Table Of ContentTINGLEY’S PROBLEM THROUGH THE FACIAL STRUCTURE
OF AN ATOMIC JBW∗-TRIPLE
7
1 FRANCISCOJ.FERNA´NDEZ-POLOANDANTONIOM.PERALTA
0
2
Abstract. Weprovethateverysurjectiveisometrybetweentheunitspheres
n of two atomic JBW∗-triples E and B admits a unit extension to a surjective
a
real linear isometry from E into B. This result constitutes a new positive
J
answertoTignley’sproblemintheJordansetting.
8
1
]
A
F 1. Introduction
.
h In a recent contribution we establishthat every surjective isometry between the
t
a unit spheres of two B(H)-spaces extends uniquely to a surjective complex linear
m orconjugatelinearsurjectiveisometrybetweenthecorrespondingspaces(see[15]).
[ This result constitutes a positive answer to Tingley’s isometric extension problem
[31] in the setting of B(H)-spaces and atomic vonNeumann algebras. Solutions to
1
Tingley’sproblemforcompactoperators,compactC∗-algebrasandweaklycompact
v
JB∗-triples have been previously obtained in [26, 14]. For additional information
2
1 onthe historicbackgroundandthe state ofthe artofTingley’sproblemthe reader
1 is referred to the introduction of [15] and to the monograph[32].
5
0 ProblemsinC∗-algebras,vonNeumannalgebrasandoperatoralgebrasareoften
. considered in the context of Banach Jordan algebras and Jordan triple systems.
1
Such studies widen the scope and often introduce new ideas and techniques not
0
7 presentintheassociativecase. TheclassofJB∗-tripleshavearichinteractionwith
1 Banach Space Theory. The spaces in this class enjoy a unique geometry which
: makes more interesting the study of certain geometric problems in a wider setting.
v
i This paper is devoted to extend the recent results in [15] to the context of atomic
X
JBW∗-triples (i.e. ℓ -sums of Cartan factors).
∞
r
a We recallthat aJB∗-triple is acomplex BanachspaceE whichcanbe equipped
with a continuous triple product {.,.,.}:E×E×E →E, which is symmetric and
linear in the first and third variables, conjugate linear in the second variable and
satisfies the following axioms
(a) L(a,b)L(x,y)=L(x,y)L(a,b)+L(L(a,b)x,y)−L(x,L(b,a)y),whereL(a,b)is
the operator on E given byL(a,b)x={a,b,x};
(b) L(a,a) is an hermitian operator with non-negative spectrum;
(c) kL(a,a)k=kak2.
2010 Mathematics Subject Classification. Primary 47B49, Secondary 46A22, 46B20, 46B04,
46A16,46E40.
Keywordsandphrases. Tingley’sproblem;extensionofisometries;atomicJBW∗-triples,Car-
tanfactors.
2 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
Examples of JB∗-triples include C∗-algebras with respect to the triple product
defined by product
1
(1) {x,y,z}= (xy∗z+zy∗x),
2
and JB∗-algebras under the triple product {x,y,z} = (x◦y∗)◦z+(z◦y∗)◦x−
(x◦z)◦y∗. The so-calledternary rings of operators (TRO’s) studied, for example,
in [25] are also examples of JB∗-triples.
A subtriple I of a JB∗-triple E is said to be an ideal or a triple ideal of E if
{E,E,I}+{E,I,E}⊆I.
A JBW∗-triple is a JB∗-triple which is also a dual Banach space (with a unique
isometric predual [2]). It is known that the second dual of a JB∗-triple is a JBW∗-
triple(compare[8]). AnextensionofSakai’stheoremassuresthatthetripleproduct
of every JBW∗-triple is separately weak∗-continuous (c.f. [2] or [19]).
Another illustrative examples of JBW∗-triples are givenby the so-calledCartan
factors. A complex Banach space is a Cartan factor of type 1 is it coincides with
the complex Banach space L(H,K), of all bounded linear operators between two
complex Hilbert spaces, H and K, whose triple product is given by (1).
Given a conjugation, j, on a complex Hilbert space, H, we can define a linear
involutiononL(H)definedbyx7→xt :=jx∗j. Atype2Cartanfactor isasubtriple
of L(H) formed by the skew-symmetric operators for the involution t; similarly, a
type 3 Cartan factor isformedbythe t-symmetricoperators. ABanachspaceX is
called a Cartan factor of type 4 or spin if X admits a complete inner product (.|.)
and a conjugation x7→x, for which the norm of X is given by
kxk2 =(x|x)+ (x|x)2−|(x|x)|2.
p
Cartanfactors oftypes5and6 (alsocalledexceptional Cartanfactors)areallfinite
dimensional. Anatomic JBW∗-tripleis aJBW∗-triple whichcanbe representedas
an ℓ -sum of Cartan factors. We refer to [7] for additional details.
