ATOMIC DECOMPOSITION OF
REAL JBW*-TRIPLES
By A. PERALTA and L. L. STACHO
Received xx April 1999]
1. Introduction
The class of complex Banach spaces called JB*-triples has been the object
of intensive investigations in the last two decades. By Kaup's Riemann mapping
theorem 14], a holomorphically symmetric bounded domain is holomorphically
equivalent to the unit ball of some complex JB*-triple and, conversely, the unit
ball of any complex JB*-triple is a symmetric domain. This fact made it possible
to apply complex geometric arguments very successfully in the study of complex
JB*-triples as well as the standard techniques of functional analysis. In particular
a deeply elaborated geometric theory of dual complex JB*-triples (the so-called
complex JBW*-triples) appeared, sheding new light to von Neumann algebras and
the Gelfand-Naimark theorem.
Recently considerable attention is paid to real JB*-triples introduced in 13]
as isometric copies of closed real subtriples of complex JB*-triples, objects which
can conveniently be studied as invariant subspaces of conjugations in complex
JB*-triples. In this manner several results concerning complex JB*-triples have
been extended to the real case 13,17,6,15].
In the complex case JBW*-triples split into the direct sum of an ideal free of
indecomposable tripotents (called atoms) and the direct sum of a family of Cartan
factors, the latter coinciding with the w*-closed linear hull of all atoms. One of the
main purposes of the present paper is the real analogue of this decomposition. For
this rst we investigate briey the natural ordering of tripotents with respect to
orthogonal decomposability. We complete the result of 6, 5, 8] proving that the
minimal tripotents of a real JBW*-triple are the support functionals of the extreme
points of the unit ball of its predual. As a key point to the atomic decomposition
we show in particular that real atoms (indecomposable tripotents in the bidual)
are the same as real minimal tripotents in the terminology of 15] and every real
atom is the real part of some complex minimal tripotent from the complexi
cation.
We conclude our work obtaining an atomic decomposition for real JBW*-triples.
2
A. PERALTA AND L.L. STACHO
2. Tripotents in real JB*-triples
By a complex JB*-triple we mean a complex Banach space V equipped
with a triple product f g : V V V ! V which is bilinear symmetric in the
outer variables and conjugate-linear in the middle one satisfying
(i) (Jordan identity) fa b fx y zgg = ffa b xg y zg ; fx fb a yg zg +
fx y fa b zgg
(ii) L(x x) is an hermitian operator on V with nonnegative spectrum, and
k fx x xgk = kxk3
for all x y z a b 2 V , where L(a b) is the linear operator de
ned by L(a b)y :=
fa b yg .
A real JB*-triple is a real Banach space E with an R -trilinear triple
product whose complexi
cation admits a (necessarily unique) complex JB*-triple
structure with norm and triple product extending those of E . It is well-known
13] that any closed real subtriple of a complex JB*-triple is a real JB*-triple. In
particular complex JB*-triples are real JB*-triples at the same time.
By a real complex] JBW*-triple we mean a real complex] JB*-triple
whose underlying Banach space is a dual Banach space in metric sense. It is known
(see 2] and 17]) that every real or complex JBW*-triple has a unique predual up
to isometric linear isomorphisms and its triple product is separately w*-continuous
in each variable (wrt. the unique weak*-topology induced by any predual).
