ALGEBRA OF MULTIPLICATION OPERATORS ON AW

October 3, 2010

15:3

WSPC/INSTRUCTION FILE

MultOpWstarfinal

Asian-European Journal of Mathematics Vol. 1, No. 1 (2010) 1–27 c World Scientific Publishing Company

DOI: 10.1142/DOI Number

THE BANACH-LIE ∗ -ALGEBRA OF MULTIPLICATION OPERATORS ON A W∗ -ALGEBRA

MAYSAA ALQURASHI King Saud University Riyadh, Saudi Arabia [email protected] NAJLA A. ALTWAIJRY King Saud University Riyadh, Saudi Arabia [email protected] C. MARTIN EDWARDS The Queen’s College Oxford, OX1 4AW, United Kingdom [email protected] CHRISTOPHER S. HOSKIN St Antony’s College Oxford, OX2 6JF, United Kingdom [email protected] Received (23 June 2010) Accepted (29 September 2010)

The hermitian part L(A)h of the Banach-Lie ∗ -algebra L(A) of multiplication operators on the W∗ -algebra A is a unital GM-space, the base of the dual cone in the dual GLspace (L(A)h )∗ of which is affine isomorphic and weak∗ -homeomorphic to the state space of L(A). It is shown that there exists a Lie ∗ -isomorphism φ from the quotient (A ⊕∞ Aop )/K of an enveloping W∗ -algebra A ⊕∞ Aop of A by a weak∗ -closed Lie ∗ -ideal K onto L(A), the restriction to the hermitian part ((A ⊕ op ∞ A )/K)h of which is a bi-positive real linear isometry, thereby giving a characterization of the state space of L(A). In the special case in which A is a W∗ -factor this leads to a further identification of the state space of L(A) in terms of the state space of A. For any W∗ -algebra A, the Banach-Lie ∗ -algebra L(A) coincides with the set of generalized derivations of A, and, as an application, a formula is obtained for the norm of an element of L(A)h in terms of a centre-valued ‘norm’ on A, which is similar to that previously obtained by non-order-theoretic methods. Keywords: W∗ -algebra; multiplication operator; generalized derivation. AMS Subject Classification: Primary 46L10. Secondary 46L60, 81P05 1

October 3, 2010

2

15:3


MultOpWstarfinal

M. Alqurashi, N.A. Altwaijry, C.M. Edwards & C.S. Hoskin

1. Introduction For elements a and b in the W∗ -algebra A, the linear mapping D(a, b) on A defined, for each element c of A, by D(a, b)c = 21 (ab∗ c + cb∗ a) is said to be a multiplication operator on A. The closed linear span L(A) of multiplication operators on the W∗ -algebra A forms a Banach-Lie ∗ -algebra. In this paper the properties of L(A) and its state space S(L(A)) are investigated. The family U (A) of partial isometries in A possesses a partial ordering defined, for elements u and v in U (A), by u ≤ v if uv ∗ u is equal to u. When a maximal element ˜ is adjoined to U (A), the partially ordered set U(A) obtained forms a complete lattice that is anti-order isomorphic to the complete lattice of weak∗ -closed faces of the unit ball A1 in A [7]. In some approaches to the theory of statistical physical systems, [8], A1 represents the set of effects of a system, the weak∗ -closed faces of A1 thereby representing the sets of effects of sub-systems. As a consequence, the complete lattice U˜(A) is anti-order isomorphic to the complete lattice of subsystems. One motivation for an investigation of L(A) is that, in a recent paper, ˜ [2], it was shown that the order structure of U(A) is closely related to the linear order structure of the hermitian part L(A)h of the Banach-Lie ∗ -algebra L(A) of multiplication operators on A. A second motivation is that, for a W∗ -algebra A, the Banach-Lie ∗ -algebra L(A) coincides with the set of generalized derivations of A. From the earliest investigations into linear mappings between C∗ -algebras it was clear that the Jordan structure of the C∗ -algebra had an important part to play. For example, it was shown by Kadison [16] that a linear isometry between unital C∗ algebras A and B that maps the unit in A to that in B is a Jordan ∗ -isomorphism. With the study of derivations of C∗ -algebras came the investigation of the Lie structure of C∗ -algebras described, for example, in the works of Kaup [18] and Upmeier [25]. Although some of the results obtained in this paper can be described in terms of the associative algebraic structure of the W∗ -algebra A, most of the methods used owe much to the Jordan and Lie traditions. This having been said it is necessarily the case that the most significant contributions come from the theory of ordered real Banach spaces and that of hermitian elements of unital complex Banach algebras. The self-adjoint part Ah of the W∗ -algebra A possesses a natural linear order stucture with respect to which it forms a unital GM-space. The hermitian part L(A)h of the Banach-Lie ∗ -algebra L(A) of multiplication operators on A also possesses a linear order structure with respect to which it too forms a unital GMspace. It is the interplay between these two order structures that lies at the heart of the investigation in this paper. For a W∗ -algebra A, with opposite algebra Aop , the direct sum A ⊕∞ Aop forms a W∗ -algebra, which, when A contains no direct summands of Type I1 , coincides with the canonical enveloping W∗ -algebra W ∗ (A) of A that has the universal property that Jordan ∗ -homomorphisms from A into a W∗ -algebra B extend uniquely to ∗ -

October 3, 2010

15:3


MultOpWstarfinal

Multiplication Operators on a W∗ -Algebra

3

homomorphisms from W ∗ (A) into B [11]. The first main result of this paper shows that the weak∗ -closed subspace K = {(z, −z) : z ∈ Z(A)}, op

of A ⊕∞ A , where Z(A) is the centre of A, is a Lie ∗ -ideal in A ⊕∞ Aop , and that there exists a Lie ∗ -isomorphism φ from the Banach-Lie ∗ -algebra (A ⊕∞ Aop )/K onto L(A). Parts of the proof of this result have appeared elsewhere in other guises. Using the theory of Glimm ideals of W∗ -algebras and the properties of a Z(A) valued ‘norm’ on A it is then shown that, when restricted to the hermitian part ((A ⊕∞ Aop )/K)h of (A ⊕∞ Aop )/K, the mapping φ is a bi-positive linear mapping that sends the order unit in the unital GM-space ((A ⊕∞ Aop )/K)h to the order unit in L(A)h and is, therefore, for the two GM-norms, an isometry. It should be observed that the GM-norm in ((A⊕∞ Aop )/K)h does not, in general, coincide with the quotient norm. As a consequence of the result stated above, it follows that the state space of L(A) can be identified with the base of the dual cone in the GL-space that is the dual of the unital GM-space ((A ⊕∞ Aop )/K)h . In the special case in which A is a W∗ -factor, it is shown that the state space S(L(A)) is affine isomorphic to the product S(A) × S(A) of two copies of the state space S(A) of A. For a W∗ -algebra A, there exists a norm (a, b) 7→ k(a, b) + Kkρ on the Banach-Lie ∗ -algebra (A ⊕∞ Aop )/K defined, for an element (a, b) in A ⊕∞ Aop , by, k(a, b) + Kkρ =

1 2

inf{kρ(a + z) + ρ(b − z)k : z ∈ Z(A)},

where ρ is the centre-valued ‘norm’ on A, the restriction of which to the unital GM-space ((A ⊕∞ Aop )/K)h coincides with the GM-norm. As an application of the main result, it is possible to show that, for a self-adjoint element (a, b) of A ⊕∞ Aop , in the unital GM-space L(A)h , kφ((a, b) + K)k =