∞
Let E and B be atomic JBW∗-triples. In our main result we prove that every
surjectiveisometryf :S(E)→S(B)admits a unique extensionto asurjectivereal
linear isometry T : E → B (see Theorem 2.9). This theorem extends the main
conclusion in [15] to the setting of atomic JBW∗-triples. In [15], we strived for
arguments essentially based on standard techniques of C∗-algebras, Geometry and
Functional Analysis. The proofs here are extended to the setting of JBW∗-triples
with new and independent techniques which could be also applied to C∗-algebras.
As in recent contributions studying Tingley’s problem on C∗-algebras and von
Neumann algebras, our arguments are based on an useful result due to L. Cheng,
Y. Dong and R. Tanaka, which asserts that every surjective isometry between the
unit spheres of two Banach spaces X and Y maps maximal proper (norm closed)
face of the unit ball of X to maximal proper (norm closed) face of the unit ball of
Y (see [6, Lemma 5.1], [27, Lemma 3.5] and [28, Lemma 3.3]).
Throughout the paper, given a Banach space X the symbols B and S(X) will
X
stand for the closed unit ball and the unite sphere of X, respectively.
ByaresultofC.M.Edwards,C.Hoskinandtheauthorsofthisnote(see [9])we
knowthatforeachnon-emptynormclosedfaceF oftheunitballB inaJB∗-triple
E
TINGLEY’S PROBLEM THROUGH THE FACIAL STRUCTURE 3
E there exists a unique compact tripotent u in E∗∗ such that
F =F =(u+E∗∗(u))∩B ,
u 0 E
where E∗∗(u) is the Peirce-zero space associated with u in E∗∗.
0
An appropriateapplicationofKadison’stransitivity theoremfor JB∗-triples([4,
Theorems 3.3 and 3.4]) proves that every maximal proper norm closed face of B
E
is of the form
(2) F =(e+E∗∗(e))∩B ,
e 0 E
for a unique minimal tripotent e in E∗∗. However this minimal tripotent e need
not be in E.
We recallthateveryatomicJBW∗-triplecoincideswiththeweak∗-closureofthe
linear span of all its minimal tripotents (see [17]).
If B is another atomic JBW∗-triple and f : S(E) → S(B) is a surjective isom-
etry, for each minimal tripotent e ∈ E, there exists a minimal tripotent u in B∗∗
satisfying
f(F )=f((e+E (e))∩B )=(u+B∗∗(u))∩B .
e 0 E 0 B
However, as in the case of Tingley’s theorem for surjective isometries between the
unit spheres of B(H)-spaces (see [15]), when dealing with maximal proper faces
in B , minimal tripotents in B∗∗ need not be in B. To avoid this difficulty, in
B
this paper we shall prove that, under the above conditions, the minimal tripotent
u belongs to B (see Theorem 2.7). The proofs in this note are based on geometric
arguments combined with Functional Analysis techniques.
2. Tingley’s problem for atomic JBW∗-triples
AnelementuinaJB∗-tripleE iscalledtripotentif{u,u,u}=u. Eachtripotent
u in E induces a Peirce decomposition of E in the form
E =E (u)⊕E (u)⊕E (u),
2 1 0
where for each i∈{0,1,2}the space E (u) is precisely the i eigenspaceof L(u,u).
i 2
Peirce arithmetic assures that {E (u),E (u),E (u)} is contained in E (u) if
i j k i−j+k
i−j+k ∈{0,1,2}, and is zero otherwise. In addition,
{E (u),E (u),E}={E (u),E (u),E}=0.
2 0 0 2
The corresponding Peirce projections, P (u): E →E (u), (i = 0,1,2) are contrac-
i i
tive and satisfy
P (u)=L(u,u)(2L(u,u)−I), P (u)=4L(u,u)(I−L(u,u)),
2 1
and P (u)=(I −L(u,u))(I−2L(u,u)) (compare [16]).
0
Anon-zerotripotenteinaJB∗-tripleE iscalledminimal (respectively,complete
or maximal) if E (e)=Ce (respectively, E (e)=0).
