\rho(a, b)=\Vert\phi(a\ominus b)\Vert$ for
Definition 2. Let
all $a,$ $b\in G$ , where $a\ominus b=a\oplus(\ominus b)$ . We call the gyrometric on
on a GGV.
$(G, \oplus, \otimes)$$\rho$$G$

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Suppose that is a gyrometric for a GGV[1, (4)] of the form
$p$

(1)

$G$

. We proved the equation$p(z, w\rangle=p(\ominus z, \ominus w)=p(w, z) , z, w\in G.$

By Proposition 15 in [1] we have(2)$\rho(x\oplus z, x\oplus w)=\rho(z, w) , x, z, w\in G.$

We call the inequality (GGV8) the gyrotriangle inequality. It is not
the triangle inequality. In general the gyrometric does not satisfy thetriangle inequality.

3. A$POSI^{t}$

TIVE CONE OF THE POSITIVE INVERTIBLE ELEMENTS

The GGV is a generalization of a gyrovector spaces [8], which isa generalization of an inner product space. The Einstein gyrovector
space and the M\"obius gyrovector space are examples of a gyrovector
space [8]. We exhibited that the positive cone of all positive invertible
elements is an example of a GGV [1].
Example 3 ([1]). Suppose that is a unital -algebra with the norm
is the set of all positive invertible elements of . Let beaxld
a positive real number. Put$A$$\Vert\cdot\Vert$$C^{*}$$A$$A_{+}^{-1}$$t$$a\oplus_{t}b=(a^{\frac{\ell}{2}}b^{t}a^{\frac{{\}}{2}})^{\frac{1}{t}}$for allThen
is a gyrocommutative gyrogroup.
The identity element 1 of as the -algebra is the identity element, the inverse of in $A.$of the gyrogroup. The inverse element
is
put
For
$a,$ $b\in A_{+}^{-1}$$(A_{+}^{-1}, \oplus_{t})$$A$$C^{*}$$\ominus a$$a^{-1}$$a$$a,$ $b\in A_{+}^{-1}$$X=(a^{\frac{t}{2}}b^{t}a^{\frac{t}{2}})^{-\frac{1}{2}}a^{\frac{t}{2}}b^{\frac{t}{2}}.$

Then $X$ is a unitary element in

$A$

and

$gyr[a, b]c=XcX^{*},\cdot a, b, c\in A_{+}^{-1}$is the gyroautomorphism generated by and$\phi\prime=log:A_{+}^{-1}arrow A_{S},$Put $r\otimes a=a^{r}$ for every
Definethe real-linear subspace of all self-adjoint elements in . The vector
spaceis the usual 1 dimensional real
vector space of the real line;is the addition of real numbers
is the scalar multiplication of real numbers. Then
is a
GGV. In fact, (GGVO) holds since $gyr[a, b]$ is a unitary transform forevery pair
Simple calculations confirm that the conditions
from (GGVO) to (GGV6) and (GGV7) hold. The condition (GGVV)
is trivial by the definition $of\oplus’and\otimes’$ . The condition (GGV8) is alsosatisfied; see [1].
$a$$a\in A_{+}^{-\lambda},$$b.$$r\in \mathfrak{W}.$$A$$\log(A_{+}^{-1})$$\oplus’,$$\otimes’\rangle=(\mathbb{R}, +, \cross)$$\oplus’$$and\otimes’$$(A_{+}^{-1}, \oplus_{t}, \otimes_{\}}\log)$$a_{\}}b\in A_{+}^{-1}$110
OSAMU HATORIThe gyrometric$\rho$

is given by the equation$\rho(a, b)=\Vert\log(a^{\frac{t}{2}}b^{-t}a^{\frac{t}{2}})^{\frac{1}{t}}\Vert$

isfor
Note that in the case where $t=1$ the metric
the Thompson metric itself. The gyromidpoint of and is given by
the equation$\rho(a, b)$$a,$ $b\in A_{+}^{-1}$

$b$

$a$

$p(a, b)=(a^{\frac{t}{2}}(a^{\frac{t}{2}}b^{-t}a^{\frac{t}{2}})^{-\frac{1}{2}}a^{\frac{t}{2}})^{\frac{1}{t}},$which coincides with the geometric mean of

$a$

and

$b$

for the case of$t=1.$In the case of the GGV of the positive cone, the gyrometric satisjfes
the usual triangle inequality (cf. [1, p. 399