Henceforth E denotes an arbitrarily xed real JB*-triple. We write
(2:1)
Eb := E iE
: x iy 7! x (;iy)
for the complexi
cation of E and the canonical conjugation of Eb . We shall
denote by k:k and f g the complex JB*-triple norm and the complex triple
product in Eb extending those of E , respectively. Notice that the conjugation is a conjugate-linear isometry of Eb which is w*-continuous if Eb is a JBW*-triple
13]. Given a real or complex JB*-triple U , the elements e 2 U with fe e eg = e
are called tripotents. We denote the set of all tripotents of U by Tri(U ) . Every
e 2 Tri(U ) induces a decomposition of U into the eigenspaces of L(e e) , the
Peirce decomposition
U = U0 (e) U1(e) U2 (e) where Uk (e) := fx 2 U : fe e xg = k2 xg
are the Peirce subspaces of U associated with the tripotent e . We have the
multiplication rules
fU0 (e) U2(e) U g = fU2 (e) U0(e) U g = 0 fUi (e) Uj (e) Uk (e)g Ui+j+k (e)
for i j k 2 f0 1 2g with the convention U` (e) := 0 (` 6= 0 1 2) . The canonical projection Pk : U ! Uk (e) called the Peirce k -projection of e is
clearly a polynomial of L(e e) . If x is an element of U , we write Q(x) for
3
A. PERALTA AND L.L. STACHO
the mapping given by Q(x)y := fx y xg . According to the Jordan identity,
Q(e)3 = Q(fe e eg) = Q(e) for every tripotent e 2 U . Hence Q(e) induces
also a decomposition
U = U 1(e) U 0(e) U ;1 (e) where U k (e) := fx 2 U : Q(e)x = kxg
into R -linear subspaces with the properties
U2(e) = U 1 (e) U ;1 (e) U 0 (e) = U1 (e) U0(e)
U j (e) U k (e) U ` (e) U jk` (e) (jk` 6= 0 j k ` 2 f0 1g) :
We write P k (e) for the natural projection of U onto U k (e) .
Two tripotents u v in U are said to be orthogonal ( u ? v in notation)
if L(u v) = L(v u) = 0 . It is well-known that u ? v () fu u vg = 0 ()
fv v ug = 0 that is if u 2 U0(v) or v 2 U0 (u) . Notice that v + z 2 Tri(U ) with
L(v + z v + z) = L(v v) + L(z z) whenever v z 2 Tri(U ) and v ? z .
Let u v 2 Tri(U ) . We say that u majorizes v in orthogonal decomposability, u v or v u in notation, if u ; v 2 Tri(U ) with u ; v ? v . Henceforth
we write min Tri(U ) := fu 2 Tri(U ) n f0g : 6 9 v 2 Tri(U ) n f0g v u 6= vg .
Notice that if u v with u v 2 U , then U2 (u) U2 (v) and U0(u) U0(v) .
Applying the algebraic argument of 5, Lemma 2.4] we get immediately that u v () Q(u)v = Q(v)u = v () U 1 (u) U 1 (v) and Q(v)u = v . Therefore, the
map u 7! U 1 (u) is a strictly monotonic mapping from Tri(U ) into the set of all
R -linear subspaces of U . In particular, e 2 min Tri(U ) if U 1(e) = Re .
Proposition 2.2. For a non zero tripotent
e in a real JBW*-triple W we have
e 2 min Tri(W ) if and only if W (e) = Re . In the latter case the Peirce space
W2 (e) is a real Hilbert space.
Proof. If e 2 Tri(W ) n f0g with W 1 (e) = Re , then trivially e 2 min Tri(W ) .
Let now e 2 min Tri(W ) . Suppose there exists x 2 W 1 (e) but x 62 Re . It
is well-known 12, Lemma 3.13] that the w*-closed complex JB*-subtriple Ab of
c generated by fe xg is a commutative complex von Neumann-*algebra (W*W
algebra) whose unit element is e when equipped with the operations u v :=
fu e vg , u := Q(e)u , (u v 2 Ab): Byn the separate w*-continuity
of the triple
o
1
1
1
1
c (e) W
c (e) W
c (e) = W
c (e) , W
c 1 (e)
product in its second variable and since W
c containing A := fa 2 Ab : a = ag .
is a w*-closed real JB*-subtriple in W
Therefore the real w*-closed JB*-subtriple of W generated by fe xg is necessarily
A . In particular A is a commutative real W*-subalgebra of Ab . Hence there is a
1
surjective linear isometry
: L1
R () ! A with (fg ) = f(f ) e (g )g (1 ) = e
for a positive Radon measure on some compact topological space . Since
;1 x 62 R1 = ;1 e , there is a -measurable set S such that (S ) (nS ) > 0 .
