Grothendieck’s inequalities for real and complex JBW*-triples

Antonio M. Peralta^{∗} and Angel Rodríguez Palacios^{†}

Abstract

We prove that, if M > 4(1 + 2\sqrt{3}) and ε > 0, if V and W are complex JBW*-triples (with preduals V_{∗} and W_{∗}, respectively), and if U is a separately weak*-continuous bilinear form on V ×W, then there exist norm-one functionals φ_{1}, φ_{2} ∈ V_{∗} and ψ_{1}, ψ_{2} ∈ W_{∗} satisfying

|U(x, y)| ≤ M ∥U∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} (∥y∥_{ψ_{2}}^{2} + ε^{2} ∥y∥_{ψ_{1}}^{2})^{\frac{1}{2}}

for all (x, y) ∈ V ×W. Here, for a norm-one functional φ on a complex JB*-triple V, ∥.∥_{φ} stands for the prehilertian seminorm on V associated to φ in [BF1]. We arrive in this “Grothendieck’s inequality” through results of C-H. Chu, B. Iochum, and G. Loupias [CIL], and a corrected version of the “Little Grothendieck’s inequality” for complex JB*-triples due to T. Barton and Y. Friedman [BF1]. We also obtain extensions of these results to the setting of real JB*-triples.

2000 Mathematics Subject Classification: 17C65, 46K70, 46L05, 46L10, and 46L70.

Introduction

In this paper we pay tribute to the important works of T. Barton and Y. Friedman [BF1] and C-H. Chu, B. Iochum, and G. Loupias [CIL] on the

∗Supported by Programa Nacional F.P.I. Ministry of Education and Science grant, D.G.I.C.Y.T. project no. PB 98-1371, and Junta de Andalucía grant FQM 0199

†Partially supported by Junta de Andalucía grant FQM 0199

\underline{2}

generalization of “Grothendieck’s inequalities” to complex JB*-triples. Of course, the Barton-Friedman-Chu-Iochum-Loupias techniques are strongly related to those of A. Grothendieck [Gro], G. Pisier (see [P1], [P2], and [P3]), and U. Haagerup [H], leading to the classical “Grothendieck’s inequalities” for C*-algebras. One of the most important facts contained in the Barton-Friedman paper is the construction of “natural” prehilbertian seminorms ∥.∥_{φ}, associated to norm-one continuous linear functionals φ on complex JB*-triples, in order to play, in Grothendieck’s inequalities, the same role as that of the prehilbertian seminorms derived from states in the case of C*- algebras. This is very relevant because JB*-triples need not have a natural order structure.

A part of Section 1 of the present paper is devoted to review the main results in [BF1], and the gaps in their proofs (some of which are also subsumed in [CIL]). We note that those gaps consist in assuming that separately weak*-continuous bilinear forms on dual Banach spaces, as well as weak*- continuous linear operators between dual Banach spaces, attain their norms. Section 1 also contains quick partial solutions of the gaps just mentioned. These solutions are obtained by applying theorems of J. Lindenstrauss [L] and V. Zizler [Z] on the abundance of weak*-continuous linear operators attaining their norms (see Theorems 1.4 and 1.6, respectively).

We begin Section 2 by proving a deeper correct version of the Barton-Friedman “Little Grothendieck’s Theorem” for complex JB*-triples [BF1, Theorem 1.3] (see Theorem 2.1). Roughly speaking, our result assures that the assertion in [BF1, Theorem 1.3] is true whenever we replace the prehilbertian seminorm ∥.∥_{ϕ} arising in that assertion with ∥.∥_{φ_{1},φ_{2}} := \sqrt{∥.∥_{φ_{1}}^{2} + ∥.∥_{φ_{2}}^{2}}, where φ_{1}, φ_{2} are suitable norm-one continuous linear functionals. It is worth mentioning that in fact our Theorem 2.1 deals with complex JBW*-triples and weak*-continuous operators, and that, in such a case, the functionals φ_{1}, φ_{2} above can be chosen weak*-continuous. Among the consequences of Theorem 2.1 we emphasize appropriate “Little Grothendieck’s inequalities” for JBW-algebras and von Neumann algebras (see Corollary 2.5 and Remark 2.7, respectively). Corollary 2.5 allows us to adapt an argument in [P] in order to extend Theorem 2.1 to the real setting (Theorem 2.9).