1 2

inf{kρ(a + z) + ρ(b − z)k : z ∈ Z(A)h },

where Z(A)h is the self-adjoint part of Z(A). A similar formula for the norm of a generalized derivation has previously been obtained by other methods [3]. The paper is organised as follows. The second section is devoted to reviewing results about ordered real Banach spaces, hermitian elements of complex unital Banach algebras, state spaces of complex unital Banach algebras, and some less wellknown properties of W∗ -algebras. For the general properties of C∗ -algebras and W∗ -algebras the reader is referred to [20, 21, 23]. In the third section the main results of the paper are proved, and, in the fourth, the application of order-theoretic methods to the description of the norm of a hermitian generalized derivation is given. 2. Preliminaries In this section some preliminary material is assembled. The first task is to review some results about partially ordered real Banach spaces. Recall that a unital GMspace E is a real Banach space, linearly partially ordered by a norm-closed cone

October 3, 2010

4

15:3


MultOpWstarfinal


E + , for which there exists an element 1 in E + such that the closed unit ball E1 in E coincides with the order interval (−1 + E + ) ∩ (1 − E + ). In this case, 1 is said to be the order unit in E [1, 26]. The elementary proof of the result below can be found in [2], Lemma 2.1. Lemma 2.1. Let E be a unital GM-space, with cone E + and order unit 1, and let F be a norm-closed subspace of E containing 1. Then, with respect to the restricted norm and cone F ∩ E + , the real Banach space F is a unital GM-space with order unit 1. Recall that a GL-space G is a real Banach space, linearly partially ordered by a norm-closed cone G+ , such that the norm is additive on G+ and the closed unit ball G1 in G coincides with the convex hull conv((G+ ∩ G1 ) ∪ (−G+ ∩ G1 )) of the set (G+ ∩ G1 ) ∪ (−G+ ∩ G1 ). In this case, the set BG of elements of G+ of norm one forms a base for the cone G+ and G1 = conv(BG ∪ −BG ). For a proof of the result below the reader is referred to [1], Proposition II.1.7. Lemma 2.2. Let E be a unital GM-space, with cone E + and order unit 1, let E ∗ be its Banach dual space, and let E ∗+ = {X ∈ E ∗ : X(T ) ≥ 0 ∀ T ∈ E + }. Then, E ∗+ is a weak ∗ -closed cone in E ∗ with respect to which the real Banach space E ∗ is a GL-space. Furthermore, the base BE ∗ of the cone E ∗+ is a weak ∗ -compact convex set given by BE ∗ = {X ∈ E ∗+ : X(1) = 1} = {X ∈ E ∗+ : kXk = 1}. Let B be a complex unital Banach algebra, with unit 1B , and let B ∗ be its Banach dual space. Then, the weak∗ -compact convex set S(B) consisting of elements X in the unit ball B1∗ in B ∗ for which X(1B ) = 1 is said to be the state space of B. For an element T in B, the numerical range V (B, T ) is defined by V (B, T ) = {X(T ) : X ∈ S(B)}. Observe that, if C is a closed subspace of B containing 1B and T , and the state space S(C) and numerical range V (C, T ) are similarly defined, then V (C, T ) = V (B, T ), and, consequently, the numerical range of T is determined by the linear span of 1B and T . The numerical radius νB (T ) is defined by νB (T ) = sup{|λ| : λ ∈ V (B, T )}.

October 3, 2010

15:3


MultOpWstarfinal


5

An element T of B is said to be hermitian if the numerical range V (B, T ) is contained in R and is said to be positive if V (B, T ) is contained in R+ . Let Bh and B + denote the sets of hermitian and positive elements of B, respectively. Observe that, for each element T in Bh , the numerical range V (B, T ) is the closed convex hull of the spectrum σB (T ) of T , and, denoting the spectral radius of T by rB (T ), it follows that kT k = νB (T ) = rB (T ). For details of this result and those quoted above the reader is referred to [4, 22]. The connection between the sets of hermitian elements in complex unital Banach algebras and partially ordered real Banach spaces is described in the next lemma, for the proof of which the reader is referred to [2], Lemma 2.3. Lemma 2.3. Let B be a complex unital Banach algebra with unit 1B and let Bh and B + be the sets of hermitian and positive elements of B, respectively. Then, Bh is a norm-closed real subspace of B, and B + forms a norm-closed cone in Bh , containing 1B , with respect to which Bh is a unital GM-space. A complex Banach space B that is also a Lie algebra, the multiplication in which is continuous in the norm topology is said to be a Banach-Lie algebra. If B also possesses a norm-continuous involution then B is said to be a Banach-Lie ∗ -algebra. For the general theory of such algebras the reader is referred to [14, 25]. A complex unital Banach algebra B is a Banach-Lie algebra relative to its Lie multiplication (S, T ) 7→ [S, T ] = ST − T S. Recall that, by [4], Lemma 2.4, for elements S and T of Bh , the element i[S, T ] also lies in Bh , and, therefore, the norm-closed subspace Bj of B, defined by Bj = Bh + iBh , is a Banach-Lie subalgebra of B every element T of which has a unique representation T = T1 + iT2 , where T1 and T2 are elements of Bh . For such an element T of Bj , define T † = T1 − iT2 . Then, the mapping T 7→ T † is a norm-continuous involution on Bj , the corresponding norm-continuous projections onto Bh being given by T 7→ 12 (T + T † ), and Bj forms a Banach-Lie ∗ -algebra.

T 7→

1 2i (T

− T † ),

October 3, 2010

6

15:3


MultOpWstarfinal


Let L be a norm-closed subspace of Bj which is closed under the involution T 7→ T † and contains 1B . For each element X in the Banach dual space L∗ of L let X † be the element of L∗ defined, for an element T in L, by X † (T ) = X(T † ). Then, the mapping X 7→ X † is a weak∗ -continuous involution on L∗ , the corresponding weak∗ -continuous projections onto the weak∗ -closed real subspace (L∗ )h of L∗ being given by X 7→ 12 (X + X † ),

X 7→

1 2i (X

− X † ).

Observe that (L∗ )h consists of those elements of L∗ that take real values on Lh and that L∗ = (L∗ )h ⊕ i(L∗ )h . Moreover, writing Lh = L ∩ B h ,

L+ = L ∩ B + ,

it follows from Lemmas 2.1 and 2.3 that Lh is a unital GM-space, the dual (Lh )∗ of which is a GL-space. In the next lemma, the proof of which is given in [2], Lemma 2.4, the relationship that exists between the two real Banach spaces (Lh )∗ and (L∗ )h is clarified. Lemma 2.4. Let B be a complex unital Banach algebra with unit 1B , let Bj be the Banach-Lie subalgebra of B equal to Bh ⊕ iBh , where Bh is the unital GM-space of hermitian elements of B, let T 7→ T † be the norm-continuous involution on Bj defined, for elements T1 and T2 in Bh by (T1 + iT2 )† = T1 − iT2 , let L be a norm-closed subspace of B containing 1B and invariant under the involution, let Lh be the unital GM-space equal to L ∩ Bh , with dual GL-space (Lh )∗ , let X 7→ X † be the adjoint weak ∗ -continuous involution on the Banach dual space L∗ of L, and let (L∗ )h be the weak ∗ -closed real subspace of L∗ consisting of elements X ˆ be the restriction for which X and X † coincide. For each element X of (L∗ )h , let X ∗ ˆ is a real linear weak -continuous isometry of X to Lh . Then, the mapping X 7→ X from the real Banach space (L∗ )h onto the GL-space (Lh )∗ , mapping the state space S(L) of L onto the base B(Lh )∗ of the cone (Lh )∗+ in (Lh )∗ . Observe that, for a W∗ -algebra A, which automatically possesses a unit 1A , the unital GM-space Ah of hermitian elements of A coincides with the self-adjoint part of A. See, for example, [4], Chapter 6. Recall that the W∗ -algebra A possesses Jordan and Lie multiplications given, for elements a and b in A by a ◦ b = 12 (ab + ba),