4
A. PERALTA AND L.L. STACHO
Then, with g1 := 1S , g2 := 1nS , ek := gk , we have e1 e2 2 Tri(W ) , e1 ? e2
and e = 1 = (g1 + g2) = e1 + e2 which contradicts the minimality of e .
The statement concerning the spin structure of the Peirce space W2 (e) holds
in a more general setting. If E is a real JB*-triple and 0 =
6 e 2 Tri(E ) with
1
E (e) = Re then, E2(e) is a real JB*-triple of rank 1 in the terminology of 15].
Then, by 15, Proposition 5.4], as metric space E2(e) is a real Hilbert space.
Remark 2.3. In 15] the term
real minimal tripotent is used for tripotents
with E 1(e) = Re in real JB*-triples. This terminology is somewhat misleading
from the viewpoint that Tri(E ) carries a natural ordering with respect to which
minimal tripotents may have dim(E 1(e)) > 1 (e.g. in CR (0 1]) ). By Proposition 2.2 above, for real JBW*-triples the terminology of 15] is appropriate. We
suggest that tripotents in real JB*-triples with E 1(e) = Re should be called
atomic tripotents in accordance with the terminology of complex atomic decompositions 7]. Indeed, as a consequence of Proposition 2.2, an atomic tripotent is
minimal (thus orthogonally indecomposable) even in the bidual embedding of the
underlying real JB*-triple.
The last part of this section is devoted to show that, analogously to the
complex case described in 5], for a real JBW*-triple, the natural ordering of its
tripotents is faithfully represented by the natural ordering of the w*-semi-exposed
faces of its unit ball and the norm-semi-exposed faces of the unit ball of its predual,
respectively. Given any real or complex Banach space X with norm k:k , we write
B (X ) and X for its closed unit ball and dual space, respectively. We denote the
standard functional norm on X by k k . For polar faces of unit balls we shall
use the notations
S 0 := ff 2 B (X ) : f (x) = 1 (x 2 S )g (S B (X ))
K0 := fx 2 B (X ) : f (x) = 1 (f 2 K )g (K B (X )) :
We say that F B (X ) is a norm-semi-exposed face of B (X ) if F = K0
for some K B (X ) . In case X has a unique predual, we write X for its
canonical predual X := fw*-continuous linear functionals of X g as a subspace
of X . Then X is identi
ed canonically with (X) as x X 3 f 7! f (x)]
for x 2 X and we say that F is a w*-semi-exposed face of B (X ) if F = S 0
for some S B (X) . We denote by Sn (B (X )) resp. S (B (X )) ] the set of all
norm-semi-exposed faces of B (X ) resp. the set of all w*-semi-exposed faces of
B (X ) if X has a unique predual].
Recently Edwards and Ruttimann 6] deduced the following order description
of tripotents in terms of faces from the analogous results in complex setting. For
the sake of completeness we include a direct independent discussion in terms of
semi-exposed faces which requires a shorter proof than that in 6].
Proposition 2.4. 6] Let
W be a real JBW*-triple. Then the map
(e) := e + B (W ) \ W0 (e)] (e 2 Tri(W ))
5
A. PERALTA AND L.L. STACHO
is an anti-order isomorphism between Tri(W ) n f0g and Sw (B (W )) n fB (W )g
(with ordinary set inclusion) and (e) = (feg0 )0 (e 2 Tri(W )) . The map
(e) := feg0 (e 2 Tri(W )) is a surjective order isomorphism between Tri(W )
and Sn (B (W)) n fB (W)g .
Proof. By 13, Proposition 4.6], the map is a bijection Tri(W ) n f0g $
Sw (B (W )) n fB (W )g . Let u v 2 Tri(W ) with 0 6= u v . In terms of the
c := W iW . Hence, by 5, Theorem 4.6],
complexi
cation, also u v in W
c0 (u)) (
b v ) := v + (B \ W
c0 (v )) :
b u) := u + (B \ W
(2:5)
(
b
b
W
W
b v ) = (v ) , from (2.5) it follows (u) =
Since it is easy to see that W \ (
b u) W \ (
b v ) = (v ) .