4. EXTENSION OF A GYROMETRIC PRESERVING $SURJEC^{r}$rION
BETWEEN CONNECTED OPEN SUBSETS OF THE POSITIVE CONES

For the case of the GGV of all invertible positive elements in a unital-algebra we have that a surjective gyrometric preserving map
between connected open subsets of the positive cones is uniquely extended to a surjective gyrometric preserving map between the whole ofthe positive cones. Note that the topology induced by the gyrometriccoincides with that induced by the metric induced by the original norm
, letof the -algebra. For$C^{*}$$\epsilon>0$$C^{*}$$B_{j}^{\epsilon}(1)=\{a\in A_{j+}^{-1} :\rho_{j}(a, 1)<\epsilon\}.$:be a bijection with $T(1)=1.$
Lemma 4. Let
Suppose that is gyrometric preserving. Then$T$

$B_{1}^{\epsilon}(1)arrow B_{2}^{\epsilon}(1)$$T$$T( \frac{1}{2}\otimes a)=\frac{1}{2}\otimes T(a) , a\in B_{1}^{\epsilon}(1)$.

By induction we have that
$T( \frac{1}{2^{n}}\otimes a)=\frac{1}{2^{n}}\otimes T(a)) a\in B_{1}^{\epsilon}(1)$for every positive integer . Then we have$n$Lemma 5. Then there exists a Jordan -isomorphism
thatand a central projection in$*$$A_{2}$

$p$

from

$J$

$A_{1}$

onto$A_{2}\mathcal{S}uch$$T(a)=pJ(a)+(1-p)J(a)^{-1}, a\in B_{1}^{\epsilon}(1)$.

Lemma 5 is the essential point where the given gyrometric preserving
map on a connected open subset of the GGV of the positive cone isextended to the whole of the GGV of $A_{1+}^{-1}:z\mapsto pJ(z)+(1-p)J(z)^{-1}$
ontodefines a gyrometric preserving surjection from
$A_{1+}^{-1}$$A_{2+}^{-1}.$

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Lemma 6. Suppose that and are surjective gyrometric preserving
maps fromonto
such that $T=S$ on an open ball. Then $T=S$
on
$S$

$T$

$A_{1+}^{-1}$$A_{2+}^{-1}$$A_{1+}^{-1}.$

By the above lemma the extension is unique.

Theorem 7. Let
be a non-empty connected open subset offor:$j=1$ , 2. Let
be a surjection. Then $TlS$ gyrometricpreserving if and only if there exists a Jordan -isomorphism form
onto
and a central projection $p\in A_{2}$ such that
$A_{j+}^{-1}$$U_{j}$$T$$U_{1}arrow U_{2}$$*$$A_{1}$$J$$A_{2}$(3)$T(z)=(b(pJ(z)+(1-p)J(z)^{-1})^{t}b)^{\frac{1}{*}}\} z\in U_{1\}}$;
where $b=(pJ(a_{0})+(1-p)J(a_{0})^{-1})^{-t}\# T(a_{0})^{t}$ for an and any$a_{0}\inU.$
is unique for any
In this case is extended to a gyrometricpreserving surjection fromonto
$a_{0}\in U_{1}$

$T$

$b$

$A_{2+}^{-1}.$$A_{1+}^{-1}$

We show a sketch proof. Let
be arbitrary. Choose a sufficiently small. Then the
with
$T’$
:
induced map
defined by$a_{0}\in U_{1}$$\epsilon>0$$\{b\in A_{1+}^{-1} :\rho_{1}(a_{0}, b)<\epsilon\}\subseteq U_{1}$$B_{1}^{\epsilon}(1)arrow B_{2}^{\epsilon}(1)$$T’(z)=\ominus {}_{t}T(a_{0})\oplus_{t}T(a_{0}\oplus_{t}z) , z\in B_{1}^{\epsilon}(1)$. Thenis a surjective gyrometric preserving map from
onto
$T’$
by Lemma 5
is extended to a surjective gyrometric preserving map
from. So is . As
ontois connected the extension is
unique up to given point . As
and ais arbitrary there is ansurjective gyrometric preserving map
from
onto
which isan extension of on. Let
and be a pairof points in . As
is connected and open, there is a continuous$[0,1]arrow
U_{1}$
map :with $\gamma(0)=a_{0}$ and $\gamma(1)=a_{1}$ . By compactnessofthere is a sequencewith $t_{0}=0,$ $t_{n}=1$ andsuch that$B_{1}^{\epsilon}(1)$$A_{1+}^{-1}$$T$$A_{2+}^{-1}$$U_{1}$$\epsilon_{0}>$$a_{0}$$a_{0}$$T_{0}$$T$$U_{1}$$B_{2}^{\epsilon}(1)$$\{b\in A_{1+}^{-1} :\rho_{1}(a_{0}, b)<\epsilon\}$$A_{2+}^{-1}$$A_{1+}^{-1}$$a_{0}$$a_{1}$$U_{1}$$\gamma$$\gamma([O, 1]$$t_{0},$$\ldots t_{n}\rangle\in[0,1]$$\epsilon_{j}>0$$\gamma([O, 1])\subset\bigcup_{j}\{z\in A_{1+}^{-1}:\rho_{1}(\gamma(t_{j}), z)<\epsilon_{j}\}\subseteq U_{j}$and$\{z\in A_{1+}^{-1}:\rho_{1}(\gamma(t_{j}), z)<\epsilon_{j}\}\cap\{z\in A_{1+}^{-1}:\rho_{1}(\gamma(t_{j+1}), z)<\epsilon_{j}\}\neq\emptyset$

