Math. Nachr. 172 (1995) 95-108

Invariant Inner Ideals in W*-algebras

By C. M. EDWARDS of Oxford, G. T. RUTTIMA" of Berne and S. Yu. VASILOVSKY of Swaziland

(Received October 23, 1992) (Revised Version June 8, 1994)

Abstract. Let H(B, a) be the JBW*-algebra of elements of a continuous W*-algebra B invariant under the *-anti-automorphism a of B of order two. Then the mapping I + I n H ( B , a) is an order isomorphism from the complete lattice of a-invariant weak* closed inner ideals in B onto the complete lattice of weak* closed inner ideals in H(B, a), every one of which is of the form eH(B, a) a(e) for some unique projection e in B with a-invariant central support. A corollary of this result completely characterizes the weak* closed inner ideals in any continuous JBW*-triple.

1. Introduction

This paper is concerned with the structure of JBW*-algebras and JBW*-triples which has interested many authors in recent years. See for example [4], [5], [6], [8], [13], [15], 1161, [17], [19] and [22]. In particular, the investigation of the inner ideal structure of JBW*-triples which was started in [9], [lo], 1111 and [12] is continued here.

In [9] the first two authors investigated the inner ideal structure in a W*-algebra B and showed that there exists an order isomorphism (e, f) H eBf from the complete lattice %(B)(B) of centrally equivalent pairs of projections in B onto the complete lattice 9 ( B ) of weak* closed inner ideals in B. In this paper this investigation is extended to the study of those weak* closed inner ideals invariant under a *-anti-automorphism u of B of order two. The intersection of such an inner ideal with the JBW*-algebra H(B, a) of elements of B invariant under CL is a weak* closed inner ideal in H ( B , u). The main result of the paper shows that, when the W*-algebra B is continuous, every weak* closed inner ideal in H ( B , a) arises in this way and is of the form eH(B, a) u(e) for some projection e in B.

The main motivation for the investigation is the identification of the weak* closed inner ideals in JBW*-triples. Recall that a tripotent u in a JBW*-triple A is said to be abelian if the sub-triple A,&) of A, the Peirce two-space of A corresponding to u, is abelian. The weak* closed linear span of the set of abelian tripotents in A is an ideal A , in A with

') Research supported by the United Kingdom Science and Engineering Research Council, the Schweizerische Nationalfonds/Fonds national suisse and the Soros Foundation.

96 Math. Nachr. 172 (1995)

complementary weak* closed ideal A: which has the property of possessing no non-zero abelian tripotents. When A and A: coincide the JBW*-triple A is said to be continuous. The important classification theorem for continuous JBW*-triples, proved by HORN and NEHER [17], shows that every continuous JBW*-triple A can be written as an M-sum of two continuous JBW*-triples A , and A,, where A , is isomorphic to a weak*-closed right ideal in a continuous W*-algebra and A , is isomorphic to a JBW*-algebra H ( B , a), where B is a continuous W*-algebra and a is a *-anti-automorphism a of B of period two. Since the weak* closed inner ideals in an M-sum of two or more JBW*-triples are precisely the M-sums of the weak* closed inner ideals in the constituents, in order to identify the weak* closed inner ideals in the continuous JBW*-triple A it is enough to identify those in JBW*-triples of the form pC with C a continuous W*-algebra and p a projection in C and those in JBW*-algebras of the form H(B, a) with the W*-algebra B continuous. In fact, in [lo], it was shown by the first two authors that every weak* closed inner ideal in pC is of the form f C g where f and g are projections in C with f majorized by p . Consequently, the main result of the paper is enough to complete the identification of the weak* closed inner ideals in continuous JBW*-triples.

The paper is organized as follows. In 0 2 definitions are given, notation is established and certain preliminary results are described. In 0 3 the properties of the JBW*-algebra H(B, a), many of which are of independent interest, are investigated and in €j 4 the main results are stated and proved. The authors are grateful to the referee for his helpful comments and, in particular, for bringing to their notice reference [2] which he used to supply a proof of Lemma 4.8 which is much shorter than that which appeared originally.

The first two authors are grateful for the support that their research has received from the United Kingdom Science and Engineering Research Council and the Schweizerischer Nationalfonds/Fonds national suisse and the third author is grateful to the Soros Foundation for a grant which supported his visit to the University of Oxford.