W \ (
Conversely, if (v) (u) then v 2 (v) (u) = u + (B (W ) \ W0(u)):
Hence v = u + x for some x 2 W0 (u) and therefore u v .
Given any e 2 Tri(W ) , the set (feg0)0 is the least w*-semi-exposed face of
B (W ) containing the tripotent e . If e 2 F 2 Sw (B (W )) then it follows u e
and (e) (u) = F . Thus (e) = (feg0 )0 .
Now we can proceed to the proof of the statement concerning the map .
Notice that in general, for any Banach space X with unique predual X ( X )
we have S0 = ((S0)0 )0 and K 0 = ((K 0)0 )0 for S B (X ) , K B (X) . Therefore
(e) = (feg0 )0 = (e)]0 and (e) = feg0 = (feg0 )0 ]0 = (e)]0 2 Sn (B (W)) n
fB (W)g for all e 2 Tri(W ) .
If u v in Tri(W ) then, as we have seen, (u) (v) and and hence
(u) = (u)]0 (v)]0 = (v) . Conversely, if (u) (v) then (u) =
(u)]0 (v)]0 = (v) implying u v .
Finally we show range() Sn (B (W)) n fB (W)g . Let K 2 Sn (B (W)) n
fB (W)g be chosen arbitrarily. By de
nition, K = S0 for some S B (W ) . Since
(S0 )0 is the smallest w*-semi-exposed face of B (W ) containing S , K 0 = (u)
for suitable u 2 Tri(W ) . It follows (K ) = (u)]0 = (K 0 )0 = (S0 )0]0 = S0 = K
that is K 2 range() .
Concerning minimal tripotents we complete this result. We are going to prove
that the minimal tripotents of a real JBW*-triple are the support functionals of
the extreme points of the unit ball of the predual.
Remark 2.6. It is well-known 12,
7 Corollary 1.2] that in a complex JB*-
triple, the Peirce projections of a tripotent are contractive. Hence, by passing to
comlexi
cation, it follows immediately that Peirce projections of tripotents in real
JB*-triples are contractive. In particular, if E is a real JB*-triple and e 2 Tri(E )
then the operator
P 1(e) := 21 (IdE + Q(e))P2(e)
is contractive, too. In 17, Lemma 2.9] it is shown that if E is a real JB*-triple,
e 2 Tri(E ) , f 2 E and kfP2(e)k = kf k then f = fP2 (e) . The following
Lemma generalizes slightly this result.
A. PERALTA AND L.L. STACHO
6
E be a real JB*-triple, u 2 Tri(E ) and f 2 E such that
f (u) = kf k = 1 . Then f = f P 1 (u) .
Proof. By 17, Lemma 2.9] we have f = P2 (u)f (:= f P2 (u)) . Let y 2 E ;1(u)
( fu u yg = y , fu y ug = ;y ). We may assume without loss of generality f (y) Lemma 2.7. Let
0 . By induction we get
n
(u + ty)3 = u + ty + O(jtj2)
(t > 0 n = 1 2 : : :):
Therefore, for t > 0 ,
n
n
n
(1 + tf (y))3 ku + tyk3 = k(u + ty)3 k = ku + ty + O(jtj2)k 1 + tkyk + O(jtj2)
1 + 3n tf (y) + O(jtj2) 1 + tkyk + O(jtj2)
3n f (y) + O(jtj) kyk + O(jtj) :
Thus, for t # 0 , we obtain f (y) 31n kyk (n = 1 2 : : :) . It follows f (y) = 0 for
every y 2 E ;1(u) . Since E2(u) = E 1(u) E ;1 (u) and f = fP2 (u) , we conclude
f = f P 1(u) .