for $j=0$ , . . . , $n-1$ . By Lemma 6 we see that $T_{j}=T_{j+1}$ onas$T_{j}=T_{j+1}$ on
is afor $j=0$ , . . . , $n-1$ . Hence we have $T_{0}=T_{n}$ . Therefore
unique extension of . This does not give a complete proof, refer forthe precise proof in [2].
Note that Hatori and Moln\’ar [3] proved the case where
and $t=1$ . Honma and Nogawa [4] proved the case of general . Notealso that the connectivity of the open sets in Theorem 7 is essential.
$A_{1+}^{-1}$$\{z\epsilon A_{1+}^{-1}:\rho_{1}(\gamma(t_{j}), z)<\epsilon_{j}\}\cap\{z\in A_{1+}^{-1}:\rho_{\lambda}(\gamma(t_{j+1}), z)<$$e_{j}\}$$T_{0}$$T$$U_{j}=A_{j+}^{-1}$
$t$

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OSAMU HATORILet $U=\{a\in A_{1+}^{-1} : \rho_{1}(a, 1)<1\}\cup\{a\in A_{1+}^{-1} : \rho_{1}(a, 10)<1\}$ . Suppose
onto itself which is not thethat is a Jordan *-isomorphism from
$T:Uarrow
U$ defined by
identity transformation. Then the map$J$$A_{1}$$T(a)=\{\begin{array}{ll}J(a) , \rho_{1}(a, 1)<1a, \rho_{1}(a, 10)<1.\end{array}$is a surjective gyrometric preserving map while it is not extended to a
by Lemma 6.
surjective gyrometric preserving map from
$A_{1+}^{-1}$

REFERENCES[1] T. Abe and O. Hatori, Generalized gyrovector spaces and a Mazur-Ulam theorem, Publ. Math. Debrecen, 87 (2015), 393-413; A Note on the Proof of
Theorem 13 in the Paper “Generalized Gyrovector Spaces and a Mazur- Ulamtheorem preprint[2] O. Hatori, Extension of Isometries in generalized gyrovector spaces of thepositive cones, preprint[3] O. Hatori and L. Moln\’ar, Isometries of the unitary groups and Thompson
isometries of the spaces of invertible positive elements in -algebras, J. Math.
Anal. Appl., 409 (2014), 158-167[4] S. Honma and T. Nogawa, Isometries of the geodesic distances for the space of
invertible positive operators and matrices, Linear Algebra Appl., 444 (2014),$C^{*}$

152-164[5] R. V. Kadison, A generalized Schwarz inequality and algebraic invariants for
operator algebras, Ann. Math. 56 (1952), 494-503.
[6] P. Mankiewicz, On extension of isometries in normed linear spaces, Bull. Acad.
Polon. Sci. S\’er. Sci. Math. Astronom. Phys., 20 (1972), 367-371.
[7] A. A. Ungar, The relativistic noncommutative nonassociative group of velocities and the Thomas rotation, Result. Math., 16 (1989), 158-179[8] A. A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity, World Scientific, (200S)[9] J. V\"ais\"al\"a, A proof of the Mazur- $m_{am}$ theorem, Amer. Math. Monthly,
110(2003), 633-635

DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, NIIGATA UNIVERSITY, NIIGATA 950-2181 JAPAN-mail address: hatori@math.sc.niigata-u.ac.jp$E$