2 0
Let x be an element in a JB∗-triple E. We will denote by E the JB∗-subtriple
x
of E generated by x, that is, the closed subspace generated by all odd powers of
the form x[1] := x, x[3] := {x,x,x}, and x[2n+1] := x,x,x[2n−1] , (n ∈ N). It is
(cid:8) (cid:9)
known that E is JB∗-triple isomorphic (and hence isometric) to a commutative
x
C∗-algebra in which x is a positive generator (cf. [21, Corollary 1.15]). Actually,
there exist a (unique) subset Sp(x) ⊂ [0,kxk] (called the triple spectrum of x)
4 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
such that kxk ∈Sp(x)∪{0} and the latter is compact, and a triple isomorphism
Ψ:E →C (Sp(x)) mapping x into the function t7→t (compare [22]).
x 0
Supposexisanorm-oneelement. Thesequence(x[2n−1])convergesintheweak∗-
topology of E∗∗ to a tripotent (called the support tripotent of x) u(x) in E∗∗ (see
[10, Lemma 3.3] or [9, page 130]).
We recall at this stage a result taken from [14].
Proposition 2.1. [14, Proposition 2.2] Let e and x be norm-one elements in a
JB∗-triple E. Suppose that e is a minimal tripotent and ke−xk = 2. Then the
identity x=−e+P (e)(x) holds. (cid:3)
0
Following the standard notation, for each ultrafilter U on an index set I, and
each family (X ) of Banach spaces, we denote by (X ) the corresponding ul-
i i∈I i U
traproduct of the X , and if X = X for all i, we write (X) for the ultrapower
i i U
of X. As usually, elements in (X ) will be denoted in the form x = [x ] , where
i U i U
(x ) is called a representing family or a representative of x, and kxk = lim kx k
i e U i
independently of the representative. The basic facts and definitions concerning
e e
ultraproducts can be found in [18].
In our concrete setting, we can reduce our attention to a family (E ) of JB∗-
i i∈I
triples. It is known that JB∗-triples are stable under ℓ -sums (see [21, p. 523]),
∞
thustheBanachspaceℓ (E )isaJB∗-triplewithpointwiseoperations. Theclosed
∞ i
subtriple c (E ) is a triple ideal of ℓ (E ), and hence (E ) =ℓ (E )/c (E ) is a
0 i ∞ i i U ∞ i 0 i
JB∗-triple (see [21]).
Our next goal is a quantitative version of the previous Proposition 2.1.
Proposition 2.2. Let e be a minimal tripotent in a JB∗-triple E. Then for each
ε > 0 there exists δ > 0 satisfying the following property: for each x ∈ S(E) with
ke−xk>2−δ we have kP (e)(e−x)k>2−ε.
2
Proof. Arguing by reduction to the absurd, we assume the existence of ε > 0
such that for each natural n there exists x ∈ S(E) with ke−x k > 2− 1 and
n n n
kP (e)(e−x )k≤2−ε, for every natural n.
2 n
Let U be a free ultrafilter over N. We consider the JB∗-triple E . Clearly the
U
element[e] isatripotentinE . Weclaimthat[e] isaminimaltripotent. Indeed,
U U U
suppose x=[x ] ∈(E ) ([e] ), then
n U U 2 U
e [x ] =P ([e] )[x ] =[P (e)(x )] =[λ e] ,
n U 2 U n U 2 n U n U
where(λ )isaboundedfamilyinC. Bycompactnessargumentsthelimitlim (λ )=
n U n
λ exists in C. By the triangular inequality we have
0
k[x ] −λ [e] k≤k[x ] −[λ e] k+k[λ e] −λ [e] k=lim|λ −λ |,
n U 0 U n U n U n U 0 U n 0
U
and hence [x ] =λ [e] ∈C[e] .
n U 0 U U
Finally,sincek[x ] k=1,and2≥k[e] −[x ] k=lim ke−x k≥lim 2−1 =
n U U n U U n U n
2. Applying Proposition 2.1 we get
[x ] =[−e] +P ([e] )[x ] =[−e+P (e)(x )] ,
n U U 0 U n U 0 n U
and thus
[P (e)(e−x )] =P ([e] )([e] −[x ] )=2[e] ,
2 n U 2 U U i U U
and
limkP (e)(e−x )k=limk2ek=2,
2 n
U U
TINGLEY’S PROBLEM THROUGH THE FACIAL STRUCTURE 5
which contradicts kP (e)(e−x )k≤2−ε, for every natural n. (cid:3)
2 n
A common tool applied in all recent studies on Tingley’s problem for non-
commutative structures is based on an appropriate description of the facial struc-
tures of the closed unit balls of the involved spaces. The justification is essentially
due to the following result established by L. Cheng, Y. Dong and R. Tanaka.