X be a Banach dual space with unique predual X , let M be
a weak*-closed subspace of X and let f 2 ext(B (X)) . Assume P is a (w*-w*)continuous contractive projection of X onto M such that f = P f . Then f jM 2
ext(B (M)) (with the usual identications X ff 2 X : f w*-continuousg ,
M ff jM : f 2 Xg ).
Proof. Suppose f jM = 21 (g + h) where g h 2 B (M) , that is g h : M ! R are
w*-continuous bounded linear functionals. Consider the functionals ge := g P and
eh := h P . Since P is (w*-w*)-continuous with kP k 1 and g h 2 B (M ) , we
have ge eh 2 B (X) . On the other hand f = f P = (f jM ) P = 21 (g + h) P =
1 e+e
h) : Since f 2 ext(B (X)) , we get f = ge = eh which entails g = h = f jM
2 (g
i.e. f jM 2 ext(B (M)) .
Lemma 2.8. Let
Proposition 2.9. Let W be a real JBW*-triple with predual
ext(B (W)) . Then ff g 2 Sn (B (X)) .
W and let f 2
Proof. Using the same argument of the proof of 17, Lemma 2.10], we see the
existence of some u 2 Tri(W ) such that f (u) = 1 . By Lemma 2.7, f = f P 1(u) .
Since the subspaces W 1 (u) and kerP 1 (u)(= W 0 (u) W ;1 (u)) are w*-closed in
W and since (see Remark 2.6) we have kP 1(u)k = 1 , we can apply Lemma 2.8
to conclude that f jW 1(u) 2 ext(B ((W 1(u)))) . It is well-known 13] that W 1 (u)
is a JB-algebra in a canonical manner. As being a w*-closed subspace of W ,
W 1 (u) has a predual and hence we can regard it as a JBW-algebra. By assumption
f jW 1(u) is a pure state (extremal point of the unit ball of the predual) of the JBWalgebra W 1(u) . Thus, by 1, Proposition 3.3], ff jW 1(u) g is a norm-exposed face
A. PERALTA AND L.L. STACHO
7
of B ((W 1(u))) , i.e. there exists a 2 W 1(u) such that f (a) = 1 and g(a) < 1
for all g 2 B ((W 1(u))) n ff jW 1(u) g . Consider any h 2 B (W) . Then hjW 1 (u) 2
B ((W 1(u))) . Thus we have the alternatives h(a) < 1 or hjW 1 (u) = f jW 1(u) .
Assume hjW 1(u) = f jW 1(u) . Then h(u) = f (u) = 1 and, by Lemma 2.7, h =
hP 1 (u) whence
h = h P 1 (u) = hjW 1 (u) ] P 1(u) = f jW 1(u) ] P 1(u) = f P 1 (u) = f :
This shows that ff g is a norm-semi-exposed face of B (W) .
As a consequence of the above proposition, if W is a real JBW*-triple and
f 2 ext(B (W)) then ff g is norm-exposed, thus (ff g0)0 = ff g is a minimalnorm-semi-exposed face of B (E) .
Corollary 2.10. The minimal tripotents of the real JBW*-triple W are exactly
the support tripotents of the extreme points of B (W) . That is f 2 extB (W) if
and only if f (u) = 1 for some u 2 min Tri(W ) . Moreover, if u 2 min Tri(W )
then f (u) = 1 for some f 2 extB (W) .
Proof. Let f 2 extB (W) . We have seen that the singleton ff g is a minimal
norm-semi-exposed face of B (W) . By Proposition 2.4, ff g = (u) and hence
f (u) = 1 for some minimal tripotent of W .
Suppose u 2 min Tri(W ) and consider any f 2 (u) . We show that ff g0 =
(u) and f 2 extB (W) .