2. Preliminaries

A Jordan *-algebra A which is also a complex Banach space such that, for all elements a and b in A,

Ila*II = ll4 3 Ila O bll 5 llall llbll and II{aaa>ll = Ila1I3,

where

(abc} = 0 (b* 0 C) + (a 0 b*) 0 c - b* 0 (a 0 C )

is the Jordan triple product on A, is said to be a Jordan C*-algebra [26] or JB*-algebra [27]. A Jordan C*-algebra which is the dual of a Banach space is said to be a Jordan W*-algebra [7] or a JBW*-algebra [27]. Examples of JB*-algebras are C*-algebras and examples of JBW*-algebras are W*-algebras in both cases equipped with the Jordan product

1 2

a 0 b = - (ab + ba) .

For the properties of C*-algebras and W*-algebras the reader is referred to [24] and for the algebraic properties of Jordan algebras to [18], [21] and [22].

Edwards/Riittimann/Vasilovsky, Invariant Inner Ideals in W*-algebras 97

Recall that the centre Z ( A ) of the JBW*-algebra A is the set of elements a in A such that, for all elements 6 and c in A,

The centre of a JBW*-algebra is an associative JBW*-algebra and, therefore, a commutative W *-algebra.

Recall that a partially ordered set 9 is said to be a lattice if, for each pair (e, f ) of elements of 8, the supremum e v f and the infimum e A f exist with respect to the partial ordering of 8. The partially ordered set B is said to be a complete lattice if, for any subset A of 8, the supremum v A and the infimum A A exist. A complete lattice has a greatest element and a least element, denoted by 1 and 0 respectively. A complete lattice together with an anti-order automorphism e H e' on 9 such that, for all elements e and f in 9, e v e' = 1, e" = e, and, if e I f, then f = e v cf A e'), is said to be orthomodular. An element z in a compIete orthomodular lattice B is said to be central if, for all elements e in 8, ( z A e') v e = (e A z ' ) v z . The set 39 of central elements of the orthomodular complete lattice 9 contains 0 and 1 and if z lies in 39' then so also does z'. A complete orthomodular lattice 9 is said to be Boolean if 58 coincides with 8. Let 8 be a complete orthomodular lattice. With the restricted order and the map z H z', 98 forms a Boolean sub-complete orthomodular lattice of 8 which is said to be the centre of 8. The central support z(e) of an element e of 8 is defined by

z(e) = V (z E a ( 8 ) : e I z} .

For details, see [23]. The set B(A) of self-adjoint idempotents, the projections, in a JBW*-algebra A forms a

complete orthomodular lattice with respect to the partial ordering defined, for elements e and f in 9 ( A ) , by e 2 f if e 0 f = e, and the mapping e H e' defined by e' = 1 - e, where 1 is the unit in A. Moreover, the centre Z(B(A) coincides with the complete orthomodular lattice 9 ( Z ( A ) ) of projections in the centre Z ( A ) of A.

Recall that a complex vector space A equipped with a triple product

(a, b, c) -+ (abc) from A x A x A to A

which is symmetric and linear in the first and third variables, conjugate linear in the second variable and satisfies the identity 3 *

[Na, b), D(c, 41 = D({abc} , d ) - D(c, (dab})

= D(u, {bcd}) - D({cda}, b ) ,

where [, 1 denotes the commutator and D is the mapping from A x A to A defined by

D(a, b) c = {abc} ,

is said to be a Jordan*-triple. When A is also a Banach space such that D is continuous from A x A to the Banach space B(A) of bounded linear operators on A, and, for each element a in A, D(a, a) is hermitian with non-negative spectrum and satisfies

IlD(a, a)ll = llal12 f

7 Math. Nachr., Bd. 172

98 Math. Nachr. 172 (1995)

then A is said to be a JB*-triple. A JB*-triple which is the dual of a Banach space is said to be a JBW*-triple. Examples of JB*-triples are JB*-algebras and examples of JBW*-triples are JBW*-algebras.

A subspace J of a Jordan*-triple A is said to be an inner ideal if { J A J } is contained in J and is said to be an ideal if { A A J ) + { M A } is contained in J .

