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Page 1
J. Korean Math. Soc. 41 (2004), No. 1, pp. 107–130
STRUCTURAL PROJECTIONS ON A JBW
∗
-TRIPLE
AND GL-PROJECTIONS ON ITS PREDUAL
Remo V. Hugli
Abstract. A JB
∗
-triple is a Banach space A on which the group
Aut(B) of biholomorphic automorphisms acts transitively on the
open unit ball B of A. In this case, a triple product {· · · } from
A × A × A to A can be defined in a canonical way. If A is also
the dual of some Banach space A
∗
, then A is said to be a JBW
∗
-
triple. A projection R on A is said to be structural if the identity
{Ra, b, Rc} = R{a, Rb, c, } holds. On JBW
∗
-triples, structural
projections being algebraic objects by definition have also some
interesting metric properties, and it is possible to give a full char-
acterization of structural projections in terms of the norm of the
predual A
∗
of A. It is shown, that the class of structural projec-
tions on A coincides with the class of the adjoints of neutral GL-
projections on A
∗
. Furthermore, the class of GL-projections on A
∗
is naturally ordered and is completely ortho-additive with respect
to L-orthogonality.
1. Introduction
The results presented here were obtained under the supervision of
and in collaboration with G. T. Ruttimann (†1999) at the department
of Mathematical Statistics at the University of Berne and C. M. Edwards
at The Queen’s College in Oxford. Their common work has ended too
early, but will persist in mathematics.
We investigate certain classes of complex Banach spaces known as
JB
∗
-triples or JBW
∗
-triples, the holomorphic properties of which entail
a variety of features, including the existence of a ternary product de-
pending canonically upon the norm on these spaces. Having its roots in
Received November 28, 2003.
2000 Mathematics Subject Classification: Primary 46L70; Secondary 17C65.
Key words and phrases: JBW
∗
-triple, structural projection, GL-projection.
Research supported in part by a grant from Schweizerischer Nationalfonds/Fonds
national suisse.
Page 2
108
Remo V. Hugli
studies begun by E. Cartan [8] on bounded symmetric domains, the the-
ory of JB
∗
-triples has received considerable attention in the last decades,
because of the deep connections that persist between them and various
fields of mathematics and mathematical physics. The works of Koecher
[39] and Loos [41] show that the finite dimensional JB
∗
-triples are pre-
cisely those complex Banach spaces, the open unit balls of which are
bounded symmetric domains, earlier classified by E. Cartan. A domain
U in a complex Banach space is said to be symmetric if, for every ele-
ment a in U, there exists a biholomorphic mapping of order two of U
onto itself having a as an isolated fixed point. In several works, the
theory of JB
∗
-triples was generalized to infinite dimensions by Braun,
Kaup and Upmeier [5], Harris and Kaup [34] and Vigue [52, 53, 54].
An early result that provided a connection between infinite dimensional
holomorphy and operator algebras was obtained by Harris, who proved
in [33] that the open unit ball of a C
∗
-algebra A is a bounded symmetric
domain.
The various types of operator algebras are sometimes considered
as non-classical probability spaces, for instance to describe quantum
mechanical systems. In this context, contractive projections represent
analogs of conditional expectations in classical probability spaces. Un-
like most other categories of operator algebras, the one consisting of
JB
∗
-triples is stable under contractive projections and, as a consequence,
the category of JBW
∗
-triples is stable under weak
∗
-continuos contrac-
tive projections. This stability property, obtained by Kaup [38] and
Stach`o [50] from holomorphic theory, generalized earlier results by Choi
and Effros [9], Effros and Størmer [26] and Friedman and Russo [28].
It is explicitly formulated for JBW
∗
-triples in Lemma 3.2 below. For
further details on these and related topics, we refer to the extended lit-
erature including [4, 8, 12, 13, 27, 31, 33, 37, 39, 41, 43, 49, 51]. An
interesting survey on the holomorphic aspects of triples is provided in
[3]. A detailed and more extended overview, containing also the history
of Jordan structures, can be found in [47] and [48].
A particular subclass of contractive projections on JBW
∗
-triples is
given by structural projections, defined by means of the triple product.
As was shown by Edwards, McCrimmon and Ruttimann [18], the set of
structural projections on a JBW
∗
-triple forms a complete lattice, iso-
morphic to the complete lattice of weak
∗
-closed inner ideals of A. For
the algebraic structure theory of Jordan
∗
-triples and Jordan
∗
-algebras,
investigated by McCrimmon [42], Neher [45] and others, inner ideals
play a pivotal role, similar to that of ideals in the theory of associative
Page 3
Structural projections
109
algebras. Further studies by Edwards and Ruttimann [21, 22] indicate
that structural projections and weak
∗
-closed inner ideals of JBW
∗
-triples
are essential also with regard to the aforementioned connections with
quantum mechanics. Using the generalized Mackey-Gleason theorem of
Bunce and Wright [6, 7], they showed that quantum decoherence func-
tionals on a W
∗
-algebra A, previously discussed in [55], can be identified
with bounded measures on the complete lattice of weak
∗
-closed inner
ideals of A as a JBW
∗
-triple, thereby showing that the properties of A
relevant for physics depend only upon the triple structure of A.
This article contains a part of the main results of [14] and [16]. We
also attempt to illustrate the connections that persist between these re-
sults and their precursors provided in [18, 19] upon which they heavily
rely, as well as some general aspects concerning JBW
∗
-triples. In Section
4, various characterizations of structural projections and weak
∗
-closed
inner ideals, involving uniqueness of weak
∗
-continuous Hahn-Banach ex-
tensions and neutral projections, are presented. This culminates in
Theorem 4.8 in which it is shown that a contractive projection on a
JBW
∗
-triple A is structural if and only if it is the adjoint of a neutral
GL-projection. The class of GL-projections can be seen as a general-
ization of L-projections. Whilst L-projections and neutral projections
have been investigated to some extend in the past [10, 11, 36, 40, 46],
GL-projections appear to be a new concept, introduced in [14]. Both
neutrality and the GL-property can be interpreted physically. The pre-
dual A
∗
of A is considered to be the normal state space of a statistical
physical system, and a contractive linear projection P on A
∗
represents
a repeatable operation on the state space [1, 2, 30, 44]. In this situa-
tion, P is neutral if and only if a state is unchanged by the operation
whenever the transition probability of the state under the correspond-
ing operation is one, and P is a GL-projection if and only if a state of
the system ‘orthogonal’ to the set of states which remain unchanged has
zero probability of being transmitted by the corresponding operation.