Proposition 2.3. ([6, Lemma 5.1], [27, Lemma 3.5] and [28, Lemma 3.3]) Let X,
Y be Banach spaces, and let T :S(X)→S(Y) be a surjective isometry. Then C is
a maximal convex subset of S(X) if and only if T(C) is that of S(Y). Then C is a
maximal proper (norm closed) face of B if and only if f(C) is a maximal proper
X
(norm closed) face of B . (cid:3)
Y
Accordingly to the notation in [11] and [13], a tripotent e in the second dual,
E∗∗, of a JB∗-triple E is said to be compact-G if there exists a norm one element
δ
a in E such that e is the support tripotent of a. A tripotent e in E∗∗ is called
compact if e=0 or it is the infimum of a decreasing net of compact-G tripotents
δ
in E∗∗.
ThenormclosedfacesoftheclosedunitballofaC∗-algebraweredeterminedby
C.A. Akemann and G.K. Pedersen in [1]. Their characterization played a decisive
roleinthe argumentspresentedin [28, 29, 30, 26]and[15]. The resultofAkemann
and Pedersen was extended to the strictly wider setting of JB∗-triples by C.M.
Edwards,C.Hoskinandtheauthorsofthisnotein[9]. Forlaterpurposeswerecall
a theorem borrowedfrom the just quoted paper.
Theorem 2.4. [9] Let E be a JB∗-triple, and let F be a non-empty norm closed
face of the unit ball B in E. Then, there exists a unique compact tripotent u in
E
E∗∗ such that
F =F =(u+E∗∗(u))∩B ,
u 0 E
where E∗∗(u) is the Peirce-zero space associated with u in E∗∗. Furthermore, the
0
mapping u 7→ F is an anti-order isomorphism from the lattice U˜ (E∗∗) of all
u c
compact tripotents in E∗∗ onto the complete lattice of norm closed faces of B . (cid:3)
E
LetebeatripotentinaJB∗-tripleE. Weshallsaythateisafinite-ranktripotent
if e can be written as a finite sum of mutually orthogonal minimal tripotents in
E. An appropriate extension of Kadison’s transitivity theorem for JB∗-triples,
establishedin [4, Theorems 3.3 and 3.4], provesthat eachfinite-rank tripotent e in
the bidual E∗∗ of E is compact, and it is further known that
(3) P (e)(E∗∗)=P (e)(E)=Ce, and P (e)(E∗∗)=P (e)(E),
2 2 1 1
where P (e) stands for the j−th Peirce projection associated with e in E∗∗. Ac-
j
cordinglyto these comments and the previous Theorem 2.4, every maximalproper
norm closed face of B is of the form
E
(4) F =(e+E∗∗(e))∩B ,
e 0 E
for a unique minimal tripotent e in E∗∗. However this minimal tripotent e need
not be in E.
We recallthat elements a,b in a JB∗-triple E are said to be orthogonal (written
a⊥b)ifL(a,b)=0(see[5,Lemma1]foradditionaldetails). ItfollowsfromPeirce
arithmetic that, for each tripotent e ∈ E, E (e) ⊥ E (e). The relation “being
2 0
orthogonal” can be applied to define a partial order in the set of tripotents in E
6 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
givenbyu≤e ife−uis atripotentorthogonalto e(see,for example,[16]or[19]).
It is knownthat in a JBW∗-triple M a tripotent e∈M is minimal if and only if it
is minimal for the order ≤.
We are now in position to extend [15, Theorem 2.3].
Theorem 2.5. Let E and B be JB∗-algebras, and suppose that f : S(E) → S(B)
is a surjective isometry. Let e be a minimal tripotent in E. Then 1 is isolated in
the triple spectrum of f(e).
Proof. Arguing by contradiction,we assume that 1 is not isolated in Sp(f(e)). We
shall identify B with C (Sp(f(e))).
f(e) 0
By Proposition 2.3, Theorem 2.4, and (4), there exists a minimal tripotent u ∈
B∗∗ such that
(5) f(F )=f((e+E (e))∩B ))=f((e+E∗∗(e))∩B ))=F =(u+B∗∗(e))∩B .
e 0 E 0 E u 0 B
For each natural n, we define xˆ ,yˆ the elements in B given by:
n n f(e)
t , if 0≤t≤1− 1 0, if 0≤t≤1− 1
xˆ (t):= at0ffine, if 1− 1 ≤t≤n1− 1 ;yˆ (t):= affine, if 1− 1 ≤t≤2n1
n n 2n n 2n
0, if 1− 1 ≤t≤1 1, if t=1.