[a, b] = 12 (ab − ba),

October 3, 2010

15:3


MultOpWstarfinal


7

with respect to which A forms a Jordan W∗ -algebra and a Banach-Lie ∗ -algebra, respectively. For the general theory of such Jordan algebras the reader is referred to [6, 11, 15, 19, 27, 28]. Let A be a W∗ -algebra with unit 1A and with centre the commutative W∗ -algebra Z(A). Recall that a non-zero ∗ -homomorphism w from Z(A) to C is said to be a character of A and that the set ∆(Z(A)) of characters of Z(A) forms the set of extreme points of the weak∗ -compact convex state space S(Z(A)) of Z(A). With the relative weak∗ -topology ∆(Z(A)) forms a hyperstonian space and the Gelfand representation theorem shows that the commutative W∗ -algebra Z(A) is isometrically ∗ -isomorphic and weak∗ -homeomorphic to the commutative W∗ algebra C(∆(Z(A))) of continuous complex-valued functions on ∆(Z(A)). For each element w of ∆(Z(A)), with kernel the maximal proper ideal ker(w), let Jw be the subset Jw = ker(w)A = {za : z ∈ ker(w), a ∈ A} of A. Then, by the Cohen-Hewitt theorem [5, 13], Jw is a norm-closed two-sided ideal in A. Ideals of this form are said to be Glimm ideals in A. The following result can be found in the proof of [9], Lemma 10. Lemma 2.5. Let A be a W ∗ -algebra with centre Z(A), and, for each element w in the character space ∆(Z(A)) of Z(A), let Jw be the corresponding Glimm ideal in A. Then, for each element a in the positive cone A+ in A, there exists an element ρ(a) in the positive cone Z(A)+ in Z(A) such that 0 ≤ a ≤ ρ(a) = inf{z ∈ Z(A) : a ≤ z}, and, for all elements w in ∆(Z(A)), the norm in the quotient C ∗ -algebra A/Jw is given by ka + Jw k = w(ρ(a)). Recall that, for each state x of a unital C∗ -algebra A, the Gelfand-Naimark-Segal construction provides a complex Hilbert space Hx , an element ξx of Hx of norm one, and a ∗ -homomorphism πx from A into the W∗ -algebra B(Hx ) of bounded linear operators on Hx such that πx (A)ξx is dense in Hx and, for all elements a in A, x(a) = hπx (a)ξx , ξx ix , where (ξ, η) 7→ hξ, ηix is the inner product in Hx . The subspace πx (A)ξ is dense in Hx for every non-zero element ξ of Hx if and only if x lies in the set ∂e S(A) of extreme points of the state space S(A), in which case πx is said to be an irreducible representation of A. The proof of the result below can be found in [10], Theorem 4.7. Lemma 2.6. Under the conditions of Lemma 2.5, for each Glimm ideal Jw in A there exists a faithful irreducible representation of the C∗ -algebra A/Jw .

October 3, 2010

8

15:3


MultOpWstarfinal


3. Multiplication Operators Let A be a W∗ -algebra, let B(A) be the complex unital Banach algebra of bounded linear operators on A, and let B(A)j be the Banach-Lie subalgebra of B(A) consisting of the complex linear span of the unital GM-space B(A)h of hermitian elements of B(A). Observe that, in order to express the results of the paper in the most convenient manner, the definitions of the Lie multiplication in A and in B(A) differ by a factor of two. As a first step towards the investigation of the Banach-Lie ∗ -algebra L(A) of multiplication operators on A the following result, the proof of much of which can be found in [25], is required. Lemma 3.1. Let A be a W ∗ -algebra, let B(A) be the complex unital Banach algebra of bounded linear operators on A, let B(A)h be the unital GM-space of hermitian elements of B(A), let B(A)j be the Banach-Lie subalgebra B(A)h ⊕ iB(A)h of B(A), and let T 7→ T † be the norm-continuous involution on B(A)j defined, for elements T1 and T2 of B(A)h , by (T1 + iT2 )† = T1 − iT2 , with respect to which B(A)j forms a Banach-Lie ∗ -algebra. Then, the following results hold. (i) An element T of B(A) lies in B(A)h if and only if, for all elements a and b in A, T (ab∗ a) = (T a)b∗ a − a(T b)∗ a + ab∗ (T a).

(3.1)

(ii) An element T of B(A) lies in iB(A)h if and only if, for all elements a and b in A, T (ab∗ a) = (T a)b∗ a + a(T b)∗ a + ab∗ (T a). (iii) An element T of B(A) lies in B(A)j if and only if there exists an element T ♮ in B(A) such that, for all elements a and b in A, T (ab∗ a) = (T a)b∗ a + a(T ♮ b)∗ a + ab∗ (T a),

(3.2)

and, when this is the case, T ♮ is unique and equal to −T † . Proof. The proofs of (i) and (ii) can be found in [12], Example 5. If T lies in B(A)j then there exist elements T1 and T2 in B(A)h such that T = T1 + iT2 . It follows that both T1 and T2 satisfy (3.1). A straightforward calculation shows that T satisfies (3.2), with T ♮ equal to −T1 + iT2 . Suppose that T lies in B(A), and that there exists an element T ♮ in B(A) such that (3.2) holds. Then, by polarization, for all elements a, b and c in A, T (ab∗ c + cb∗ a) = (T a)b∗ c + cb∗ (T a) + a(T ♮ b)∗ c + c(T ♮ b)∗ a + ab∗ (T c) + (T c)b∗ a.