An element u in a JBW*-triple A is said to be a tripotent if (UUU} is equal to 11. The set of tripotents in A is denoted by %(A). For each tripotent u in the JBW*-triple A the weak* continuous conjugate linear operator Q(u) and the weak* continuous linear operators Pf(u) , j = 0, 1,2, are defined by

Q(u)a = {uau), P f ( u ) a = Q ( u ) ~ ,

P ~ ( u ) = ~ ( D ( u , U) - Q ( u ) ~ ) , P ~ ( u ) = I - ~ D ( u , U) + Q ( u ) ~ .

The linear operators P f ( u ) , j = 0, 1,2, are projections onto the eigenspaces Aj(u) of D(u, u) corresponding to eigenvalues j / 2 and

A = A&) 0 A,(u) 0 A,(u)

is the Peirce decomposition of A relative to u. For i, j , k = 0, 1,2, A,(u) is a sub-JBW*-triple such that

{Ai(u) A,(u) A ~ ( u ) ) A i - j + k ( U )

when i - j + k = 0,1, or 2, and (0) otherwise, and

{ A 2 ( 4 A&) A ) = {AOM A2(4 A3 = ( 0 ) . Moreover, Ao(u) and A,(u) are inner ideals in A and, with respect to the product (a, b) + {aub}, unit u and involution a --f {uau}, A,(u) is a JBW*-algebra. A pair u, v of elements of %(A) is said to be orthogonal if u is contained in A,(u). For two elements u and u of @(A), write u I u if {UVU} = u or, equivalently, if v - u is tripotent orthogonal to u. This defines a partial ordering on %(A) with respect to which @(A) with a greatest clement adjoined forms a complete lattice. An element u in %(A) is maximal in (%(A), I ) if and only if Ao(u) coincides with { O } . For each element u of %!(A) there is a maximal tripotent u such that u 5 u. A tripotent u is said to be abelian if the sub-JBW*-triple A2(u) has the property that, for all a, b,c and d in A2(u), [D(a, b), D(c,d)]A,(u) = 0 and is said to be unitary if A 2 ( u ) coincides with A. A maximal tripotent u is said to be faithful if, for each non-zero ideal I in A, the subtriple I n A,(u) is non-zero. For details the reader is referred to PI, [41, [W, [171 and [201.

When A is a W*-algebra, %(A) coincides with the set of partial isometries in A.

Lemma 2.1. Let A be a W*-algebra and let u and u be partial isometries in A . Then the

(i) u and v are orthogonal elements of %(A); (ii) u*u = uu* = 0.

Proof. If (i) holds then, since v lies in A,(u),

following conditions are equivalent:

(1 - uu*) u(l - u*u) = v

EdwardslRuttimannlVasilovsky, Invariant Inner Ideals in W*-algebras 99

and, therefore,

uu*uu*u = uu*u + uu*u

Multiplying by the idempotent uu*, it follows that

uu*u = 0 .

Therefore

(u*u)* (u*u) = u*uu*u = 0

and u*u = 0. Similarly, uu*u = 0 and hence

(uu*) (uu*)* = uu*uu* = 0

which implies that uu* = 0. Therefore (ii) holds. Conversely, if (ii) holds, then

u = uu*uu*u + (1 - uu*) uu* u + uu*v(l - u*u) + (1 - uu*) u ( l - u*u)

= (1 - uu*)u(l - u*u)

and u lies in A,(u) as required.

Recall that a subspace L of a Banach space X is said to be an M-summand [l] of X if there exists a subspace L' of X such that every non-zero element a in X admits a unique decomposition

a = a , + a 2 ,

where a, lies in L, a2 lies in L' and

l l 4 = max {lblIl> lla*Il)

In this case we write

X = L@,L'

and X is said to be the M-sum of L and L'. The set of M-summands in a dual Banach space forms a complete Boolean orthomodular lattice under the inclusion ordering and the mapping L H L'.

Let I be a weak* closed ideal in the JBW*-triple A and let

I' = n { A , ( ~ ) : % ( I ) ) . Then 1' is also a weak* closed ideal in A. Moreover,

A = I @ M I '

and the weak* closed ideals in A coincide with its M-summands ([3], [4], [22]). Consequently, the M-summands in a W*-algebra or in JBW*-algebra A coincide with its weak* closed ideals. Moreover, the mapping z w z 0 A is an order isomorphism from the complete Boolean lattice %Y(A) of central projections in A onto the complete Boolean lattice of M-summands in A.

7'

100 Math. Nachr. 172 (1995)

3. The JBW*-algebra H(B, a)

In this section the properties of the particular JBW*-algebra of interest are studied.