By the de
nition of (u) , f (u) = 1 . Hence ff g0 2 Sw (B (W )) and, ff g0 (u)]0 = (u) (since, as a folklore result in Banach space geometry, S T B (X ) implies S 0 T 0 and S0 = ((S0)0 )0 ). By Proposition 2.4, (u) is a maximal
w*-semi-exposed face of B (W ) and (u) is a minimal norm-semi-exposed face
of B (W ) with (u) = (u)]0 . It follows ff g0 = (u) . Assume f = (f1 + f2 )=2
where f1 f2 2 B (W) . Since u 2 Tri(W ) , kuk = 1 , and kfk k 1 we have
jfk (u)j 1 (k = 1 2) , necessarily
1 = f (u) = f1(u) = f2(u)
kf jW2(u)k = kf1jW2(u)k = kf2 jW2(u)k = 1 :
By Lemma 2.7, f = f P 1 (u) and fi = fi P 1 (u) ( i = 1 2 ). Since u 2 min Tri(W )
we have P 1 (u)(W ) = Ru entailing f1 = f2 = f .
3. Atomic decomposition
As previously W denotes a real JBW*-triple. First we establish a relationc.
ship between the minimal tripotents of W and those of its complexi
cation W
The term (complex) minimal tripotent is well-established in the literature of
complex JB*-triples: if V is a complex JB*-triple then u 2 Tri(V ) n f0g is called
a (complex) minimal tripotent if
(3:1)
V2(u) = Cu
A. PERALTA AND L.L. STACHO
8
without any reference to the ordering of orthogonal decomposability. Notice
that, if V is a complex JB*-triple and u 2 Tri(V ) , then V2 (u) = Cu is equivalent
to V 1 (u) = Ru . Similar arguments to that in the proof of Proposition 2.2 show
that indeed min Tri(V ) = fu 2 Tri(V ) : V2 (u) = Cug in any complex JBW*triple V .
e 2 Tri(W ) be a tripotent with W 1 (e) = Re . Then e is the
c.
sum of at most 2 orthogonal (minimal) tripotents in W
c there is nothing to prove. If e is not minimal,
Proof. If e is minimal in W
W2 (e) is a real spin factor (see last paragraph of the proof of Proposition2.2).
c2 (e) = W2 (e) iW2 (e) is a complex spin factor. It is well-known 16]
Hence W
Lemma 3.2. Let
that in a complex spin factor each element of the real part is the sum of two
orthogonal (complex) minimal tripotents being conjugates of each other. Thus
c2 (e))( Tri(W
c)) , W
c2 (e)]2 (v ) = Cv and v ? v . In
e = v + v where v 2 Tri(W
c2 (v ) W
c2 (e) whence even W
c2 (v ) =
c ) . Therefore, W
particular v e in Tri(W
c2 (e)]2 (v ) = Cv .
W
c of any real JBW*-triple W deBy 7, Theorem 2], the complexi
cation W
composes into the orthogonal direct sum (or which is the same `1 -direct sum)
c ) and Nb := fx 2 W
c : L(a a)x =
of the JBW*-ideals Ab := Spanw min Tri(W
0 (a 2 Ab)g: Here min Tri(Nb ) = and
Ab = SpanC w
F
2F
cg :
F for F := f minimal w*-closed complex ideals of W
Since the conjugation preserves the triple product and is w*-continuous,
c)) =min Tri(W
c ), (Ab) = Ab and (Nb ) = Nb . Hence W decomposes
(min Tri(W
into the orthogonal direct sum
W = Ab 1 Nb Ab := fx 2 Ab : (x) = xg
Nb := fx 2 Nb : (x) = xg :
We set F0 := fF 2 F : (F ) = F g and x any maximal subset F1 of F n F0
with the property (F ) ? F (F 2;F1 ) . We de
ne F;1 := f (F ) : F 2 F1g . By
15, Lemma 6.2] we have Ab = 1 F;1 F0 F1 :
Remarks 3.3.
c or e = v + (v )
(a) If e 2 min Tri(W ) by Lemma 3.2, e is either minimal in W
c ) such that v ? (v ) . It is well-known 8] that
for some v 2 min Tri(W
c ) belongs to a unique minimal w*-closed complex ideal
each v 2 min Tri(W
c . Therefore, given any e 2 min Tri(W ) , there
(complex Cartan factor) of W
c and a complex-minimal tripotent
exists a complex Cartan factor F 2 F of W
v 2 F such that either e = v 2 F , or F = (F ) , e = v + (v) , or F ? (F ) ,
e = v + (v) .