In Section 5, we elaborate a list of properties, characterizing the
set GL(A
∗
) of GL-projections on the predual of a JBW
∗
-triple, which,
thereby are of independent interest. Section 6 provides investigations
into the order structure and further specific features of GL(A
∗
). Exam-
ples of GL-projections showing that GL(A
∗
) encompasses well known
classes of contractive projections can be found in [14, 16]. However,
we confine ourselves here to constructing a set of particular examples
arising in a simple manner from the theory developed.
Page 4
110
Remo V. Hugli
2. JB
∗
- and JBW
∗
-triples
Let A be a complex Banach space with open unit ball D
A
and denote
by Aut(D
A
) the group of all biholomorphic automorphisms of D
A
. If
Aut(D
A
) acts transitively on D
A
, the Banach space A can be shown to
be a JB
∗
-triple. If, furthermore, A is the dual of some Banach space,
denoted A
∗
, then A is said to be a JBW
∗
-triple. By arguments involving
holomorphy, it can be shown that every JB
∗
-triple A has a canonically
determined algebraic structure, given by a triple product {···} from
A × A × A to A. By the results of [37], this allows a characterization of
JB
∗
-triples in an axiomatic setting, using only algebraic properties of the
triple product and its relations to the intrinsic Banach space structure
of A. A complex Banach space A is a JB
∗
-triple if and only if there
exists a mapping {. . .} from A × A × A to A satisfying the following
conditions.
(i) The expression {a, b, c} is linear in the variables a and c and
conjugate linear in the variable b.
(ii) For all elements a, b, c, d and e in A, the following identity, referred
to as the Jordan triple identity is valid.
D(a,b){c, d, e} = {D(a,b)c, d,e}+{c, d, D(a,b)e}−{c, D(b,a)d, e},
where D(a,b) denotes the linear operator on A, defined by
D(a,b)c = {a, b, c}.
(iii) For every element a of A, the operator D(a,a) is hermitian in that,
for every real t, the linear operator exp(itD(a,a)) is an isometry
of A.
(iv) The spectrum σ(D(a,a)) of D(a,a) is non-negative, and the norm
D(a,a) of D(a,a) is equal to a
2
.
The properties (i) and (ii) can be used to develop a purely algebraic
theory of triples, which, in this case are referred to as Jordan
∗
-triples.
For more details on the algebraic features of Jordan
∗
-triples see for ex-
ample [41, 42, 45]. However, since we are interested also in analytic
structure, we will always assume here that A is a JB
∗
-triple or a JBW
∗
-
triple. Notice that, if two Banach spaces with a triple product, ver-
ifying the above axioms, are isomorphic as JB
∗
-triples, then they are
isometrically isomorphic as Banach spaces and vice versa. Furthermore,
a JB
∗
-triple is anisotropic in that zero is the only element a of A for
which {a, a, a} is zero. For these and further results on the connections
of JB
∗
-triples and JBW
∗
-triples to holomorphy of Banach spaces and
Page 5
Structural projections
111
related topics the reader is referred to the extended literature, e.g. to
[3, 13, 17, 20, 32, 33, 34, 41, 52, 53, 54].
The results in [33] imply that every C
∗
-algebra A is an example of
a JB
∗
-triple, the holomorphic automorphisms of which are essentially
Mobius transformations. The triple product is defined, for elements a,
b and c in A, by
{a, b, c} =
1
2
(ab
∗
c + cb
∗
a).
Similarly, if A is a JB
∗
-algebra with Jordan product ◦ : A × A → A and
involution
∗
: A → A, then it is a JB
∗
-triple, equipped with the triple
product
{a, b, c} =
1
2
(a ◦ (b
∗
◦ c) + c ◦ (b
∗
◦ a) − b
∗
◦ (a ◦ c)).
Note that the passage from JB
∗
-triples to JBW
∗
-triples is analogous to
that from C
∗
-algebras to W
∗
-algebras, or from JB
∗
-algebras to JBW
∗
-
algebras. Therefore, if in addition the space A, described in these ex-
amples, is dual to some Banach space, then A is a JBW
∗
-triple. If a,
b and c are elements of of a complex Hilbert space H with hermitian
scalar product < ., . >, then H is a JBW
∗
-triple, with the triple product
defined by
{a, b, c} =
1
2
(< a,b > c+ < c,b > a).
There are JB
∗
-triples which are not isomorphic to subtriples of neither
of the aforementioned examples. Such exceptional triples are provided
by the Cartan factors M
8
3
, the space of 3 × 3 hermitian matrices over
the Cayley numbers O and M
8
1,2
, the space of 1 × 2 matrices over O,
sometimes referred to as the bi-Cayley triple. For more details see for
example [23, 25, 45].
3. Weak
∗
-closed inner ideals and structural projections
A subspace B of a JB
∗
-triple A is said to be a subtriple of A if B is
closed under the triple product, i.e. if
{B, B, B} ⊆ B,
and B is said to be an inner ideal of A if
{B, A, B} ⊆ B.
A projection P is an idempotent linear mapping on a normed vector
space E. When P is continuous and of norm one, it is said to be con-
tractive. Recall that, in this case, the dual (PE)
∗
of the range PE of
Page 6
112
Remo V. Hugli
P is isometrically isomorphic to the range P
∗
E
∗
of the adjoint P
∗
of P.
Therefore, whenever P is a contractive projection, (PE)
∗
and P
∗
E
∗
will
be identified. A projection R on a JB
∗
-triple A is said to be structural
if, for all elements a, b and c of A, the following relation in connection
with the triple product is satisfied.
R{a, Rb, c} = {Ra, b, Rc}.
Obviously, the range RA of a structural projection R is necessarily an
inner ideal of A. As a deep result, elaborated in [18], a converse holds
for weak
∗
-closed inner ideals of JBW
∗
-triples.