Lemma 3.1. Let B be a V-a lgebra , let a be a *-anti-automorphism of B of order two and let H(B, a ) be the set of elements of B which are invariant under a. Then H(B, a ) is a J B W *-algebra.

P r o of. Clearly H(B, a) is a weak* closed subspace of B and, for elements a and b in B,

1

2

1 2

a(a 0 b) = - (a@) a(a) + a(a) a@))

_ - - (ba + ab) = a 0 b .

This completes the proof.

Lemma 3.2. Let B be a W *algebra and let o! be a *-anti-automorphism of B of order two. Then the centre Z ( H ( B , a)) of the J B W *-algeba H(B, a) of elements of B invariant under c( coincides with the intersection of the centre Z ( B ) of the W *-algebra B and H(B, a).

Proof. Let e be a central projection in H(B, a). Then

1 2

0 = eo ( 1 - e) = - (e(1 - e) + (1 - e ) e ) = e ( l - e) = (1 - e ) e .

Since e is a central projection in H(B, a), for each element b in H(B, a),

e 0 (b 0 (1 - e)) = b 0 (e 0 ( 1 - e)) = (1 - e) 0 (e 0 b) = 0 .

It follows that

(eb(1 - e)} = 0.

However, for each element a in B, the element ea(1 - e) + ( 1 - e)a(a)e lies in H ( B , a). Therefore, for each element a in B,

0 = e(ea(1 - e) + ( 1 - e)a(a)e) (1 - e)

+ ( 1 - e) (ea(1 - e) + (1 - e)a(a)e)e

= ea(1 - e) + ( 1 - e )a (a )e .

It follows that ea = eae and hence that e is a central projection in B. Since Z ( H ( B , a)) is weak* generated by its projections, it follows that Z(H(B , a)) is contained in Z(B). On the other hand, for a in Z ( B ) and in H(B, a) and, for all b and c in H (B, a),

1 4

a 0 (b 0 c) = - (a(bc + cb) + (bc + cb) a) = b 0 (a 0 c ) ,

from which it follows that a lies in Z(H(B , a)).

Edwards/Riittimann/Vasilovsky, Invariant Inner Ideals in W*-algebras 101

Lemma 3.3. Under the conditions of Lemma 3.2: (i) Let B ( B ) be the complete orthomodular lattice of projections in the W *-algebra-B and

let %B(B) be its centre. Then, ~ l ~ ( ~ ) and alge(B) are order automorphisms. (ii) Let B(H(B, a)) be the complete orthomodular lattice of projections in H(B, a) and let

S B ( H ( B , a ) ) be its centre. For each element z in ZTB(H(B, a)), the mapping 011,~ is a *-anti-automorphism of order two of the weak* closed ideal zB in B. Moreover, the JBW *-algebra H(B, a) is the M-sum of the weak* closed ideal H(zB, a1J and the weak* closed ideal H (( 1 - z) B, al(l - z)B)).

Proof. The proofs are straightforward applications of the definiton of a.

Recall that, according to [9], a pair (e, f) of projections in the W*-algebra B is said to be centrally equivalent if the central supports of e and f coincide. The set W(B(B)) of centrally equivalent pairs of elements of B ( B ) forms a complete lattice and the mapping (e, f) I+ eBf is an order isomorphism from %?(B(B)) onto the complete lattice Y ( B ) of weak* closed inner ideals in B.

Lemma 3.4. Let B be a W *-algebra and let a be a *-anti-automorphism of B of order two. (i) Let (e, f) be apair of centrally equivalant projections in B. Then the weak* closed inner

ideal eBf is a-invariant if and only i f f = a(e). (ii) Let e be a projection in B. Then the pair (e, a(e)) is centrally equivalent if and only if

the central support z(e) of e lies in the complete Boolean lattice %"B(B, a) of a-invariant central projections in B.

(iii) Thepartially orderedset B(B, a) ofprojections e in B such that z(e) lies in %""B(H(B, a)) forms a complete lattice and the mapping e H eBa(e) is an order isomorphism from B ( B , a) onto the complete lattice of a-invariant weak* closed inner ideals in B.