A. PERALTA AND L.L. STACHO
9
(b) If W is reexive, there is no in
nite sequence e1 e2 : : : of pairwise orthogonal
non-zero tripotents in W (because their span would be a subspace isomorphic
to c0 ). Therefore, by spectral decomposition, any element is a nite linear
combination of a family of minimal tripotents in a reexive real JB*-triple.
(c) According to Kaup's classi
cation in 15, Theorem 4.1], the real form of a
complex Cartan factor of type < 4 can be written as
fx 2 L(H K ) : Px = xg
;
P 2 LR L(H K ) L(H K )
P2 = P
such that the real-linear projection P is w*-continuous and maps nite rank
operators to nite rank operators.
H K be complex Hilbert spaces and let H1 H , K1 K
be subspaces. Suppose Z is a real subtriple of L(H K ) (with the triple product fx y zg := (xyz + zy x)=2 ). Then min Tri(Z1 ) min Tri(Z ) for the real
subtriple Z1 := fz 2 Z : z(H ) K1 z (K ) H1g .
Proof. Let e 2 min Tri(Z1) . Since the Peirce projection P2 (e) has the form
P2 (e)z = Q(e)2z = ee ze e ( z 2 Z ), it is easy to prove that Z2 (e) Z1 .
Assume e = e1 + e2 where e1 e2 are orthogonal tripotents in Z . Then
e1 e2 2 Z2 (e) Z1 (since e e1 e2 in Z ). However by the minimality of e in
Z1 , either e1 = 0 or e2 = 0 .
Lemma 3.4. Let
Corollary 3.5. If dim(H1 ) dim(K1 ) < 1 then any element of
Z1 is a nite
real-linear combination from min Tri(Z ) \ Z1 .
Proof. Observe that dim(Z1) dim(H1)dim(K1) < 1 . Thus, according to Remarks 3.3 (b) , any element of Z1 is a nite real-linear combination of minimal
tripotents of Z1 and the latter are automatically minimal tripotents of Z .
Now we are in a position to prove our main result.
Theorem 3.6. Let
(3:7)
W be a real JBW*-triple. Then
W = A 1 N
where A := SpanwR min Tri(W ) , and N := fx 2 W : L(a a)x = 0 (a 2 A)g
are w*-closed ideals in W and min Tri(N ) = . In terms of the complexications
c := W iW , Ab := A iA , Nb := N iN we have Ab = SpanwC min Tri(W
c)
W
c : L(a a)x = 0 (a 2 Ab)g .
and Nb = fx 2 W
Proof. According to Remark 3.3 (a) , if e 2 min Tri(W ) then either e 2 F for
some Cartan factor F 2 F0 or e 2 F (F ) for some F 2 F1 . Observe that
e + (e) 2 min
Trifx + (x) : x 2 F g whenever F 2 F1 and e 2 min Tri(F ) . Since
w
F = SpanR min Tri(F ) for any Cartan factor F 2 F and since the conjugation
is w*-continuous, (3.7) is immediate in this case.