October 3, 2010

15:3


MultOpWstarfinal


9

It follows from [25], Proposition 22.4, that T ♮ satisfies (3.2) with T ♮♮ equal to T . Let T1 and T2 be the elements of B(A) defined by T1 = 21 (T − T ♮ ),

1 T2 = − 2i (T + T ♮ ).

Then, for k equal to 1 or 2, and, for all elements a and b in A, Tk (ab∗ a) = (Tk a)b∗ a − a(Tk b)∗ a + ab∗ (Tk a) and, by (i), both T1 and T2 lie in B(A)h . It follows that T = T1 − iT2 lies in B(A)j . Moreover, T † = T1 + iT2 = 12 (T − T ♮ ) − 12 (T + T ♮ ) = −T ♮ , as required. For elements a and b of the W∗ -algebra A, let D(a, b) be the element of B(A) defined, for each element c of A, by D(a, b)c = 12 (ab∗ c + cb∗ a). Then D(a, b) is said to be a multiplication operator on A. Since the identity operator idA on A is equal to D(1A , 1A ), it follows from [2], Theorem 3.1 that the normclosure L(A) of the complex linear span of the family of multiplication operators on A forms a Banach-Lie ∗ -subalgebra of the Banach-Lie ∗ -algebra B(A)j , that is said to be the Banach-Lie ∗ -algebra of multiplication operators on A. By Lemmas 2.1 and 2.3, the hermitian part L(A)h of L(A) is a unital GM-subspace of the unital GM-space B(A)h of hermitian elements of B(A). Moreover, by Lemma 2.4, the state space of L(A) can be identified with the base B(L(A)h )∗ of the dual cone (L(A)h )∗+ in the GL-space (L(A)h )∗ dual to L(A)h . As a consequence of Lemma 3.1, rather more can be said. Lemma 3.2. Let A be a W ∗ -algebra, let B(A) be the complex unital Banach algebra of bounded linear operators on A, let B(A)h be the unital GM-space of hermitian elements of B(A), let B(A)j be Banach-Lie subalgebra of B(A) equal to B(A)h ⊕ iB(A)h , and let L(A)h be the hermitian part of the Banach-Lie ∗ -algebra L(A) of multiplication operators on A. Then, the following results hold. (i) B(A)h = L(A)h = linR {D(a, a) : a ∈ A}. (ii) B(A)j = L(A) = linC {D(a, a) : a ∈ A} = linC {D(a, b) : a, b ∈ A}. Proof. A simple calculation shows that, for elements a, b, and c in A, D(a, a)(cb∗ c) = (D(a, a)c)b∗ c − c(D(a, a)b)∗ c + cb∗ (D(a, a)c),

October 3, 2010

10

15:3


MultOpWstarfinal


and it follows from Lemma 3.1(i) that D(a, a) is an element of the unital GM-space B(A)h . Therefore, by [2], Theorem 3.1(ii), L(A)h = linR {D(a, a) : a ∈ A} ⊆ B(A)h .

(3.3)

Let T be an element of B(A)h . Then, by [25], Proposition 22.24 and [24], Theorem 3.10, there exist self-adjoint elements a, a1 , a2 , . . . an , and b1 , b2 , . . . bn in A such that n X (3.4) [Taj , Tbj ], T = Ta − i j=1

where the bounded linear operator Ta on A is defined, for elements c in A, by Ta c = 12 (ac + ca).

(3.5)

Observe that, Ta = 2i (D(1A , ia) − D(ia, 1A )) = 21 (D(1A + a, 1A + a) − D(a, a) − D(1A , 1A )), (3.6) and n X

[Taj , Tbj ] =

j=1

=

1 2

n X

(D(aj , bj ) − D(bj , aj )) j=1 n X (D(aj , aj ) + D(bj , bj ) − 2i j=1

− D(aj + ibj , aj + ibj )).

(3.7)

It follows from (3.4)-(3.7) that T lies in linR {D(a, a) : a ∈ A}. Therefore, using (3.3), linR {D(a, a) : a ∈ A} is norm-closed and (i) holds. It follows that B(A)j = B(A)h ⊕ iB(A)h = L(A)h ⊕ iL(A)h = L(A) = linC {D(a, a) : a ∈ A}}. Since, for elements a and b in A, D(a, b) = 21 (D(a + b, a + b) − D(a, a) − D(b, b)) −

i 2 (D(ia

+ b, ia + b) − D(a, a) − D(b, b)),

it follows that linC {D(a, a) : a ∈ A} and linC {D(a, b) : a, b ∈ A} coincide, thereby completing the proof. Recall that, for two complex Banach spaces E and F , the complex Banach space E ⊕∞ F is the direct sum of E and F endowed with the norm k . k∞ defined, for elements a in E and b in F , by k(a, b)k∞ = max{kak, kbk}. The proof of the following result is a straightforward verification.

October 3, 2010

15:3


MultOpWstarfinal


11

Lemma 3.3. Let A be a W ∗ -algebra, with self-adjoint part Ah and positive cone A+ , and let Aop be the complex Banach space A endowed with the same linear structure and involution as that in A, but with multiplication (a, b) 7→ a ·op b defined by a ·op b = ba. Then, Aop is a W ∗ -algebra and the complex Banach space A ⊕∞ Aop , endowed with the multiplication and involution, defined, for elements a, b, c, and d in A, by ((a, b), (c, d)) 7→ (a, b)(c, d) = (ac, b ·op d) = (ac, db), and (a, b) 7→ (a, b)∗ = (a∗ , b∗ ), is a W ∗ -algebra with self-adjoint part the unital GM-space Ah ⊕∞ Ah and positive cone A+ ⊕∞ A+ . The W∗ -algebra Aop defined above is said to be the W∗ -algebra opposite to A. It is now possible to prove the first main result. This allows the Banach-Lie ∗ -algebra L(A) of multiplication operators on the W∗ -algebra A to be identified. Theorem 3.4. Let A be a W ∗ -algebra, with opposite algebra Aop and centre Z(A), let A ⊕∞ Aop be the W ∗ -algebra defined in Lemma 3.3, let K be the subspace of A ⊕∞ Aop defined by K = {(z, −z) : z ∈ Z(A)}, and, for elements a and b in A, let θ(a, b) be the mapping from A to itself defined, for each element c in A, by θ(a, b)c = 21 (ac + cb). Then, the following results hold. (i) K is a weak ∗ -closed Lie ∗ -ideal in A ⊕∞ Aop , and the mapping θ : (a, b) → θ(a, b) is a positive norm-continuous Lie ∗ -homomorphism from the W ∗ -algebra A ⊕∞ Aop onto the Banach-Lie ∗ -algebra L(A) of multiplication operators on A, with kernel K. (ii) The quotient space (A ⊕∞ Aop )/K, endowed with the quotient norm, multiplication, and involution defined, for elements (a1 , b1 ) + K, (a2 , b2 ) + K, and (a, b) + K by k(a, b) + Kk = inf{k(a, b) + (z, −z)k : (z, −z) ∈ K}, = inf{max{ka + zk, kb − zk} : z ∈ Z(A)}, [(a1 , b1 ) + K, (a2 , b2 ) + K] = [(a1 , b1 ), (a2 , b2 )] + K = ([a1 , a2 ], −[b1 , b2 ]) + K,

October 3, 2010

12

15:3


MultOpWstarfinal


and (a + K, b + K)† = (a∗ , b∗ ) + K, respectively, forms a Banach Lie ∗ -algebra, and there exists a norm-continuous Lie ∗ -isomorphism φ from (A ⊕∞ Aop )/K onto L(A), defined, for each element (a, b) + K in (A ⊕∞ Aop )/K by φ((a, b) + K) = θ(a, b). (iii) The real vector space ((A ⊕∞ Aop )/K)h of self-adjoint elements of (A ⊕∞ Aop )/K, endowed with the quotient cone ((A ⊕∞ Aop )/K)+ = {(a, b) + K : a, b ∈ A+ }, forms a partially ordered real Banach space the restriction of φ to which is a norm-continuous positive linear mapping onto the real Banach space L(A)h mapping the element (1A , 1A ) + K to the order unit idA in L(A)h . Proof. Let (a, b) be an element of A ⊕∞ Aop . Then, the mapping θ(a, b) from A to itself is clearly linear, and, for an element c in A, kθ(a, b)ck ≤ 12 (kak + kbk)kck ≤ k(a, b)k∞ kck.