Theorem 3.1. Every weak
∗
-closed inner ideal I of a JBW
∗
-triple A
is the range of a unique structural projection R on A.
By this theorem, the set S(A) of structural projections on A can be
directly identified with the set I(A) of weak
∗
-closed inner ideals of A.
Observe, that the intersection of a family of inner ideals is again an
inner ideal, and that I(A) is naturally ordered by set inclusion, with
least element {0} and greatest element A. Therefore, I(A) and S(A)
form complete lattices which are isomorphic.
It was shown in [38] and [50] that on the range RA of a contrac-
tive projection R on a JB
∗
-triple A, a triple product can be defined in
a canonical way, such that RA becomes a JB
∗
-triple in is own right,
and, hence, that the category of JB
∗
-triples is stable under contrac-
tive projections. By standard arguments of functional analysis, this
result translates immediately to the case of JBW
∗
-triples and contrac-
tive projections which are also weak
∗
-continuous. This is described in
the following lemma.
Lemma 3.2. Let P be a contractive projection on the predual A
∗
of
the JBW
∗
-triple A, and let P
∗
be the adjoint of P. Then, with respect
to the triple product {···}
P
∗
A
from P
∗
A×P
∗
A×P
∗
A to P
∗
A, defined,
for elements a, b, and c in P
∗
A, by
{a, b, c}
P
∗
A
= P
∗
{a, b, c},
the range P
∗
A of P
∗
is a JBW
∗
-triple with predual PA
∗
.
It has to be pointed out here that, by the results in [4], the predual
of a JBW
∗
-triple is unique up to isometry, and, furthermore, that the
triple product is separately weak
∗
-continuous, a fact which is of great
importance whenever a property specific to JBW
∗
-triples and their triple
product is to be verified.
Page 7
Structural projections
113
In what follows, some algebraic and analytic relations on a JB
∗
-triple
A are presented, which, when A is a JBW
∗
-triple, appear to be in close
connection with the geometry of its predual A
∗
. A pair a,b of elements
of A is said to be orthogonal, denoted a⊥b if the linear operator D(a,b)
is identically zero on A. It can be shown that this relation is symmetric
and remains valid when passing to a JB
∗
-triple containing A as a norm
closed subtriple. The algebraic annihilator H
⊥
and the kernel Ker(H)
of a non-empty subset H of A are defined, respectively, to be the sets
Ker(H) = {a ∈ A : {H, a, H} = {0}},
H
⊥
= {a ∈ A : a⊥b ∀b ∈ H} =
⋂
b∈H
{b}
⊥
.
An element u in A is said to be a tripotent if {u, u, u} equals u. Every
non-zero tripotent is of norm one. Furthermore, the annihilator {u}
⊥
of
a tripotent u is an inner ideal of A. Hence, the annihilator H
⊥
of any
subset H consisting of tripotents is an inner ideal [37, 41]. If a further
tripotent v of A is such that
u ⊥ (v − u),
then, u is said to be less than or equal to v, denoted u ≤ v. This rela-
tion provides a partial order on the set U(A) of all tripotents in A. A
tripotent u in a JBW
∗
-triple A is σ-finite, if any set of pairwise orthog-
onal tripotents all of which are less than or equal to u is of countable
cardinality. The set of all σ-finite tripotents of A is denoted by U
σ
(A).
By the results of [35] Lemma 3.11, a JBW
∗
-triple A is the norm-closure
of the linear span of the set U(A), and by [27], for each element x in the
predual A
∗
of A, there exists a tripotent e
A
(x) of A that is the smallest
of all elements u of U(A) such that u(x) equals x . The tripotent e
A
(x)
is said to be the support tripotent of x. Furthermore, by [24] a tripotent
u of A is σ-finite if and only if it is equal to e
A
(x) for some element x
in A
∗
.
In the subsequent considerations, a linearized, weak
∗
-closed version
of the concept of support tripotent, will be of importance. For a non-
empty subset G of A
∗
, the support space s(G) of G is defined to be the
smallest weak
∗
-closed subspace of A containing the support tripotent
e
A
(x) of all elements x in G. Some general properties of the support
space s(G) of G can be found in [14]. However, the case when G is the
range PA
∗
of a contractive projection P is of particular interest. This
situation has been investigated in [19], the results of which are based
Page 8
114
Remo V. Hugli
on similar considerations in [29]. For a proof of (i), (ii) and (iii) in the
following lemma, see [19] Lemma 5.1 and [29] Proposition 2.2, and for
(iv) see [14] Lemma 3.2.
Lemma 3.3. Under the conditions of Lemma 3.2, the following results
hold.
(i) The support space s(PA
∗
) of the norm-closed subspace PA
∗
of A
∗
is a weak
∗
-closed subtriple of A.
(ii) The space s(PA
∗
) ⊕
∞
s(PA
∗
)
⊥
is a weak
∗
-closed subtriple of A
in which s(PA
∗
) and s(PA
∗
)
⊥
are weak
∗
-closed ideals.
(iii) The weak
∗
-closed subspace P
∗
A of A is contained in s(PA
∗
) ⊕
∞
s(PA
∗
)
⊥
, and the restriction φ of the M-projection from s(PA
∗
)⊕
∞
s(PA
∗
)
⊥
onto s(PA
∗
) is a weak
∗
-continuous isometric triple iso-
morphism from the JBW
∗
-triple P
∗
A endowed with the triple
product {···}
P
∗
A
onto the sub-JBW
∗
-triple s(PA
∗
) of A.
(iv) The inverse φ
−1
of φ is the restriction of P
∗
to s(PA
∗
), the predual
of which can be identified with PA
∗
, and the pre-adjoints φ
∗
and
(φ
−1
)
∗
are the identity mappings on PA
∗
.
Corollary 3.4. Under the conditions of Lemma 3.3, if either P
∗
A
is contained in s(PA
∗
) or s(PA
∗
) is contained in P
∗
A then P
∗
A and
s(PA
∗
) coincide.
Proof. This is immediate from Lemma 3.3, (iii) and (iv).