2n
Clearlyxˆ ,yˆ ∈S(B)andxˆ ⊥yˆ . Weclaimthatyˆ ∈(u+B∗∗(u))∩B =F .
n n n n n 0 B u
Indeed, the support tripotent of f(e) is bigger than or equal to u in B∗∗, that is,
f(e) = u+P (u)(f(e)) in B∗∗ (see (5)). Thus, the support tripotent of yˆ also is
0 n
bigger than or equal to u in B∗∗, that is, yˆ =u+P (u)(yˆ )∈F .
n 0 n u
Therefore, denoting by x = f−1(−xˆ ) ∈ S(E) and y = f−1(yˆ ) ∈ F =
n n n n e
(e+E∗∗(e))∩B =(e+E (e))∩B , we have
0 E 0 E
1=kxˆ +yˆ k=kyˆ −(−xˆ )k=ky −x k,
n n n n n n
1
2− =kf(e)+xˆ k=ke−x k,
n n
n
and y =e+P (e)(y ), for every natural n.
n 0 n
Applying Proposition 2.2 we can find a natural n such that
0
3
kP (e)(e−x )k> >1.
2 n0 2
Finally, the inequalities
3
1≥kP (e)(y −x )k=kP (e)(e+P (e)(y )−x )k=kP (e)(e−x )k> ,
2 n0 n0 2 0 n0 n0 2 n0 2
give the desired contradiction. (cid:3)
In our list of ingredients to extend the results in [15], the next goalis a general-
ization of [15, Lemma 2.4].
Lemma 2.6. Let E be a JB∗-triple. Then every minimal tripotent u in E∗∗\E is
orthogonal to all minimal tripotents in E.
Proof. Suppose,contrarytowhatwewanttoprove,thatthereexiste∈Esuchthat
uis notorthogonaltoe. TheatomicpartofE∗∗ ispreciselythe JBW∗-subtripleof
E∗∗ generatedbyallminimaltripotents inE∗∗ (see[16,Theorem2])andcoincides
withanℓ -sumofacertainfamilyofCartanfactors(compare[17,Proposition2]).
∞
We are in position to apply [14, Section 3, page 16, (X.1) and (X.2)] to assure
that one of the following statements holds:
TINGLEY’S PROBLEM THROUGH THE FACIAL STRUCTURE 7
(a) Thereexistsaquadrangle(v ,v ,v ,v )(thatis,v ⊥v ,v ⊥v ,v ∈E∗∗(v ),
1 2 3 4 1 3 2 4 j 1 j+1
v ∈ E∗∗(v ) for every j ∈ {1,2,3}, v ∈ E∗∗(v ), v ∈ E∗∗(v ) and v =
j+1 1 j 1 1 4 4 1 1 4
2{v ,v ,v }) of minimal tripotents in a Cartanfactor contained in the atomic
1 2 3
partofE∗∗ andcomplexnumbersα,β,γ,δ suchthate=v ,|α|2+|β|2+|γ|2+
1
|δ|2 =1, αδ =βγ and u=αe+βv +δv +γv ;
2 3 4
(b) There exists a trangle (e,v,e˜) (i.e. e⊥e˜, e,e˜∈ E∗∗(v), v ∈ E∗∗(e), v ∈ E∗∗(e˜)
2 1 1
and e = {v,e˜,v}) with v a rank-2 tripotent and e˜ a minimal tripotent in a
CartanfactorcontainedintheatomicpartofE∗∗ andcomplexnumbersα,β,δ
such that |α|2+2|β|2+|δ|2 =1, αδ =β2 and u=αe+βv+δe˜.
By assumptions e 6⊥ u, and thus δ 6= 1. The minimality of e in E can be
combined with Kadison’s trasitivity theorem (compare [4, Theorems 3.3 and 3.4]
and (3)) to deduce that P (e)(E∗∗)⊆E.