A. PERALTA AND L.L. STACHO
10
Let F 2 F0 . If F is reexive, that is if F is of Type 4 or nite dimensional (in
particular of Type 5,6) then Remark 3.3 (b) ensures (3.6). Assume F is of Type 3 . Then, by Remark 3.3 (c) , we may assume without loss of generality that F is
a w*-closed real subtriple of the form F = fx 2 L(H K ) : (x) = xg for some
complex Hilbert spaces H K and a w*-continuous (complex-linear) projection
mapping nite rank operators to nite rank operators. Also the conjugation maps nite rank operators to nite rank operators. Thus Z := fz 2 F : (z) = zg
is a w*-closed real subtriple of L(H K ) . Given any element z of Z , there is a
net (zi : i 2 I ) consisting of nite rank operators converging to z in w*-sense in
L(H K ) . Then fi := ( 21 + 12 )( 21 + 12 )zi w! z and fi 2 fz 2 Z : rank(z) < 1g
(i 2 I ) . Applying the Corollary of the previous lemma with the nite dimensional
spaces Hi := fi (H ) respectively Ki := fi(K ) in place of H1 resp. K1 , we see
that any term fi is a nite real-linear combination from min Tri(F ) . The proof
of (3.6) and hence the proof of (3.7) is complete.
Acknowledgement
The authors were supported by the Bilateral Spanish-Hungarian Research
Cooperation grant No. TET E-3/97, the Programa Nacional F.P.I. of the Ministry
of Education of Spain and the grant OTKA T26532 of the Hungarian Scienti
c
Research Foundation.
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2.
3.
4.
5.
6.
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8.
9.
10.
REFERENCES
E. Alfsen and F.W. Shultz, 'State spaces of Jordan algebras', Acta Math., 140 (1978),
155-190.
T.J. Barton and R.M. Timoney, 'Weak*-continuity of Jordan triple products and applications', Math. Scand., 59 (1986), 177-191.
T.-C. Dang and B. Russo, 'Real Banach Jordan triples', Proc. Amer. Math. Soc., 112(1)
(1994), 135-145.
C.M. Edwards, 'On the facial structure of a JB-algebra', J. London Math. Soc. Ser.(2), 19
(1979), 335-344.
C.M. Edwards and G.T. Ruttimann, 'On the facial structure of the unit balls in a JBW*triple and its predual', J. London Math. Soc. Ser.(2), 38 (1988), 317-332.
C.M. Edwards and G.T. Ruttimann, 'The facial and inner ideal structures of a real JBW*triple', Mathematisches Nachrichten, to appear.
Y. Friedman and B. Russo, 'Structure of the predual of a JBW*-triple', J. Reine u. Angew.
Math., 356 (1985), 67-89.
Y. Friedman and B. Russo, 'Solution of the contractive projection problem', J. Funct. Anal.,
60, (1985), 56-79.
Y. Friedman and B. Russo, 'The Gelfand-Naimark theorem for JB*-triples', Duke Math. J.
53 (1989), 139-148.
L. Harris, 'Bounded symmetric homogeneous domains in innite dimensional spaces', Lecture Notes in Math. No. 364, Springer Verlag, Berlin, 1973.
11
11.
12.
13.
14.
15.
16.
17.
18.
A. PERALTA AND L.L. STACHO
H. Hanche-Olsen and E. Strmer, 'Jordan Operator Algebras', Pitman, London, 1984.
G. Horn, 'Klassikation der JBW*-Tripel vom Typ I', Ph.D. Thesis, Tubingen, 1984.
J.-M. Isidro, W. Kaup, and A. Rodriguez, 'On real forms of JB*-triples', Manuscripta
Math., 86 (1995), 311-335.
W. Kaup, 'A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces', Math. Z., 183 (1983), 503-529.
W. Kaup, 'On real Cartan factors', Manuscripta Math., 92 (1997), 191-222.
O. Loos, Bounded Symmetric Domains and Jordan Pairs, Lecture Note of the UCI, Irwine,
1977.
J. Martinez and A. Peralta, 'Separate weak*-continuity of the triple product in dual real
JB*-triples', Math. Z.,to appear.
H. Upmeier, 'Symmetric Banach Manifolds and Jordan C*-algebras', Mathematics Studies
No. 104 (Notas de Matematica ed. by L. Nachbin), North Holland, Amsterdam, 1985.
Departamento de Analisis Matematico
Universidad de Granada
18071 Granada, SPAIN
E-mail: [email protected]
Bolyai Institute,
University of Szeged
6720 Szeged, HUNGARY
E-mail: [email protected]