Proof. (i) It is clear that the weak* closed inner ideal eBa(e) is a-invariant. Conversely, if eBf is winvariant then it follows that the weak* closed inner ideals eBf and a(f) Ba(e) coincide. By [9], Theorem 3.10, since ( e , f ) is a centrally equivalent pair, e I aCf) and f I a(e). Applying Lemma 3.3. (i),

a(e) 5 a'Cf) = f I a(e)

and the proof is complete. (ii) Notice that, by Lemma 3.3, for any element e in B(B), the central support z(a(e)) of

a(e) coincides with a(z(e)). Consequently, the pair (e, a(e)) lies in W(B(B)) if and only if z(e) is a-invariant.

(iii) Observe that 0 and 1 lie in P(B, a) and if (ej)jGA lies in B(B, a), then, since

it follows that v ej lies in B ( B , a) which therefore forms a complete lattice. Moreover, the

mapping e H (e, a(e)) is an order isomorphism from Y ( B , a) onto a sub-lattice of V(B(B)) . It follows from (i) and (ii) that the mapping (e, a(e)) H eBa(e) is an order isomorphism from this sub-lattice onto the complete lattice of a-invariant weak* closed inner ideals in B.

j s A

102 Math. Nachr. 172 (1995)

Notice that the intersection eH(B, a) a(e) of the a-invariant weak* closed inner ideal eBa(e) in B with the JBW*-algebra H(B, a ) is a weak* closed inner ideal in H(B, a). One purpose of the paper is to prove the converse 'of this statement when B is a continuous W*-algebra. The following result considerably simplifies the proof.

Lemma 3.5. Let B be a W *-algebra, let u be a *-anti-automorphism of B of order two and let H(B, c1) he the JBW *-algebra of elements of B invariant under a. Then there exists an element z in Ti??(B) such that

2 + @(Z) I 1,

and the centre Z((1 - z - u(z)) B ) of the weak*-closed ideal (1 - z - a(z)) B in coincides with the centre Z(H((1 - z - a(z)) B, al( l -z-m(z))B)) o f the JBW*-algebra H((1 - z - .(z)) B, 4 ( l - z - n ( z ) ) B ) .

Proof. The existence of z follows from [14], Lemma 7.3.4. Furthermore, the element z + a@) is contained in both H(B, a) and in the centre of B. Lemma 3.2 completes the proof.

4. The main theorem

This section is concerned with the proof of the following result.

Theorem 4.1. Let B be a continuous W*-algebra, let a be a *-anti-automorphism of B of order two, and let J be a weak* closed inner ideal in the JBW *-algebra H(B, a) of a-invariant elements of B. Then there exists a projection e in B such that

J = eH(A,a)a(e)

A corollary of this result follows immediately.

Theorem 4.2. Let B be a continuous W*-algebra, let a be a *-anti-automorphism of B of order two and let H(B, a ) be the J B W *-algebra of elements of B invariant under a.

(i) The mapping e H eH(B, a ) a(e) is an order isomorphism from the complete lattice Y ( B , a ) ofprojections in B with a-invariant central supports onto the complete lattice of weak* closed inner ideals in H(B, a).

(ii) The mapping I H I A H(B,a ) is an order isomorphism from the complete lattice of u-invariant weak* closed inner ideals in B onto the complete lattice of weak* closed inner ideals in H(B, a )

Proof. By Lemma 3.4 it only remains to show that the mapping

e +P eH(B, a) a(e)

is onto the complete lattice of weak* closed inner ideals in H ( B , a). Let J be a weak* closed inner ideal in H ( E , a) and let f be a projection in B such that J coincides with f H ( B , a) a(f). Then

fH@, a) a(f ) = f W f ) n H(B, 4 ,

Edwards/Ruttimann/Vasilovsky, Invariant Inner Ideals in W*-algebras 103

and since f B a ( f ) is an a-invariant weak* closed inner ideal in B, there exists an element e in Y(B ,a ) such that eBa(e) coincides with f B a ( f ) . It follows that J coincides with eH(B, a) a(e) as required.

The proof of Theorem 4.1 has several steps which will be made through a series of lemmas some of which have independent interest and hold in greater generality than that required by the theorem.

Lemma 4.3. Let 3 be a W *-algebra, let a be a *-antomorphism of B of order two, let J be a weak* closed inner ideal in the JBW *-algebra H(B, a) of elements of B invariant under a, and let z be an element of %“Y(H(B, a)). Then J n H(zB, ~ 1 , ~ ) is a weak* closed inner ideal in H(zB, a(zB) and J is the M-sum of J n H(zB, a ) and J n H ( ( l - z ) B,

Proof. This follows from [12], Lemma 2.1.