(3.8)

It follows that θ(a, b) is an element of B(A). Observe that, for elements c and d in A, θ(a, b)(cd∗ c) = 12 (acd∗ c + cd∗ cb) = 21 ((ac + cb)d∗ c − c(a∗ d + db∗ )∗ c + cd∗ (ac + cb)) = (θ(a, b)c)d∗ c + c(−θ(a∗ , b∗ )d)∗ c + cd∗ (θ(a, b)c). From Lemma 3.1(iii), it can be seen that θ(a, b) lies in B(A)j and that θ(a, b)† = θ(a∗ , b∗ ). Therefore, by Lemma 3.2(ii), θ(a, b) lies in the Banach-Lie ∗ -algebra L(A). It follows that the mapping θ is well-defined, is clearly linear from A ⊕∞ Aop to L(A) and, by (3.8), is norm-continuous. Let X be an element of the state space S(B(A)) of the unital complex Banach algebra B(A), and, for each element (a, b) in A ⊕∞ Aop , let xX ((a, b)) = X(θ(a, b)). Then, xX is a linear functional on A ⊕∞ Aop and, by (3.8), for all elements (a, b) of A ⊕∞ Aop , |xX ((a, b))| = |X(θ(a, b))| ≤ kθ(a, b)k ≤ k(a, b)k∞ .

(3.9)

xX ((1A , 1A )) = X(idA ) = 1,

(3.10)

Moreover,

and it follows from (3.9)-(3.10) that xX is a state of the W∗ -algebra A ⊕∞ Aop . Since states of W∗ -algebras are hermitian and positive, for all elements (a, b) in

October 3, 2010

15:3


MultOpWstarfinal


13

(A ⊕∞ Aop )h , X(θ(a, b)) is real, and, in particular, for all elements (a, b) in (A ⊕∞ Aop )+ , X(θ(a, b)) is non-negative. Hence, if (a, b) is hermitian then the numerical range of θ(a, b) is real and θ(a, b) lies in L(A)h , and if (a, b) is positive then θ(a, b) lies in L(A)+ . It follows that θ is a positive linear mapping from the W∗ -algebra A ⊕∞ Aop into the Banach-Lie ∗-algebra L(A). Let (a1 , b1 ) and (a2 , b2 ) be elements of A ⊕∞ Aop . Observe that, in the W∗ -algebra A ⊕∞ Aop , [(a1 , b1 ), (a2 , b2 )] = 21 ((a1 , b1 )(a2 , b2 ) − (a2 , b2 )(a1 , b1 )) = 12 ((a1 a2 , b1 ·op b2 ) − (a2 a1 , b2 ·op b1 )) = 21 (a1 a2 − a2 a1 , b1 ·op b2 − b2 ·op b1 ) = ([a1 , a2 ], [b1 , b2 ]op ),

(3.11)

where [ . , . ]op denotes the Lie multiplication in the W∗ -algebra Aop . Moreover, for all elements c in A, using (3.11), [θ(a1 , b1 ), θ(a2 , b2 )]c = θ(a1 , b1 )θ(a2 , b2 )c − θ(a2 , b2 )θ(a1 , b1 )c = 14 (a1 (a2 c + cb2 ) + (a2 c + cb2 )b1 −a2 (a1 c + cb1 ) − (a1 c + cb1 )b2 ) =

1 2 ([a1 , a2 ]c

+ c[b2 , b1 ]) = 12 ([a1 , a2 ]c + c[b1 , b2 ]op )

= θ([a1 , a2 ], [b1 , b2 ]op )c = θ([(a1 , b1 ), (a2 , b2 )])c, from which it follows that θ is a Lie homomorphism. Consequently, θ is a normcontinuous positive Lie ∗ -homomorphism from A ⊕∞ Aop into L(A). Let T be an element of L(A)h . Then, as in the proof of Lemma 3.2, there exist elements a, a1 , a2 , . . . an , and b1 , b2 , . . . bn in Ah such that

T = Ta − i

n X [Taj , Tbj ],

(3.12)

j=1

where the bounded linear operator Ta on A is defined by (3.5). Let the self-adjoint element b in A be defined by

b = − 2i

n X j=1

[aj , bj ].

October 3, 2010

14

15:3


MultOpWstarfinal


Then, using (3.12), for each element c in A, θ(a + b, a − b)c = 21 ((a + b)c + c(a − b)) n X i ((aj bj − bj aj )c − c(aj bj − bj aj )) = Ta c − 4 j=1

= Ta c −

i 4

n X

(aj (bj c + cbj ) + (bj c + cbj )aj

j=1

−bj (aj c + caj ) − (aj c + caj )bj ) n X [Taj , Tbj ]c = T c. = Ta c − i j=1

It follows that θ(a + b, a − b) is equal to T . Since L(A)h linearly generates L(A), the positive Lie ∗ -homomorphism θ maps onto L(A). It can be seen that, for each element z in Z(A) and c in A, θ(z, −z)c = 12 (za − az) = 0, and it follows that K is contained in the Lie ∗ -ideal that is the kernel of θ. Conversely, if (a, b) is an element of A ⊕∞ Aop for which θ(a, b) is equal to zero then, for each element c in A, 0 = 2θ(a, b)c = ac + cb. Choosing c equal to 1A shows that a is equal to −b, and, hence, that b lies in the centre Z(A) of A, which implies that the kernel of θ is contained in K. Therefore, the kernel of θ and K coincide, and, since the centre of A is weak∗ -closed in A, it follows that K is weak∗ -closed in A ⊕∞ Aop , thereby completing the proof of (i). Using the properties of quotient mappings, the proofs of (ii) and (iii) follow immediately from the corresponding properties of θ, noticing that in the proof of (i) it is shown that θ maps (A ⊕∞ Aop )h onto L(A)h . Observe that, in Theorem 3.4, although L(A)h with the operator norm, positive cone L(A)+ , and order unit idA forms a unital GM-space, there is no reason to think that the same applies to (A ⊕∞ Aop )/K with the quotient norm. Before continuing the investigation, a slight deviation will be made in order to mention a further consequence of Theorem 3.4. Let B be a Lie algebra and let (B+ , B− ) be a pair of subspaces of B with direct sum B such that [B+ , B+ ] ⊆ B+ ,

[B+ , B− ] ⊆ B− ,

[B− , B− ] ⊆ B+ .

Then (B+ , B− ) is said to be a multiplicative grading of B. In this case, B+ forms a Lie subalgebra of A. The following result is easily verified.

October 3, 2010

15:3


MultOpWstarfinal


15

Lemma 3.5. Let A be a W ∗ -algebra, with opposite algebra Aop , let A ⊕∞ Aop be the W ∗ -algebra defined in Lemma 3.3, and let (A ⊕∞ Aop )+ = {(a, −a) : a ∈ A},

(A ⊕∞ Aop )− = {(a, a) : a ∈ A}.

Then, the mapping a 7→ (a, a) is an isometric Jordan ∗ -isomorphism from A onto (A ⊕∞ Aop )− , and ((A ⊕∞ Aop )+ , (A ⊕∞ Aop )− ) forms a multiplicative grading of A ⊕∞ Aop . The preceding result leads to the following corollary of Theorem 3.4. Corollary 3.6. Let A be a W ∗ -algebra, let L(A) be the Banach-Lie ∗ -algebra of multiplication operators on A, and, for each element a in A, let Ta and Sa be the bounded linear operators on A defined, for an element c in A, by Ta c = 12 (ac + ca),

Sa c = 21 (ac − ca).