4. Geometric structure and GL-projections
Being defined algebraically, structural projections have also interest-
ing geometric and topological properties. Before stating further alge-
braic results, some generalities concerning a normed vector space E and
its dual E
∗
and projections on these spaces are to be described. Let E
1
and E
∗
1
be the closed unit balls in E and E
∗
, respectively. For subsets G
of E and H of E
∗
, let G
◦
and H
◦
denote the topological annihilators of
G in E
∗
and H in E, respectively. When G and H are R-homogeneous
let G
♯
and H
♯
be the subsets of E
∗
and E consisting of elements that
attain their norms on G and H, respectively. To be precise,
G
♯
= {a ∈ E
∗
: a = sup{|a(x)| : x ∈ G ∩ E
1
}},
H
♯
= {x ∈ E : x = sup{|a(x)| : a ∈ H ∩ E
∗
1
}}.
Page 9
Structural projections
115
A contractive projection P on E is said to be neutral if, whenever an
element x of E has the property that
Px = x ,
then x equals Px. A proof of the following lemma, characterizing neu-
trality, is given in [14], some parts of which can also be found in [19] and
[46].
Lemma 4.1. Let E be a complex Banach space and let P be a contrac-
tive projection on E. Then, the following conditions on P are equivalent.
(i) The projection P is neutral.
(ii) Every weak
∗
-continuous linear functional on the range P
∗
E
∗
of
the adjoint P
∗
of P has a unique weak
∗
-continuous Hahn-Banach
extension to E
∗
.
(iii) Every contractive projection S on E having the property that the
range S
∗
E
∗
of its adjoint S
∗
coincides with P
∗
E
∗
is neutral.
(iv) The set (P
∗
E
∗
)
♯
coincides with the range PE of P.
Furthermore, if the contractive projection P is neutral and S is a further
contractive projection such that P
∗
E
∗
and S
∗
E
∗
coincide then P and S
coincide.
The results presented next, which predates Theorem 3.1, were proved
in [19].
Theorem 4.2. Let A be a JBW
∗
-triple with predual A
∗
and let R
be a structural projection on A. Then, the following results hold.
(i) The projection R is contractive and weak
∗
-continuous. In partic-
ular, there exists a contractive projection P on A
∗
, such that R
equals the adjoint P
∗
of P.
(ii) The projection P on A
∗
, as in (i), is neutral.
The neutrality of the pre-adjoint P of R is therefore necessary for R
to be structural. On the other hand, it is easy to construct an example of
a neutral projection P on a the predual of a JBW
∗
-triple, the adjoint P
∗
of which is not structural. This raises the question of whether there are
further non-trivial conditions upon P which, together with neutrality,
would ensure that P
∗
is structural. Such a condition, in terms of the
algebraic structure of A and P
∗
, is given in the following theorem from
[19].
Theorem 4.3. Let P be a neutral projection on the predual A
∗
of a
JBW
∗
-triple A, and let P
∗
be the adjoint of P. If the range P
∗
of P
∗
is
a subtriple of A, then P
∗
is structural.
Page 10
116
Remo V. Hugli
Theorem 4.3 and Theorem 4.2 combine to give the following result.
Corollary 4.4. Let A be a JBW
∗
-triple with predual A
∗
. Let I(A)
denote the family of weak
∗
-closed inner ideals in A, let S(A) denote the
family of structural projections on A, and let N(A
∗
) denote the family
of neutral projections on A
∗
with the property that P
∗
A is a subtriple
of A. Then the following results hold.
(i) The mapping P ↦→ P
∗
is a bijection from N(A
∗
) onto S(A).
(ii) The mapping R ↦→ RA is a bijection from S(A) onto I(A).
In proving the above theorems, and in view of Lemma 4.1, the proofs
of the following two lemmas, which can be found in [18, 19], become
evident.
Lemma 4.5. Let A be a JBW
∗
-triple, with predual A
∗
, and let J be
a subtriple of A. Then, the following conditions are equivalent.
(i) J is a weak
∗
-closed inner ideal in A.
(ii) The set J
♯
of elements of A
∗
attaining their norm on J is a subspace
of A
∗
.
(iii) Every element of the predual J
∗
of J has a unique weak
∗
-continuous
Hahn-Banach extension as a weak
∗
-continuous linear functional on
A.
Lemma 4.6. Let A be a JBW
∗
-triple, with predual A
∗
, let R be a
structural projection on A, and let J be the weak
∗
-closed inner ideal
RA in A. Then, the following results hold.
(i) R is the unique structural projection with range J.
(ii) R is contractive and weak
∗
-continuous.
(iii) The kernel Ker(J) of J coincides with the kernel Ker(R) of R.
(iv) The predual J
∗
of J coincides with J
♯
which consists of the ele-
ments x in A
∗
the support tripotent e
A
(x) of which lies in J.
(v) J coincides with the support space s(J
∗
) of J
∗
.
Hence, the uniqueness of weak
∗
-continuous linear functionals on J
has a precise expression in terms of algebraic conditions upon J. Recall
that, in contrast to structurality, neutrality is a purely metric property
of a projection P on A
∗
. Therefore, it would be desirable to replace the
condition that P
∗
A be a subtriple by a condition using only the norm of
A
∗
. A solution of this problem involves the relation of L-orthogonality
between elements of A
∗
.
A pair x,y of elements in a normed vector space E is said to be
L-orthogonal, denoted x y, if
x + y = x − y = x + y .
Page 11
Structural projections
117
The L-complement F of a non-empty subset G of E is defined by
G = {x ∈ E : x y ∀y ∈ G}.
A contractive projection P on E is said to be a GL-projection if the
L-complement (PE) of its range is a subset of the kernel KerP of P.
Hence, GL-projections provide a generalisation of L-projections which
are defined to be those linear projections P for which (PE) coincides
with KerPE.
By [29] Lemma 2.5, for every tripotent u of A, there exists a family
{x
i
}
i∈I
of pairwise L-orthogonal elements in A
∗
, such that u equals the
weak
∗
-limit
∑
i∈I
e
A
(x
i
). The relations between support tripotents and
L-orthogonality, described in the following lemma, can be found in [24]
and [27]. See also [36].