It follows from Lemma 3.5 and Lemma 4.3 that, in order to study a weak* closed inner ideal J in H(B, a), it is sufficient to consider two cases, the first being that when the central projection z in Lemma 3.5 satisfies the condition that z + a(z) = 1, and the second being that in which the centre Z ( B ) of B coincides with the centre Z(H(B , a)) of H ( B , a). The next result shows that the first case is easily resolved.

Lemma 4.4. Under the condition of Lemma 4.3, suppose that there exists a centralprojection z in the W *-algebra B such that

z + a(z) = 1 *

Let J be a weak* closed inner ideal in the JBW*-algebra H(B,a). Then there exists a projection e in B such that

J = eH(B, a) a(e).

Proof. Notice that H(B, a) coincides with the set {za + a(z) a(a): a E A}. Define the set by

J ” = { za : za + a ( z ) a ( a ) ~ J } .

Then J”is easily seen to be a weak* closed inner ideal in the W*-algebra zB. By [9], Theorem 3.16, it follows that there exist projections f and g in zB such that

J = fzBg = f i g .

Using Lemma 3.3 (i), since f ; g I z, it follows that aCf), a(g) I M ( Z ) and hence, since z and a(z) are orthogonal, that

fBa(f) = 44 Bg = fBa(g) = a(g) Bf = (0) . Therefore

J = {fag + a(z) a ( f a g ) : a E B}

= {fag + a(g) = { ( f + d g ) ) (a + &a ( g + 4f)) a E B )

= (f + 4 g ) ) H(B? 4 .(f + 4 g ) )

a - 1 : a E B )

= eH(B, a ) a(e) ,

104 Math. Nachr. 172 (1995)

and the proof is complete.

It is clear that a weak* closed inner ideal J in the JBW*-algebra H(B, a), which is not in general a sub-algebra, is a sub-triple. The next result needs to make use of the properties of JBW*-triples.

Lemma 4.5. Under the conditions of Lemma 4.3 and the assumption that the centre Z(B) of the W *-algebra B is contained in H(B, a), suppose that J is a weak* closed inner ideal in H ( B , a) and that there exist weak* closed ideals J , and J , in J such that

J = J , @y J 2 .

Then there exists an element w of S Y ( H ( B , a)) such that,

J , = J n H(wB, a ( w B ) , J , = J n H ( ( 1 - W ) B, ~ t l ( ~ - ~ ) B ) .

Proof. Let P , and P , be the M-projections from J onto J , and J , respectively, let u be a maximal tripotent in J , and let Plu = u , and P,u = uz. Then, by Peirce arithmetic, u, is a maximal tripotent in J , , uz is a maximal tripotent in J z , and u1 and 24, are orthogonal and satisfy

u1 + u, = u .

For each element a in B,

b = {ul(ulauz + u2a(a) u,) u,} E { J H ( B , a) J } c J .

Therefore

{ U l b U d E { J l J J J = ( 0 ) .

However, since, by Lemma 2.1, ufu, = u3u1 = uzu: = ulur = 0, it follows that

0 = 4(u1bu2} = 2(u,b*u, + u,&*u,) = ul(u~(ulau2 + uza(a) u l ) 4) u2 + u 1 ( u ~ ( u 1 a u 2 + u2a(a) ul) u r ) u2

+ U2(U~(UlUU2 + u2a(u) U J uT) u, + u2(u:(u,au, + u 2 4 4 u1) uT) U ]

= ulau2 + u,a(a)u, . Multiplying on the left by uluT it follows that, for all elements a in B ulau2 = 0 and therefore that uTu,Bu,uf = { O } . By [24], Theorem 1.10.7, the central support projections of the projections uTul and u2uf are orthogonal and there exists an element w of S”B(H(B, a)) such that

u:u, I w , u2u; s 1 - w .

Therefore, by Lemma 3.3 (i),

u,u t = GI(u:ul) I .(W) = w ,

Edwards/Riittimann/Vasilovsky, Invariant Inner Ideals in W*-algebras 105

and, similarly, u”;, I 1 - w. Let a be an element in J , . Then, using the maximality of u1

wa = w(u,uTauru, + (1 - uluf)au:ul + u,ura(l - uluT)) = a .