Then, both Ta and Sa lie in L(A) and if L(A)+ = {Sa : a ∈ A},

L(A)− = {Ta : a ∈ A},

then (L(A)+ , L(A)− ) is a multiplicative grading of L(A). Proof. Using the notation of Theorem 3.4 and Lemma 3.5, observe that, for an element (a, −a) in (A ⊕∞ Aop )+ , θ(a, −a) = Sa , and, for an element (a, a) in (A ⊕∞ Aop )− , θ(a, a) = Ta . Since θ is a Lie ∗ -homomorphism onto L(A), the result follows immediately. It is now possible to return to the main theme of the paper. Recall that, for an element a in the W∗ -algebra A, the absolute value |a| of A is defined to be the 1 square root (a∗ a) 2 of the positive element a∗ a of A. The first result extends Lemma 2.5. Lemma 3.7. Let A be a W ∗ -algebra, with self-adjoint part Ah and centre Z(A), let A+ and Z(A)+ be the weak ∗ -closed cones of positive elements in A and Z(A), respectively, and, for each element w of the character space ∆(Z(A)) of Z(A), let ker(w) be the maximal proper ideal in Z(A) that is the kernel of w, and let Jw be the Glimm ideal ker(w)A of A. Then, the following results hold. (i) For each element a in A, there exists an element ρ(a) in Z(A)+ , such that, in the unital GM-space Ah , |a| ≤ ρ(a) = inf{z ∈ Z(A)+ : |a| ≤ z},

(3.13)

October 3, 2010

16

15:3


MultOpWstarfinal


and, for each element w in ∆(Z(A)), ka + Jw k = w(ρ(a)),

(3.14)

where a + Jw 7→ ka + Jw k is the norm in the quotient C ∗ -algebra A/Jw . (ii) For elements a and b in A and z in Z(A): (a) (b) (c) (d) (e) (f )

ρ(za) = |z|ρ(a); ρ(a + b) ≤ ρ(a) + ρ(b); ρ(ab) ≤ ρ(a)ρ(b); ρ(a∗ a) = ρ(a)2 ; ρ(a∗ ) = ρ(a); kρ(a)k = kak.

(iii) For each element a in Ah , ρ(a) = inf{z ∈ Z(A)+ : −z ≤ a ≤ z}. Proof. From Lemma 2.5, it can be seen that (i) holds when a is positive. Therefore, for an arbitrary element a in A, |a| ≤ ρ(|a|) = ρ(a), and, for an element w of ∆(Z(A)), using the properties of the C∗ -algebra A/Jw , w(ρ(a)) = w(ρ(|a|)) = k|a| + Jw k = k|a + Jw |k = ka + Jw k, thereby completing the proof of (i). Notice that, for each element w in ∆(Z(A)) and each element z in Z(A), since, z = (z − w(z)1A ) + w(z)1A ,

(3.15)

where z − w(z)1A lies in ker(w) which is contained in Jw , z + Jw = w(z)(1A + Jw ).

(3.16)

Therefore, using (3.14), (3.16), and the properties of the norm in the quotient C∗ algebra A/Jw , w(ρ(za)) = kza + Jw k = k(z + Jw )(a + Jw )k = |w(z)|ka + Jw k = w(|z|)w(ρ(a)) = w(|z|ρ(a)), w(ρ(a + b)) = k(a + b) + Jw k ≤ ka + Jw k + kb + Kw k = w(ρ(a) + ρ(b)), w(ρ(ab)) = kab + Jw k ≤ ka + Jw kkb + Jw k = w(ρ(a)ρ(b)), w(ρ(a∗ a)) = ka∗ a + Jw k = ka + Jw k2 = w(ρ(a)2 ), and w(ρ(a∗ )) = ka∗ + Jw k = ka + Jw k = w(ρ(a)).

October 3, 2010

15:3


MultOpWstarfinal


17

Since the commutative W∗ -algebra Z(A) separates points in its character space, (ii)(a)-(ii)(e) follow immediately. Using the Gelfand representation theorem, the fact that characters take non-negative values on positive elements of Z(A), and (3.14), kρ(a)k = sup{|w(ρ(a))| : w ∈ ∆(Z(A))} = sup{w(ρ(a)) : w ∈ ∆(Z(A))} = sup{ka + Jw k : w ∈ ∆(Z(A))k ≤ kak,

(3.17)

since the quotient mapping from A onto A/Jw is contractive. On the other hand, by (3.13), kak = k|a|k ≤ kρ(|a|)k = kρ(a)k.

(3.18)

and, from (3.17)-(3.18), (ii)(f) holds. Finally, suppose that a lies in the unital GM-space Ah . Then, using (3.13) and the fact that a lies in the order interval [−|a|, |a|] in Ah , −ρ(a) ≤ −|a| ≤ a ≤ |a| ≤ ρ(a) and inf{z ∈ Z(A)+ : −z ≤ a ≤ z} ≤ ρ(a).

(3.19)

+

Suppose that z is an element of Z(A) such that, in the unital GM-space Ah , −z ≤ a ≤ z. Then, for each element w of ∆(Z(A)), using the positivity of the quotient mapping from A onto A/Jw , − (z + Jw ) ≤ a + Jw ≤ z + Jw .

(3.20)

Hence, by (3.16) and (3.20), −w(z)(1A + Jw ) ≤ a + Jw ≤ w(z)(1A + Jw ), and, using (3.14) and the fact that the unit ball in the self-adjoint part of the unital C∗ -algebra A/Jw is the order interval [−1A + Jw , 1A + Jw ], w(ρ(a)) = w(ρ(|a|)) = k|a| + Jw k = k|a + Jw |k = ka + Jw k ≤ w(z). It again follows from the Gelfand representation theorem that ρ(a) ≤ z.

(3.21)

The proof is therefore completed by (3.19) and (3.21). Recall that, by Theorem 3.4, every element of the Banach-Lie ∗ -algebra L(A) of multiplication operators on the W∗ -algebra A is of the form θ(a, b) for some elements a and b in A. Moreover, under the conditions of Lemma 3.7, using Lemma 2.5, there exists an irreducible representation of the W∗ -algebra A with kernel Jw . Before giving a proof of the second main result of the paper, the following lemma is needed.

October 3, 2010

18

15:3


MultOpWstarfinal


Lemma 3.8. Let A be a W ∗ -algebra with centre Z(A), for each element w of the character space ∆(Z(A)) of Z(A), let Jw be the Glimm ideal ker(w)A of A, let πw be an irreducible representation of A on the complex Hilbert space Hw with kernel Jw , and, for an element ξ of unit norm in Hw , let xξ be the state of the W ∗ -algebra B(Hw ) of bounded linear operators on Hw defined, for an element d of B(Hw ) by xξ (d) = hdξ, ξiw , where (ξ, η) 7→ hξ, ηiw is the inner product in Hw . For elements ξ and η of Hw of unit norm and, for each element θ(a, b) of the Banach Lie ∗ -algebra L(A) of multiplication operators on A, let Xw,ξ,η (θ(a, b)) = 21 (xη (πw (a)) + xξ (πw (b))). Then, Xw,ξ,η is an element of the state space S(L(A)) of L(A). Proof. Observe that, since, for each element z of Z(A), xη (πw (a + z)) + xξ (πw (b − z)) = xη (πw (a)) + xξ (πw (b)), by Theorem 3.4(ii), the mapping Xw,ξ,η is a well-defined linear functional on L(A). Let xξ,η be the bounded linear functional on the W∗ -algebra B(Hw ) defined, for an element d in B(Hw ), by xξ,η (d) = hdξ, ηiw , and let {xξ,η }′ be the weak∗ -closed face of the unit ball B(Hw )1 in B(Hw ) defined by {xξ,η }′ = {d ∈ B(Hw )1 : xξ,η (d) = 1} = {d ∈ B(Hw )1 : dξ = η, d∗ η = ξ}.