Lemma 4.7. Let A be a JBW
∗
-triple with predual A
∗
. Then, for ele-
ments x and y in A
∗
, the conditions x y and e
A
(x)⊥e
A
(y) are equivalent.
Furthermore, in this case, e
A
(x)y equals zero.
We now state a main result, established in [14], which resolves the
problem posed above. The proof requires a certain characterization of
GL-projections on A
∗
, provided only in Theorem 5.3, and is therefore
postponed to the end of Section 5.
Theorem 4.8. Let A be a JBW
∗
-triple, with predual A
∗
, and let R
be a linear projection on A. Then R is a structural projection if and
only if there exists a neutral GL-projection P on A
∗
with adjoint equal
to R.
5. Characterizations of GL-projections on A
∗
The set GL(A
∗
) of GL-projections on the predual A
∗
of a JBW
∗
-triple
A not only proves to be useful for characterizing structural projections,
but has some further interesting properties of its own. As a first goal
of this section, Theorem 5.3 provides several characterizations of GL-
projections among the set of contractive projections on A
∗
. Its proof is
based on the following technical lemma.
Lemma 5.1. Let A be a JBW
∗
-triple, with predual A
∗
, and let G
be a non-empty subset of A
∗
, having L-orthogonal complement G
3
and
support space s(G). Then, the following results hold.
(i) The support space s(G
3
) of G
3
coincides with the weak
∗
-closed
inner ideal s(G)
⊥
, and the predual (s(G)
⊥
)
♯
of s(G)
⊥
coincides
with G
3
.
Page 12
118
Remo V. Hugli
(ii) The kernel Ker(s(G)
⊥
) of s(G)
⊥
coincides with the topological
annihilator (G
3
)
◦
of G
3
.
(iii) The L-orthogonal complement G
3
of G is contained in the topo-
logical annihilator s(G)
◦
of s(G).
Proof. (i) Let H denote the subset {e
A
(x) : x ∈ G} of U
σ
(A). Since
taking the annihilator reverses the order of set inclusion, it is clear that
s(G)
⊥
is a subset of H
⊥
. Conversely, let b be an element in H
⊥
, i.e.
{e
A
(x), b, a} = 0 for all x ∈ G and c ∈ A. Linearity and the weak
∗
-
continuity of the triple product implies that, for all c ∈ s(G), the product
{c, b, a} is also zero. Therefore, b lies in s(G)
⊥
. This shows that the
annihilator s(G)
⊥
of the weak
∗
-closed subtriple s(G) is given by
(5.1)
s(G)
⊥
= H
⊥
=
⋂
x∈G
{e
A
(x)}
⊥
,
and, in particular, that s(G)
⊥
is a weak
∗
-closed inner ideal of A. These
results hold more generally for any weak
∗
-closed subtriple B of A and
H equal to U
σ
(B). Here, only the special case when B is equal to s(G)
is needed. By Lemma 4.7, an element x of A
∗
lies in G
3
if and only if,
for all elements y in G, the tripotent e
A
(x) lies in {e
A
(y)}
⊥
, which by
equation (5.1), is true if and only if e
A
(x) lies in the weak
∗
-closed inner
ideal s(G)
⊥
, and by Lemma 4.6 (iv), if and only if x lies in (s(G)
⊥
)
♯
,
completing the proof of (i).
(ii) By Lemma 4.6 (iii), the kernel Ker(s(G)
⊥
) of the weak
∗
-closed
inner ideal s(G)
⊥
coincides with the kernel of the structural projection
onto s(G)
⊥
, which, by Lemma 4.6 (iv), itself coincides with ((s(G)
⊥
)
♯
)
◦
.
The result follows from (i).
(iii) Observe that s(G) is contained in Ker(s(G)
⊥
). Using (i) and
Lemma 4.6 (iv),
G
3
= (s(G)
⊥
)
♯
= Ker(s(G)
⊥
)
◦
⊆ s(G)
◦
,
as required.
Corollary 5.2. Under the conditions of Lemma 5.1, the following
results hold.
(i) The L-orthogonal complement G
3
of a non-empty subset G of A
∗
is a norm-closed subspace of A
∗
.
(ii) Let F, G and H be mutually L-orthogonal subspaces of A
∗
. Then,
the subspace F ⊕ G is L-orthogonal to H.
Page 13
Structural projections
119
Proof. The proof of (i) follows from Lemma 5.1(i). To prove (ii),
observe that both F and G are contained in the closed subspace H
3
,
from which it follows that the subspace F ⊕ G is contained in H
3
.
It is now possible to establish various characterizations of GL-projec-
tions among the contractive projections using the support space of their
ranges.
Theorem 5.3. Let A be a JBW
∗
-triple, with predual A
∗
, let P be
a contractive projection on A
∗
, with adjoint P
∗
, and let s(PA
∗
) be the
support space of the range PA
∗
of P. Then, the following conditions
are equivalent.
(i) P is a GL-projection.
(ii) The range P
∗
A of P
∗
is contained in the kernel Ker(s(PA
∗
)
⊥
) of
the weak
∗
-closed inner ideal s(PA
∗
)
⊥
.
(iii) s(PA
∗
) is contained in P
∗
A.
(iv) s(PA
∗
) coincides with P
∗
A.
(v) s(PA
∗
) contains P
∗
A.
(vi) The topological annihilator s(PA
∗
)
◦
of s(PA
∗
) is contained in the
kernel Ker(P) of P.
(vii) s(PA
∗
)
⊥
is contained in the weak
∗
-closed inner ideal (P
∗
A)
⊥
.
(viii) s(PA
∗
)
⊥
coincides with (P
∗
A)
⊥
.
Proof. (i) ⇔ (ii) This follows from Lemma 5.1.
(iii) ⇔ (iv) ⇔ (v) This follows from Corollary 3.4.
(ii) ⇒ (v) By Lemma 3.3 (iii), for each element a in P
∗
A, there exist
elements b in s(PA
∗
) and c in s(PA
∗
)
⊥
such that a is equal to b + c. It
follows from (ii) that
0 = {c, a, c} = {c, b + c, c} = {c, b, c} + {c, c, c} = {c, c, c},
and, by the anisotropy of A, c is equal to 0. It follows that a lies in
s(PA
∗
), as required.