Therefore J , is contained in J n H(wB, aIWB) and, similarly, J , is contained in J n H((1 - w) B, a/( , -,,,,.). Conversely, if a is contained in both J and H(wB, a[,.), writing a , for Pl (a ) and a , for P,(a), then wa, = a , and (1 - w) a, = a,. Hence,

in WJA,

a , = wa, + wa, = w(al + a,) = wa = a .

It follows that a lies in J , and the proof is complete.

Lemma 4.6. Under the conditions of Lemma 4.3, let J be a weak* closed inner ideal in the JBW *-algebra H(B, a). Suppose that there exist projections el and e , in the W *-algebra B such that

J = J i @ M J z ,

where, f o r j = 1,2, J j denotes the weak* closed inner ideal ejH(B, a) u(ej). Then there exists a projection e in B such that

J = eH(B, a ) a(e) .

Proof. By Lemma 3.5, there exists an element z of ?Z”B(B) such that z + a(z) I 1 and such that the centre of the weak* closed ideal (1 - z - a(z)) B in B coincides with the centre of the JBW* algebra H((1 - z - a(z)) B, C I I ( , - ~ - ~ ( ~ ) ~ ) . Writing zo = 1 - z - a(z) and using Lemma 3.3 (ii), it follows that

= H ( ( z + @‘(z)) B, a l ( z+a(z ) )B) @M H ( z O B , alzoB)

where, by Lemma 4.4,

H ( ( z + a(z)) B, @ l ( z + ~ ( z ) ) B ) = eH( ( z + a(z)) B, a l ( z+a(z ) )B) a(e)

= (e(z + ~ ( 4 ) ) H(B , a) a(e(z + 4)) for some element e of B ( B ) with e I z + a(z). Moreover,

J n H(z0B, alzoB) = z0J = z0J , @M z O J , .

By Lemma 4.5, there exists an element w in d B ( H ( z , B , alzoB)) such that

zoJ , = wzoJ, = (wzoe,) H(B, a) a(wzoel)

z0J2 = (zo - w) zoJ, = ((zo - w) zoe2) H(B, a) a((zO - w) zoez) .

J = (e(z + a(z)) H(B, a ) a(e(z + a(z))) OM (wzoel) H(B , a) a(wzoel)

and

Therefore

@M ((zo - w) zoe,H(B, 4 - 4 zoez) = (e(z + a(z)) + wzOel + (zo - w) zoez) H(B, 4

x a(e(z + a(z)) + wzOel + (zo + w)zoez)

and the proof is complete.

106 Math. Nachr. 172 (1995)

It follows from this result that in order to prove Theorem 4.1 it is only necessary to prove the theorem in the special case in which the centre of H ( B , a ) coincides with the centre of B.

Recall that, by [17], Lemma 2.4, for a weak* closed inner ideal J in H(B, a ) and a maximal element u in %(J), there exist weak* closed ideals J 1 and J , in J and maximal elements u1 in & ( J J and u, in %(J,) such that u1 is unitary, u2 is faithful,

u = u1 + u , ,

J = J , @,,, J 2 . and

It should be noted that this decomposition is not in general unique. The next result shows that when J , coincides with J the proof of Theorem 4.1 is complete.

Lemma 4.7. Let B be a W *-algebra, let a be a *-anti-automorphism of B of order two and let H(B, a ) be the JBW *-algebra of elements of B invariant under a. Suppose that the centre Z ( B ) of B is contained in H(B, IX). Let J be u weuk* closed inner ideal in H(B, ct) and let u he a unitary tripotent in J . Then J coincides with uu*H(B, ci) ci(uu*).

Proof. Since J is a weak* closed inner ideal in H ( B , a), for each element u in %(J),

J,(u) = H(B, IX), (u) = B2(u) n H(B, a ) .

If u is unitary, then

J = J ~ ( u ) = H(B, a)2 (u) = UU*H(B, a) U*U = uu*H(B, a) U ( U U * )

as required.

It follows from Lemma 4.5 and Lemma 4.6 that it now only remains to consider the situation in which in the weak* closed inner ideal J in H(B, a) possesses a faithful maximal tripotent. Notice that it has not yet been necessary to assume that the W*-algebra B is continuous. The authors are grateful to the referee for providing the proof of the following lemma.