(3.22)

Observe that, by choosing two orthonormal bases for Hw containing ξ and η, respectively, the face {xξ,η }′ contains a unitary element. Then, by the transitivity theorem [17], for each positive real number ǫ, there exists an element c in A, of norm less than 1 + ǫ, such that πw (c) is contained in {xξ,η }′ . Therefore, using (3.22), the Cauchy-Schwarz inequality in Hw , and the fact that the mapping πw is contractive, |Xw,ξ,η (θ(a, b))| = 12 |(hπw (a)πw (c)ξ, ηiw + hπw (b)ξ, πw (c)∗ ηiw )| = 21 |(hπw (ac + cb)ξ, ηiw )| = |hπw (θ(a, b)c)ξ, ηiw | ≤ kπw (θ(a, b)c)kkξkw kηkw ≤ kθ(a, b)kkck < kθ(a, b)k(1 + ǫ), from which it follows that the linear functional Xw,ξ,η is bounded with norm less than or equal to one. However, since Xw,ξ,η (idA ) = 1, it can be seen that Xw,ξ,η is a state of the Banach-Lie ∗ -algebra L(A).

October 3, 2010

15:3


MultOpWstarfinal


19

Recall that an element a in the self-adjoint part Ah of a W∗ -algebra A is positive if and only if, for some positive real number α, such that kak is less than or equal to α, it follows that kα1A − ak ≤ α,

(3.23)

and, if this is the case then (3.23) holds for all such positive real numbers α. The proof of the second main result of the paper is now accessible. Theorem 3.9. Let A be a W ∗ -algebra, with opposite algebra Aop and centre Z(A), let A ⊕∞ Aop be the W ∗ -algebra defined in Lemma 3.3, let K be the weak ∗ -closed Lie ∗ -ideal in A ⊕∞ Aop , defined by K = {(z, −z) : z ∈ Z(A)}, that is the kernel of the positive norm-continuous Lie ∗ -homomorphism θ from the W ∗ -algebra A ⊕∞ Aop onto the Banach-Lie ∗ -algebra L(A) of multiplication operators on A defined, for elements a, b, and c in A, by θ(a, b)c = 21 (ac + cb), and let φ be the corresponding positive norm-continuous Lie ∗ -isomorphism from the quotient Banach-Lie ∗ -algebra (A ⊕∞ Aop )/K, equipped with the quotient ordering, onto the Banach-Lie ∗ -algebra L(A) of multiplication operators on A. Then, the following results hold. (i) The inverse mapping φ−1 is positive. (ii) There exists a norm (a, b) + K 7→ k(a, b) + KkGM on the self-adjoint part ((A ⊕∞ Aop )/K)h of (A ⊕∞ Aop )/K, defined by k(a, b) + KkGM = inf{λ ∈ R+ : (a, b) + K ∈ λB}, where B is the quotient ball, given by B = ((−(1A , 1A ) + (A ⊕∞ Aop )+ ) ∩ ((1A , 1A ) − (A ⊕∞ Aop )+ )) + K, in the real Banach space ((A ⊕∞ Aop )/K)h , with respect to which ((A ⊕∞ Aop )/K)h forms a unital GM-space with order unit (1A , 1A ) + K, such that the restriction of φ to ((A⊕∞ Aop )/K)h is a bi-positive real linear isometry onto the unital GM-space L(A)h of hermitian elements of L(A). Proof. (i) It follows from Theorem 3.4 that φ maps the self-adjoint part ((A ⊕∞ Aop )/K)h of (A ⊕∞ Aop )/K onto L(A)h . Let a and b be elements of Ah such that θ(a, b) is an element of L(A)+ . Let ρ be the mapping from A to Z(A)+ defined in Lemma 3.7, and let z0 be the element of Z(A)h defined by z0 = kbk1A − ρ(kbk1A − b).

(3.24)

By Lemma 2.6, there exists a character w of Z(A) with corresponding Glimm ideal Jw of A and an irreducible representation πw of A on a Hilbert space Hw , having

October 3, 2010

20

15:3


MultOpWstarfinal


kernel Jw . For an element ξ of norm one in the Hilbert space Hw , let xξ be the state of the W∗ -algebra B(Hw ) of bounded linear operators on Hw defined, for an element d in B(Hw ) by xξ (d) = hdξ, ξiw , where (ξ, η) 7→ hξ, ηiw is the inner product in Hw . Then, using (3.14), (3.15), and (3.24), for any element ξ in Hw of unit norm, xξ (πw (z0 )) = xξ (kbk1B(Hw ) − πw (ρ(kbk1A − b))) = kbk − xξ (w(ρ(kbk1A − b))1B(Hw ) ) = kbk − w(ρ(kbk1A − b)) = kbk − k(kbk1A − b) + Jw k = kbk − kπw (kbk1A − b)k,

(3.25)

using Lemma 2.6 and the fact that faithful representations of a C∗ -algebra are isometric. From the definition of the norm in B(Hw ) and (3.25), given a positive real number ǫ, there exists an element ξǫ in Hw of norm one such that kπw (kbk1A − b)k ≤ |hπw (kbk1A − b)ξǫ , ξǫ iw | + ǫ = kbk − xξǫ (πw (b)) + ǫ,

(3.26)

since kbk1A −b is a positive element of A. Therefore, by (3.24)-(3.26), for any element ξ in Hw of unit norm, xξ (πw (a + z0 )) ≥ xξ (πw (a)) + kbk − kπw (kbk1A − b)k ≥ xξ (πw (a)) + xξǫ (πw (b)) − ǫ = 2Xw,ξǫ ,ξ (θ(a, b)) − ǫ.

(3.27)

where Xw,ξǫ ,ξ is the state of L(A) defined in Lemma 3.8. Since θ(a, b) is positive, which implies that its numerical range lies in R+ , and since ǫ is arbitrary, it can be seen from (3.27) that, for all elements ξ in Hw of unit norm, xξ (πw (a + z0 )) ≥ 0.

(3.28)

Similarly, by (3.25), for all elements ξ in Hw of unit norm, xξ (πw (b − z0 )) = xξ (πw (b)) − xξ (πw (z0 )) = hπw (b)ξ, ξiw − kbk + kπw (kbk1A − b)k ≥ hπw (b)ξ, ξiw − kbk + hπw (kbk1A − b)ξ, ξiw = 0.

(3.29)

It follows from (3.28)-(3.29) that, for all elements w of ∆(Z(A)), the elements πw (a+z0 ) and πw (b−z0 ) of B(Hw ) are positive. Therefore, using the norm criterion for positivity of elements of W∗ -algebras (3.23), it follows that, for all real numbers α ≥ kπw (a + z0 )k, kα1B(Hw ) − πw (a + z0 )k ≤ α.