(v) ⇒ (vi) By taking topological annihilators this is immediate.
(vi) ⇒ (i) By Lemma 5.1(iii),
(PA
∗
)
3
⊆ s(PA
∗
)
◦
⊆ Ker(P),
as required.
(v) ⇒ (vii) By taking algebraic annihilators this is immediate.
(vii) ⇒ (ii) Observing that s(PA
∗
)
⊥⊥
is contained in Ker(s(PA
∗
)
⊥
),
P
∗
A ⊆ (P
∗
A)
⊥⊥
⊆ s(PA
∗
)
⊥⊥
⊆ Ker(s(PA
∗
)
⊥
),
as required.
(iv) ⇒ (viii) ⇒ (vii) These trivially hold.
Page 14
120
Remo V. Hugli
We are finally in the position to establish a proof of Theorem 4.8.
Proof. If P is a neutral GL-projection, then Theorem 5.3 and Lemma
3.3 show that the range P
∗
A of P
∗
is a subtriple of A, and by Corollary
4.4, the adjoint P
∗
of P is structural.
Conversely, let R be a structural projection on A. Then, by Corollary
4.4 there exists a unique neutral projection P on A
∗
, such that P
∗
is
equal to R. By Theorem 5.3 it is enough to show that s(PA
∗
) is a subset
of P
∗
A. That this is indeed the case can be seen either from Lemma
4.6 (v), or from the following, perhaps more illustrative calculations.
Firstly, since P
∗
is structural, P
∗
A is a subtriple, even an inner ideal of
A. Consider an element x in the range PA
∗
of P. By Lemma 3.3 it can
be seen that
P
∗
e
A
(x) = {P
∗
e
A
(x), P
∗
e
A
(x), P
∗
e
A
(x)}
= P
∗
{P
∗
e
A
(x), P
∗
e
A
(x), P
∗
e
A
(x)},
and, furthermore, that the element P
∗
e
A
(x) is equal to e
A
(x) + v for
some v in the annihilator (s(PA
∗
))
⊥
of the support space s(PA
∗
) of
PA
∗
. Using the structurality of P
∗
and the orthogonality of v and
e
A
(x) it can be seen that
P
∗
e
A
(x) = {P
∗
e
A
(x), e
A
(x), P
∗
e
A
(x)}
= {e
A
(x) + v, e
A
(x), e
A
(x) + v}
= {e
A
(x), e
A
(x), e
A
(x)} + 2{v, e
A
(x), e
A
(x)}
+ {v, e
A
(x), v}
= e
A
(x).
Hence, the support tripotent e
A
(x) of every x in PA
∗
is contained in the
range P
∗
A of P
∗
and, consequently, the support space s(PA
∗
) of PA
∗
is a subspace of P
∗
A, and the proof is complete.
6. Order structure of GL-projections
The set of all projections on a vector space E is ordered in a natural
way. For projections P and Q on E the relation P ≤ Q, is said to
hold if PE ⊆ QE. This relation is not a partial order, although it
is transitive and reflexive, but not in general antisymmetric, i.e. the
conditions P ≤ Q and Q ≤ P do not imply that P and Q coincide.
When P and Q are confined to a subset F of all projections on E,
the antisymmetric property of ≤ restricted to F is equivalent to the
Page 15
Structural projections
121
condition that every subspace F of E is the range of at most one element
of F. A first consequence of Theorem 6.1, a main result of [16], is that
this occurs when F is chosen to be equal to GL(A
∗
).
Theorem 6.1. Let A be a JBW
∗
-triple with predual A
∗
, and let
P be a contractive projection on A
∗
. Then, there exists a unique GL-
projection S on A
∗
such that the range SA
∗
of S coincides with the
range PA
∗
of P, and the range S
∗
A of its adjoint S
∗
coincides with the
support space s(PA
∗
) of PA
∗
. Furthermore, S is given by Pφ
∗
, where φ
is the contractive, weak
∗
-continuous projection from s(PA
∗
)⊕s(PA
∗
)
⊥
to s(PA
∗
).
Proof. Let φ be the isometric triple isomorphism from P
∗
A onto
s(PA
∗
) defined in Lemma 3.3. Then, the mapping φP
∗
is a weak
∗
-
continuous linear mapping from A onto s(PA
∗
). Observe that, by
Lemma 3.3 (iv), for all elements a in A,
(φP
∗
)
2
a = φ(P
∗
φ)P
∗
a = (φP
∗
)a,
and φP
∗
is a contractive projection onto P
∗
A. Let S be the contractive
projection on A
∗
such that S
∗
coincides with φP
∗
. Then, again using
Lemma 3.3 (iv),
SA
∗
= (Ker(S
∗
))
◦
= (Ker(φP
∗
))
◦
= (Ker(P
∗
))
◦
= PA
∗
,
as required. It remains to show that S is a GL-projection. However,
from Lemma 3.3 (iii) it can be seen that
S
∗
A = s(PA
∗
) = s(SA
∗
) ⊆ Ker(s(SA
∗
)
⊥
),
and it follows from Lemma 5.1(ii) that
(SA
∗
)
3
= (Ker(s(SA
∗
)
⊥
))
◦
⊆ (S
∗
A)
◦
= Ker(S),
as required. In order to obtain uniqueness, suppose that Q and S are
GL-projections on A
∗
such that QA
∗
and SA
∗
coincide. It follows from
Theorem 5.3 that
Q
∗
A = s(QA
∗
) = s(SA
∗
) = S
∗
A,
and, hence, that
Ker(Q) = (Q
∗
A)
◦
= (S
∗
A)
◦
= Ker(S).
Since Q and S have the same range and kernel, Q and S coincide.
Corollary 6.2. The set GL of all GL-projections on the predual A
∗
of a JBW
∗
-triple A is a partially ordered set, the least element of which
is the zero projection and the greatest element of which is the identity
projection on A
∗
.
Page 16
122
Remo V. Hugli
Proof. The relation ≤ is clearly transitive and reflexive, that ≤ is also
antysimmetric follows immediately from Theorem 6.1.