1
In the case (a), we have v ,v ∈ E∗∗(e) = P (e)(E∗∗) = P (e)(E) ⊆ E because
2 4 1 1 1
e ∈ E. Another application of Kadison’s trasitivity theorem (3) shows that v ∈
3
E∗∗(v )=P (v )(E∗∗)=P (v )(E)⊆E because v ∈E. Therefore, v ,v ,v and
1 2 1 2 1 2 2 1 2 3
v belong to E and hence u=αe+βv +δv +γv ∈E, which is impossible.
4 2 3 4
In case (b) we have v ∈ E∗∗(e) = P (e)(E∗∗) = P (e)(E) ⊆ E because e ∈
1 1 1
E. Since v is a rank-2 tripotent, Kadison’s trasitivity theorem (3) proves that
e˜ ∈ E∗∗(v) = P (v)(E∗∗) = P (v)(E) ⊆ E because v ∈ E. This shows that
1 1 1
u=αe+βv+δe˜∈E leading us to a contradiction. (cid:3)
Our next result is the realcore of the study of surjective isometries between the
unit spheres of two atomic JBW∗-triples.
Theorem 2.7. LetE and B beatomic JBW∗-triples, and supposethat f :S(E)→
S(B) is a surjective isometry. Then, for each minimal tripotent e in E there exists
a uniqueminimal tripotent u in B such that f(e)=u. Moreover, there exists a real
linear surjective isometry T :E (e)→B (u) such that
e 0 0
f(e+x)=u+T (x),
e
for every x ∈ B , and the restriction of f to the maximal norm-closed face
E0(e)
F =e+B is an affine function.
e E0(e)
Proof. Combining Proposition2.3, Theorem 2.4, and (4), we find a minimal tripo-
tent u∈B∗∗ such that
(6) f(F )=f((e+E (e))∩B ))=f((e+E∗∗(e))∩B ))=F =(u+B∗∗(e))∩B .
e 0 E 0 E u 0 B
We claim that u ∈ B. If on the contrary u ∈ B∗∗\B, Lemma 2.6 implies that
u⊥v for every minimal tripotent v ∈B. Theorem 2.5 proves that 1 is an isolated
point in the triple spectrum of f(e).
As before, we shall identify B with C (Sp(f(e))). Since 1 is isolated in
f(e) 0
Sp(f(e)), the elementwˆ =χ (f(e))is atripotentinB (actually wˆ isthe support
{1}
tripotent of f(e)). Having in mind that f(e)∈F , we have f(e)=u+P (u)(f(e))
u 0
and hence u≤wˆ.
Since B is atomic, we can find a minimal tripotent wˆ in B satisfying wˆ ≤ wˆ.
0 0
ByLemma2.6andtheassumptionswehaveu⊥wˆ ,andthusu≤wˆ−wˆ . Clearly,
0 0
2=kf(e)+wˆ k=kf(e)−(−wˆ )k=ke−f−1(−wˆ )k
0 0 0
and, by Proposition 2.1 [14, Proposition 2.2] or Proposition 2.2, we have
w =f−1(−wˆ )=−e+P (e)(w ).
0 0 0 0
8 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
Having in mind that f(e)−wˆ ∈ (u+B∗∗(u))∩B , we deduce that the element
0 0 B
z = f−1(f(e)−wˆ ) belongs to (e+E∗∗(e))∩B , and thus z = e+P (e)(z), and
0 0 E 0
therefore
1=kf(e)k=kf(e)−wˆ +wˆ k=k(f(e)−wˆ )−(−wˆ )k=kz−w k
0 0 0 0 0
=ke+P (e)(z)−(−e+P (e)(w ))k=k2e+P (e)(z−w )k=2,
0 0 0 0 0
which is impossible.
We havetherefore shown that u∈B. We shallnow mimic the arguments in the
proof of [26, Proposition 3.1], details are enclosed for completeness reasons. Since
f e+B =f((e+E (e))∩B )=F =(u+B (u))∩B =u+B ,
(cid:0) E0(e)(cid:1) 0 E u 0 B B0(u)
denoting by T the translation with respect to x (i.e. T (x) = x +x ), the
x0 0 x0 0
mapping f = T−1| ◦f| ◦T | is a surjective isometry from B onto
e u Fu Fe e BE0(e) E0(e)
B . Mankiewicz’s theorem (see [23]) implies the existence of a surjective real
B0(u)
linear isometry T :E (e)→B (u) such that f =T | and hence
e 0 0 e e S(E0(e))
f(e+x)=u+T (x), for all x in B .