Lemma 4.8. Let B be a W *-algebra, let a be a *-anti-automorphism of B of order two and suppose that the centre Z ( B ) o f B is contained in the J B W *-algebra H ( B , GI) of a-inuariant elements of B. Let A be a W *-algebra which, as a J B W *-algebra, is isomorphic to H ( B , a). Then A is Commutative.

Proof. Let z be the maximal central abelian projection in H(B, a). Then, by Lemma 3.2, z lies in Z ( B ) and, moreover, the W*-algebra (1 - z ) B does not contain any non-zero central abelian projections. Since the centre Z((1 - z ) B ) of the W*-algebra (1 - z ) B is contained in the JBW*-algebra H(( l - z ) B, I X ~ ( ~ - = ) ~ ) , it follows from [2], Proposition 2.6, that H((1 - z) B, ~ t l ( ~ - ~ ) ~ ) does not have any direct summands which are isomorphic as an JBW*-algebra to a W*-algebra. This provides a contradiction unless z is equal to the identity in which case B is commutative. It follows that A is commutative and the proof is complete.

Finally, it is necessary to impose the continuity condition on the W*-algebra B and hence on the JBW*-algbra H ( B , a).

Edwards/Riittimann/Vasilovsky, Invariant Inner Ideals in W*-algebras 107

Lemma 4.9. Let B be a continuous W *-algebra, let a be a *-anti-automorphism of B of order two and let H ( B , a) be the J B W *-algebra of elements of B invariant under a. Suppose that the centre Z ( B ) of B is contained in H(l3, a). Then there are no non-trivial weak* closed inner ideals in H ( B , a ) which possess a faithful maximal tripotent.

Proof. Let J be a weak* closed inner ideal in H ( B , a) and let u be a faithful maximal tri- potent in J . By [9], Lemma 3.2, the Peirce two-space B2(u) is a W*-algebra with respect to the multiplication (a, b) H a . b = au*b with unit u and involution a H at = ua*u. Moreover, the mapping ~ I I ~ ~ ( ~ ) is clearly a *-anti-automorphism of B,(u) of order two such that

H ( l 3 2 ( ~ ) , a) = H ( B , cO2 (u) = J2(u)

since J is a weak*-closed inner ideal in H ( B , a). Now suppose that z is a central projection in the W*-algebra B2(u) such that z + a(z) 4 u. Then z is a partial isometry in B and since

zu*z = z ,

zu* = zu*zu* = z(u*zu*) = zz* .

uz*u = z ,

it follows that

Therefore zu* is a projection in B and

0 = z . B,(u) . .(Z) = ZU*UU*BU*UU*~(Z) = Z U * B ~ ( Z U * ) .

By [24], Proposition 1.10.7, there exists a central projection w in B such that

zu* 5 w ,

zz* = zu* = zu*w = a(.(zu*)) w = a((1 - w) a(zu*)) w = zu*(l - w) w = 0 ,

a(zu*) I 1 - w . But then

and it follows that z is zero. By Lemma 3.5, the centre Z(B,(u)) of the W*-algebra B,(u) is contained in the JBW*-algebra H(B,(u), a) which, from above, coincides with J2(u) . By [17], Lemma 2.6, there exists a non-commutative W*-algebra A which, when regarded as a JBW*-algebra, is isomorphic to the JBW*-algebra H(B,(u), a). But this contradicts Lemma 4.8 and the proof is complete.

4.7 and 4.9. Proof of Theorem 4.1. This now follows immediately from Lemmas 3.5, 4.4, 4.5, 4.6,

The main theorem can now be applied to the identification of inner ideals in continuous JBW*-triples. The characterization of continuous JBW*-triples used in the next result is that shown by HORN and NEHER [17].

Theorem 4.10. Let A be a continuous J B W *-triple and let q be an isomorphism f rom the J B W *-triple A onto the J B W *-triple H ( B , a) ow pC, where B and C are continuous W *-algebras, a is a *-anti-automorphism of B of order two andp is a projection in C. Then,

f o r each weak* closed inner ideal J in A, there exist a projection e in B with a-invariant central support and apair (L g ) ofprojections in C with f p such that q ( J ) coincides with eH(B, CO 4 4 OM fCg.

and Theorem 4.1 above. Proof. This is immediate from [12], Lemma 2.1, [17], Theorem 1.20, [lo], Theorem 4.2

108 Math. Nachr. 172 (1995)

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