It is not known wether GL(A
∗
) is a lattice, let alone a complete lattice.
Nevertheless it is possible to provide criteria for the existence of suprema
and infima of subsets of GL(A
∗
) under certain conditions, one of which
is mutual orthogonality of the elements in the subset considered. For
a result concerning the supremum and infimum of pairs of commuting
GL-projections, see [16]. The set of all linear projections on a normed
vector space E inherits naturally any relation which may, or may not,
persist between the ranges PE and QE of linear projections P and Q. In
this sense, the relations P
Q and P 2 Q are said to hold if PE
QE
and PE 2 QE respectively. The relation x 2 y denotes M-orthogonality
between elments x and y of E, which is defined to hold if x ± y
equals max{x , y }. The following theorem connects geometric and
algebraic orthogonality of GL-projections on A
∗
and their adjoints.
Theorem 6.3. Let P and Q be GL-projections on the predual A
∗
of
a JBW
∗
-triple A. Then, the following conditions are equivalent:
(i) P 3 Q,
(ii) P
∗
⊥ Q
∗
,
(iii) P
∗
2 Q
∗
.
Furthermore, if these conditions hold, then PQ = QP = 0.
Proof. (i)⇔ (ii) Suppose that PA
∗
is L-orthogonal to QA
∗
, i.e. every
element x in PA
∗
is L-orthogonal to every element y in Q(A
∗
). Since,
by Lemma 4.7, the conditions x y and e
A
(x)⊥e
A
(y) are equivalent, it
can be seen that
{e
A
(x) : x ∈ PA
∗
} ⊥ {e
A
(y) : y ∈ QA
∗
}.
By the linearity and the weak
∗
-continuity of the triple product, it follows
that
s(PA
∗
) ⊥ s(QA
∗
),
and, therefore, by Theorem 5.3 (iv), that
P
∗
A ⊥ Q
∗
A.
The converse can be proved by reversing the argument.
(ii) ⇔ (iii) By the GL-property of P and Q, the subspaces P
∗
A and
Q
∗
A are weak
∗
-closed subtriples, for which, by [15] Theorem 4.4, the
relations ⊥ and 2 coincide. Hence (ii) and (iii) are also equivalent.
Page 17
Structural projections
123
Since P and Q are GL-projections, it follows from Lemma 3.3 and
Theorem 5.3 that
P
∗
A ⊆ KerQ
∗
, Q
∗
A ⊆ KerP
∗
,
and, hence, that both PQ and QP are equal to zero.
Lemma 6.4 provides a preliminary version of Theorem 6.5 for a finite
family of pairwise L-orthogonal GL-projections. It also shows that, in
this case, a converse of the statemtent of Theorem 6.5 holds.
Lemma 6.4. Let A be a JBW
∗
-triple with predual A
∗
and let P
1
,P
2
,
...,P
n
be pairwise L-orthogonal contractive projections on A
∗
. Then
their sum
∑
n
k=1
P
k
, the range of which is
⊕
n
k=1
P
k
A
∗
, is a GL-projection
if and only if P
1
, P
2
,...,P
n
are GL-projections.
Proof. By the mutual L-orthogonality of the projections P
1
, P
2
,...,
P
n
, it follows from Theorem 6.3 that
(6.1)
(
n
∑
k=1
P
k
)A
∗
=
⊕
k=1
P
k
A
∗
.
Suppose that P
1
,...,P
n
are GL-projections. The proof proceeds by
induction. Consider the case, in which n is equal to 2. From Theorem
6.3 it can be seen that, for GL-projections P and Q, the sum
(P
∗
+ Q
∗
)A = P
∗
A ⊕ Q
∗
A.
is an M-sum, and, hence, that, for all elements a in A,
||(P
∗
+ Q
∗
)a|| = max{P
∗
a ,||Q
∗
a||} ≤ ||a||.
Therefore, (P +Q)
∗
, is a contractive projection and, consequently, P +Q
is also contractive. Since P and Q are GL-projections, by definition
(PA
∗
)
3
⊆ KerPA
∗
, (QA
∗
)
3
⊆ KerQA
∗
.
Therefore,
((P + Q)A
∗
)
3
= (PA
∗
⊕ QA
∗
)
3
= (PA
∗
)
3
∩ (QA
∗
)
3
⊆ KerP ∩ KerQ ⊆ Ker(P + Q),
and, hence, P + Q is a GL-projection. If the statement is true for GL-
projections P
1
,...,P
k
and if P
k+1
is a GL-projection, such that for all
j equal to 1,...,k,
P
j
3 P
k+1
,
Page 18
124
Remo V. Hugli
then
∑
k
j=1
P
j
is a GL-projection. By Corollary 5.2 (i), the L-complement
(P
k+1
A
∗
)
3
of P
k+1
A
∗
is a subspace of A
∗
. It follows that
(
k
∑
j=1
P
j
) 3 P
k+1
.
(6.2)
Hence, putting P equal to
∑
k
j=1
P
j
and Q equal to P
k+1
, it can be seen
from above that
∑
k+1
j=1
P
j
is a GL-projection.
On the other hand, suppose that
∑
n
j=1
P
j
is a GL-projection. This
sum of projections is itself a projection, if and only if, for j equal to
1,...,n with j = k,
P
∗
j
A ⊆ KerP
∗
k
.
(6.3)
Furthermore, by the mutual L-orthogonality of P
1
,...,P
n
, Theorem 6.3
implies that, for distinct j and k from 1,...,n,
s(P
j
A
∗
) ⊆ s(P
k
A
∗
)
⊥
.
(6.4)
Let x be an element in of P
j
A
∗
. Then, (6.4) and (6.3) show that, if k is
different from j,
e
A
(x) ∈ s(P
j
A
∗
) ⊆ s(P
k
A
∗
)
⊥
⊆ KerP
∗
k
.
But by assumption and Theorem 5.3, the support tripotent e
A
(x) of x
lies in (
∑
n
k=1
P
∗
k
)A, and, hence,
e
A
(x) =
n
∑
k=1
P
∗
k
(e
A
(x)) = P
∗
j
(e
A
(x)).