e E0(e)
In particular f(e)=u. Now, since
f| =T | ◦f ◦T−1| =T | ◦T ◦T−1| ,
Fe u BB0(u) e e Fe u BB0(u) e e Fe
we deduce that f| is real affine function. (cid:3)
Fe
We recall now some terminology taken from [3]. Let K(H,H′) be the space of
all compact linear operators between two complex Hilbert spaces. We shall write
K(H) instead of K(H,H). If C is a Cartan factor of type j ∈{1,2,3,4,5,6},we
j
define K =K(H,H′)for C =L(H,H′), K =C ∩K(H)for j =2,3,andin the
1 1 j j
remaining casesK =C , K =C , andK =C . The JB∗-triples K ,K ,...,K
4 4 5 5 6 6 1 2 6
are called elementary JB∗-triples. Suppose E = ∞C is an atomic JBW∗-triple,
i i
whereeachC is a Cartanfactor. Itis knownthaLtthe c -sumK(E)= c0K is a
i 0 i i
weaklycompactJB∗-tripleandatripleidealofE withK(E)∗∗ =E (seeL[3,Remark
2.6]). JB∗-triplesofthe formK(E)arecalledweaklycompactJB∗-triples(see[3]).
Itis further knownthat everyelement x in a weaklycompactJB∗-triple K(E) can
be written as a norm convergent (possibly finite) sum x = λ e , where e are
X n n n
n=1
mutually orthogonal minimal tripotents in K(E) (and in E), and (λ )⊆R+ with
n 0
(λ )→0 (see [3, Remark 4.6]).
n
Now, just a final technical step is separating us from our main goal.
Proposition 2.8. Let E and B be atomic JBW∗-triples, and suppose that f :
S(E)→S(B) is a surjective isometry. Then the following statements hold:
(a) For each minimal tripotent e ∈ E we have T (v) = f(v) for every minimal
e
tripotent v ∈ E (e), where T : E (e) → B (f(e)) is the surjective real linear
0 e 0 0
isometry given by Theorem 2.7;
(b) Lete ,...,e bemutuallyorthogonalminimaltripotentsinE,andletλ ,...,λ
1 n 1 n
be positive real numbers with max λ =1. Then
j j
n n
f λ e = λ f(e );
X j j X j j
j=1 j=1
(c) f maps S(K(E)) onto S(K(B));
TINGLEY’S PROBLEM THROUGH THE FACIAL STRUCTURE 9
(d) For each minimal tripotent e ∈ E we have f(u) = T (u) for every non-zero
e
tripotent u∈E (e);
0
(e) Letv ,...,v bemutuallyorthogonalnon-zerotripotentsinE,andletλ ,...,λ
1 n 1 n
be positive real numbers with max λ =1. Then
j j
n n
f λ v = λ f(v ).
X j j X j j
j=1 j=1
Proof. (a)LetvbeaminimaltripotentinE (e). ByTheorem2.7f(e),f(−e),f(v)
0
and f(−v) are minimal tripotents in B with kf(e)−f(−e)k = k2ek = 2. Lemma
2.1 implies that f(−e)=−f(e). Similarly, f(−v)=−f(v). By hypothesis
kf(e)±f(v)k=kf(e)−f(∓v)k=ke±vk=1.
By [12, (6) in page 360] we have
cp({f(e)})={y ∈B :ky±f(e)k≤1}=B .
B0(e)
In particular f(v)∈B .
B0(e)
The mapping T is a surjective real linear isometry between JBW∗-triples. It
e
follows from [20, Theorem 4.8] that T preserves the symmetrized triple product
e
hx,y,zi:= 1({x,y,z}+{z,x,y}+{y,z,x}). InparticularT preservescubesofthe
3 e
formx[3] ={x,x,x}andmapsminimaltripotentstominimaltripotents. Therefore
T (v) is a minimal tripotent in B (e).
e 0
Again by Theorem 2.7 we have
f(v)+T (e)=f(e+v)=f(e)+T (v).
v e
Since f(v)⊥f(e), we deduce that
f(v)=P (f(v))(f(v)+T (e))=P (f(v))(f(e)+T (v))=P (f(v))(T (v)).
2 v 2 e 2 e
SincekT (v)k=kvk=1,Lemma1.6in[16]impliesthatT (v)=f(v)+P (f(v))T (v).
e e 0 e
Finally, since T (v) is a minimal tripotent we get P (f(v))T (v)=0, T (v)=f(v).
e 0 e e
(b) Let e ,...,e be mutually orthogonal minimal tripotents in E, and let
1 n
λ ,...,λ be positiverealnumberswithmax λ =1. Wemayassumeλ =1. Let
1 n j j 1
T be the surjective real linear isometry given by Theorem 2.7. We deduce from
e1
the just quoted Theorem and the statement in (a) that
n n n n
f λ e =f(e )+T λ e =f(e )+ λ T (e )= λ f(e ).