October 3, 2010

15:3


MultOpWstarfinal


21

Therefore, since πw is isometric on A/Jw , for real numbers α ≥ k(a + z0 ) + Jw k, kα1A − (a + z0 ) + Jw k ≤ α. Hence, by Lemma 3.7(i),(ii)(f), for all real numbers α ≥ sup{w(ρ(a + z0 )) : w ∈ ∆(Z(A))} = kρ(a + z0 )k = ka + z0 k, and, for all elements w of ∆(Z(A)), w(ρ(α1A − (a + z0 ))) ≤ αw(1A ). which, by the Gelfand representation theorem, implies that ρ(α1A − (a + z0 )) ≤ α1A , and, hence, that kα1A − (a + z0 )k = kρ(α1A − (a + z0 ))k ≤ α. Again using the norm criterion for positivity (3.23), it follows that a+z0 is a positive element of A. Similarly, b − z0 is positive in A. Therefore, (a, b) + K = (a + z0 , b − z0 ) + K = φ−1 (θ(a, b)) is a positive element of ((A ⊕∞ Aop )/K)h , as required. (ii) Using Theorem 3.4, it can now be seen that the restriction of the Lie ∗ isomorphism φ to the partially ordered real vector space ((A ⊕∞ Aop )/K)h is bipositive and maps the element (1A , 1A ) + K in ((A ⊕∞ Aop )/K)+ to the order unit idA in the unital GM-space L(A)h . It follows from [1], Propositions II.1.2-3, that, when endowed with the norm (a, b)+K 7→ k(a, b)+KkGM , the partially ordered real vector space ((A⊕∞ Aop )/K)h forms a unital GM-space which φ maps isometrically onto the unital GM-space L(A)h . It is now possible to identify the state space of the Banach-Lie ∗ -algebra of multiplication operators on the W∗ -algebra of bounded linear operators on a complex Hilbert space. In fact, the theorem proved below applies to any W∗ -factor A. Recall that, for convex sets S1 and S2 , a mapping Φ from S1 to S2 is said to be affine if, for all elements x and y in S1 and α in the real interval [0, 1], Φ(αx + (1 − α)y) = αΦ(x) + (1 − α)Φ(y). Theorem 3.10. Let A be a W ∗ -factor, let L(A) be the Banach-Lie ∗ -algebra of multiplication operators on A, let S(A) and S(L(A)) be the state spaces of A and L(A), respectively, and, for elements a and b in A, let θ(a, b) be the element of L(A) defined, for an element c of A, by, θ(a, b)c = 12 (ac + cb).

October 3, 2010

22

15:3


MultOpWstarfinal


Then, there exists an affine isomorphism Φ from the convex set S(A) × S(A) onto S(L(A)) defined, for elements x and y in S(A) and a and b in A, by Φ(x, y)(θ(a, b)) = 12 (x(a) + y(b)). Proof. Since A is a W∗ -factor, by Theorem 3.4, the kernel K of the Lie homomorphism θ from the W∗ -algebra A ⊕ Aop onto L(A) is given by

∗

-

K = {(α1A , −α1A ) : α ∈ C}. Notice that, for an element (x, y) in S(A) × S(A), Φ(x, y)(θ(a + α1A , b − α1A )) = 12 (x(a + α1A ) + y(b − α1A )) = 21 (x(a) + y(b)) = Φ(x, y)(θ(a, b)) Therefore the linear functional Φ(x, y) on L(A) is well-defined. As in the proof of Theorem 3.9, if θ(a, b) is positive, there can be found α in R such that a + α1A and b − α1A are positive in A, in which case, since x and y are positive linear functionals on A, Φ(x, y)(θ(a, b)) = Φ(x, y)(θ(a + α1A , b − α1A )) = 21 (x(a + α1A ) + y(b − α1A )) ≥ 0.

(3.30)

Moreover, Φ(x, y)(idA ) = Φ(x, y)(θ(1A , 1A )) = 21 (x(1A ) + y(1A )) = 1,

(3.31)

It follows from (3.30)-(3.31) that the restriction of Φ(x, y) to the unital GM-space L(A)h is an element of the base B(L(A)h )∗ of the cone (L(A)h )∗+ in (L(A)h )∗ which, by Lemma 2.4, can be identified with the state space S(L(A)) of L(A). It is clear that the mapping Φ from S(A) × S(A) to S(L(A)) is affine and injective. For a state X of L(A), define the linear functionals xX and yX on A, for each element a in A, by xX (a) = 2X(θ(a, 0)),

yX (a) = 2X(θ(0, a)).

Then, using (3.9), the linear functional xX is bounded with norm less than or equal to one, and xX (1A ) = 2X(θ(1A , 0)) = X(idA ) = 1. It follows that xX and, similarly, yX are elements of S(A). Furthermore, for elements a and b in A, Φ(xX , yX )(θ(a, b)) = 21 (xX (a) + yX (b)) = X(θ(a, 0)) + X(θ(0, b)) = X(θ(a, b)), which implies that Φ(xX , yX ) = X. Consequently, the mapping Φ is surjective, and the proof is complete.

October 3, 2010

15:3


MultOpWstarfinal


23

4. An application In this section the mapping ρ from a W∗ -algebra A to the positive cone Z(A)+ in its centre Z(A), introduced in Lemma 3.7, is exploited in order to define a norm on the Banach-Lie ∗ -algebra (A ⊕∞ Aop )/K, introduced in Theorem 3.4 which, when restricted to its self-adjoint part, coincides with the GM-norm defined in Theorem 3.9. This has previously been computed by rather different methods. See [3], Theorem 4.1.20. Lemma 4.1. Let A be a W ∗ -algebra, with opposite algebra Aop and centre Z(A), let A ⊕∞ Aop be the W ∗ -algebra defined in Lemma 3.3, let K be the weak∗ -closed Lie ∗ -ideal in A ⊕∞ Aop defined by K = {(z, −z) : z ∈ Z(A)}, let (A ⊕∞ Aop )/K be the corresponding quotient Banach-Lie ∗ -algebra, and let ρ be the mapping from A to the positive cone Z(A)+ in the self-adjoint part Z(A)h of Z(A), defined, for an element a in A, by ρ(a) = inf{z ∈ Z(A)+ : |a| ≤ z}. Then, the mapping (a, b) + K 7→ k(a, b) + Kkρ defined by k(a, b) + Kkρ =

1 2

inf{kρ(a + z) + ρ(b − z)k : z ∈ Z(A)}

is a norm on (A ⊕∞ Aop )/K such that, for an element (a, b) + K in the self-adjoint part ((A ⊕∞ Aop )/K)h of (A ⊕∞ Aop )/K, k(a, b) + Kkρ =

1 2

inf{kρ(a + z) + ρ(b − z)k : z ∈ Z(A)h }.

Proof. A straightforward calculation shows that the mapping (a, b) + K 7→ k(a, b) + Kkρ is a semi-norm on the Lie ∗ -algebra (A ⊕∞ Aop )/K. Let a and b be elements of A such that inf{kρ(a + z) + ρ(b − z)k : z ∈ Z(A)} = 0. Then, there exist elements z1 , z2 , . . . in Z(A) such that, for n equal to 1, 2, . . ., kρ(a + zn ) + ρ(b − zn )k