Consequently, the support space s(P
j
A
∗
) of P
j
A
∗
is a subset of P
∗
j
A
and, by Theorem 5.3 (iii), P
j
is a GL-projection. This completes the
proof.
Theorem 6.5. For every family {P
i
}
i∈I
of pairwise L-orthogonal GL-
projections on A
∗
, the supremum
∨
i∈I
P
i
exists as a GL-projection, the
range of which is
⊕
i∈I
P
i
n
. Furthermore,
∨
i∈I
P
i
is explicitly given, for
elements x in A
∗
as the norm limit
(
∨
i∈I
P
i
)x =
∑
i∈I
P
i
x.
Proof. Let I
f
denote the directed set of all finite subsets of I, ordered
by set inclusion. By Lemma 6.4, for every finite subset F in I
f
, there
Page 19
Structural projections
125
exists a GL-projection P
F
, given by
P
F
=
∑
i∈F
P
i
.
In particular, P
F
is contractive. Using the mutual L-orthogonality of
{P
i
}
i∈F
, it can be seen that, for all elements x in A
∗
,
∑
i∈F
P
i
x
=
∑
i∈F
P
i
x = P
F
x ≤ x .
(6.5)
Let sup
F∈I
f
∑
i∈F
P
i
x be denoted by k. Then, for every positive
number ε, there exists a finite subset F
ε
of I, such that
k − ε ≤
∑
i∈F
ε
P
i
x ≤ k.
(6.6)
Consider an element F in I
f
such that F ∩F
ε
is empty. Using again the
L-orthogonality of {P
i
}
i∈I
and relation (6.6),
∑
j∈F
P
j
x
≤
∑
i∈F
ε
P
i
x +
∑
j∈F
P
j
x − k + ε
=
∑
j∈F∪F
ε
P
j
x − k + ε
≤ k − k + ε = ε.
It follows that {P
F
x}
F∈I
f
is a Cauchy net with respect to the norm
topology on A
∗
. Hence, the norm limit lim
F∈I
f
P
F
x of this net exists
for every x in A
∗
. By (6.5), and the uniform boundedness principle, the
mapping x ↦→ Px, defined by
Px = lim
F∈I
f
P
F
x
is a linear contractive projection on A
∗
. In particular, the range PA
∗
of
P is given by
PA
∗
= lin
⋃
i∈I
P
i
A
∗
n
,
(6.7)
which, by Theorem 6.3 coincides with
⊕
i∈I
P
i
A
∗
n
. It remains to verify
that P is a GL-projection. Since, for each i in I, P
i
is a GL-projection,
Page 20
126
Remo V. Hugli
the L-complement (P
i
A
∗
)
3
of P
i
is a subset of KerP
i
. It follows that
(PA
∗
)
3
= ( lin
⋃
i∈I
P
i
A
∗
n
)
3
⊆ (lin
⋃
i∈I
P
i
A
∗
)
3
⊆ (
⋃
i∈I
P
i
A
∗
)
3
=
⋂
i∈I
P
i
A
3
∗
⊆
⋂
i∈I
KerP
i
= {x ∈ A
∗
: P
i
(x) = 0 ∀i ∈ I}
⊆ KerP.
Hence P is a GL-projection. If Q is any upper bound of the family
{P
i
}
i∈I
, then the range QA
∗
of Q must contain lin
⋃
i∈I
P
i
A
∗
n
as a sub-
space. The proof is completed by Corollary 6.2 and equation (6.7).
We close with an application of the above results, which allows us to
construct examples of GL-projections on A
∗
.
Theorem 6.6. Let {x
i
}
i∈I
be a family of mutually L-orthogonal el-
ements in the predual A
∗
of a JBW
∗
-triple A. Then, there exists a
GL-projection P on the norm closed subspace lin{x
i
: i ∈ I}
n
, and P
and its adjoint P
∗
are explicitly given, for elements y in A
∗
and a in A,
by
Py =
∑
i∈I
e
A
(x
i
)(y) x
i
, P
∗
a =
∑
i∈I
x
i
(a) e
A
(x
i
),
where the sums are to be understood as norm and weak
∗
-limits respec-
tively.
Proof. Consider first the case in which I consists of one element.
By the Hahn-Banach theorem and Theorem 6.1, every one dimensional
subspace of A
∗
is the range of a unique GL-projection. More explicitly,
it must be shown that, for an element x of norm one in A
∗
, with support
tripotent e
A
(x) in A, the mappings P
x
and P
∗
x
, which are given by
P
x
y = e
A
(x)(y)x, P
∗
x
a = a(x)e
A
(x),
meet our requirement in this case. Clearly P
x
is a linear mapping with
range Cx. This mapping is a projection since
P
x
(P
x
y) = P
x
(e
A
(x)(y)x) = e
A
(x)(y)P
x
x
= e
A
(x)(y)e
A
(x)(x)x = e
A
(x)(y)x
= P
x
y.
The projection P
x
is contractive since, for all elements y in A
∗
,
P
x
y = e
A
(x)(y)x ≤ e
A
(x) y x = y .
Page 21
Structural projections
127
Furthermore, if the elements x and y in A
∗
are L-orthogonal, then it
follows from Lemma 4.7 (ii) that
P
x
y = e
A
(x)(y)x = 0,
and, hence, that P
x
is a GL-projection. The uniqueness of P
x
is a
consequence of Theorem 6.1 . The dual projection P
∗
x
of P
x
is defined
for all elements a in A and y in A
∗
by
(P
∗
x
a)(y) = a(x)e
A
(x)(y).
Hence, P
∗
x
a is equal to a(x)e
A
(x). When I is a general set, an application
of Theorem 6.5 leads to the desired proof.
Further examples of GL-projections are provided in [14] and [16],
where it is shown, in particular, that certain projections on a JBW
∗
-
triple A, known as Peirce projections, are the adjoints of GL-projections.
For a description of Peirce projections see for example [17, 20, 41, 45].
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Page 24
130
Remo V. Hugli
School of Mathematics
Trinity College
Dublin 2, Ireland
E-mail: hugli@maths.tcd.ie