X j j 1 e1 X j j 1 X j e1 j X j j
j=1 j=2 j=2 j=1
(c) We have already commented that every element in K(E) can be approxi-
matedinnormbyanelementoftheform n λ e ,wheree ,...,e aremutually
Pj=1 j j 1 n
orthogonalminimaltripotentsinE,andλ ,...,λ arepositiverealnumbers. Con-
1 n
sequently,elementsinS(K(E))canbe approximatedinnormby finite sumsofthe
n
form λ e , with e ,...,e and λ ,...,λ as above and max λ = 1. If we
Pj=1 j j 1 n 1 n j j
n n
observethat,byTheorem2.7and(b), f λ e = λ f(e )∈S(K(B)),we
X j j X j j
j=1 j=1
conclude that f(S(K(E))) ⊆ S(K(B)). Applying the same argument to f−1 we
get f(S(K(E)))=S(K(B)).
10 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
(d) Let u be a non-zero tripotent in E (e), where e is a minimal tripotent in
0
E. We may assume that u is not minimal, otherwise the desired statement follows
from(a). Thus, since E is atomic, we can find a minimal tripotent v in E (e) with
0
u≥v. By Theorem 2.7 we have
f(e)+T (u)=f(e+u)=f(e+v+(u−v))=f(v)+T (e)+T (u−v)
e v v
=(by (a))=f(v)+f(e)+T (u−v)=f(e)+f(v+u−v)=f(e)+f(u),
v
which proves the desired statement.
(e) Under the assumptions there exists k ∈ {1,...,n} with λ = 1. Since E is
k
atomic, we can find a minimal tripotent e ≤ v . Let T be the surjective real
k k ek
linear isometry given by Theorem 2.7. We deduce from the just quoted Theorem
and the statement in (d) that
n
f λ v =f(e )+T (v −e )+ λ v
X j j k ek k k X j j
j=1 j6=k
=f(e )+T ((v −e ))+ λ T (v )
k ek k k X j ek j
j6=k
n
=f(e +v −e )+ λ f(v )= λ f(v ).
k k k X j j X j j
j6=k j=1
(cid:3)
We can establish now our main result.
Theorem 2.9. Let f :S(E)→S(B) be a surjective isometry, where E and B are
atomic JBW∗-triples. Then there exists a (unique) real linear isometry T :E →B
such that f =T .
|S(E)
Proof. Let K(E) and K(B) denote the ideals of E and B generated by the mini-
mal tripotents in E and B, respectively. We deduce from Proposition 2.8(c) that
f(S(K(E)))=S(K(B))andf| :S(K(E))→S(K(B)) isasurjectiveisom-
S(K(E))
etry. WeobservethatK(E)andK(B)areweaklycompactJB∗-triplesinthesense
of [3, 14], so by [14, Theorem 2.5] there exists a surjective real linear isometry
S :S(K(E))→S(K(B)) satisfying f(x)=S(x), for every x∈S(K(E)).
The mapping T = S∗∗ : K(E)∗∗ = E → K(B)∗∗ = B is a surjective real linear
isometry and a weak∗ continuous mapping. By construction T(x) = S(x) = f(x),
for every x∈S(K(E)).
We claim that
(7) T(w)=f(w), for every non-zero tripotent w∈E.
Since E is atomic, we can find a minimal tripotent e ≤ w. The mapping T :
e
E (e)→B (f(e))givenbyTheorem2.7isasurjectivereallinearisometrybetween
0 0
(atomic) JBW∗-triples. By [24, Proposition 2.3(1.)] T also is weak∗-continuous.
e
By construction T (v) = f(v) = S(v) = T(v) for every finite-rank tripotent v ∈
e
E (e). Since everytripotent in anatomic JBW∗-triple canbe approximatedin the
0
weak∗-topology by a net of finite-rank tripotents, we deduce from the above that
T(w)=T (w) for every tripotent in E (e). Now, by Theorem 2.7 and Proposition
e 0
2.8(d) or (e), we have
f(w)=f(e)+T (w−e)=f(e)+T(w−e)=T(e)+T(w−e)=T(w),
e