Section 3 contains the main results of the paper, namely the “Big Grothendieck’s inequalities” for complex and real JBW*-triples (Theorems 3.1 and 3.4, respectively). Indeed, given M > 4(1 + 2\sqrt{3}) (respectively, M > 4(1 + 2\sqrt{3}) (1 + 3\sqrt{2})^{2}), ε > 0, V, W complex (respectively, real) JBW*- triples, and a separately weak*-continuous bilinear form U on V × W, there exist norm-one functionals φ_{1}, φ_{2} ∈ V_{∗} and ψ_{1}, ψ_{2} ∈ W_{∗} satisfying

|U(x, y)| ≤ M ∥U∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} (∥y∥_{ψ_{2}}^{2} + ε^{2} ∥y∥_{ψ_{1}}^{2})^{\frac{1}{2}}

for all (x, y) ∈ V × W.

The concluding section of the paper (Section 4) deals with some applications of the results previously obtained. We give a complete solution to a gap in the proof of the results of [R1] on the strong* topology of complex JBW*-triples, and extend those results to the real setting. We also extend to the real setting the fact proved in [R2] that the strong* topology of a complex JBW*-triple W and the Mackey topology m(W, W_{∗}) coincide on bounded subsets of W. From this last result we derive a Jarchow-type characterization of weakly compact operators from (real or complex) JB*-triples to arbitrary Banach spaces.

1 Discussing previous results

We recall that a complex JB*-triple is a complex Banach space E with a continuous triple product {., ., .} : E × E × E → E which is bilinear and symmetric in the outer variables and conjugate linear in the middle variable, and satisfies:

1. (Jordan Identity) L(a, b){x, y, z} = {L(a, b)x, y, z} − {x, L(b, a)y, z} + {x, y, L(a, b)z} for all a, b, c, x, y, z in E, where L(a, b)x := {a, b, x};

2. The map L(a, a) from E to E is an hermitian operator with nonnegative spectrum for all a in E;

3. ∥{a, a, a}∥ = ∥a∥^{3} for all a in E.

Complex JB*-triples have been introduced by W. Kaup in order to provide an algebraic setting for the study of bounded symmetric domains in complex Banach spaces (see [K1], [K2] and [U]).

If E is a complex JB*-triple and e ∈ E is a tripotent ({e, e, e} = e) it is well known that there exists a decomposition of E into the eigenspaces of L(e, e), the Peirce decomposition,

E = E_{0}(e) ⊕ E_{1}(e) ⊕ E_{2}(e),

where E_{k} := \{x ∈ E : L(e, e)x = \frac{k}{2}x\}. The natural projection P_{k}(e) : E → E_{k}(e) is called the Peirce k-projection. A tripotent e ∈ E is called complete if E_{0}(e) = 0. By [KU, Proposition 3.5] we know that the complete tripotents in E are exactly the extreme points of its closed unit ball.

By a complex JBW*-triple we mean a complex JB*-triple which is a dual Banach space. We recall that the triple product of every complex JBW*- triple is separately weak*-continuous [BT], and that the bidual E^{∗∗} of a complex JB*-triple E is a JBW*-triple whose triple product extends the one of E [Di].

Given a complex JBW*-triple W and a norm-one element φ in the predual W_{∗} of W, we can construct a prehilbert seminorn ∥.∥_{φ} as follows (see [BF1, Proposition 1.2]). By the Hahn-Banach theorem there exists z ∈ W such that φ(z) = ∥z∥ = 1. Then (x, y) ↦ φ {x, y, z} becomes a positive sesquilinear form on W which does not depend on the point of support z for φ. The prehilbert seminorm ∥.∥_{φ} is then defined by ∥x∥_{φ}^{2} := φ {x, x, z} for all x ∈ W. If E is a complex JB*-triple and φ is a norm-one element in E^{∗}, then ∥.∥_{φ} acts on E^{∗∗}, hence in particular it acts on E.

In [BF1, Theorem 1.4], J. T. Barton and Y. Friedman claim that for every pair of complex JB*-triples E, F, and every bounded bilinear form V on E × F, there exist norm-one functionals φ ∈ E^{∗} and ψ ∈ F^{∗} such that the inequality

|V (x, y)| ≤ (3 + 2 \sqrt{3}) ∥V ∥ ∥x∥_{φ} ∥y∥_{ψ}  (1.1)

holds for every (x, y) ∈ E × F. This result is called “Grothendieck’s inequality for JB*-triples”. However, the beginning of the Barton-Friedman proof assumes that the two following assertions are true.

1. For E, F and V as above, there exists a separately weak*-continuous extension of V to E^{∗∗} × F^{∗∗}.

2. Again for E, F and V as above, every separately weak*-continuous extension of V to E^{∗∗} × F^{∗∗} attains its norm (at a couple of complete tripotents).

We have been able to verify Assertion 1, but only by applying the fact, later proved by C-H. Chu, B. Iochum and G. Loupias [CIL, Lemma 5], that every bounded linear operator from a complex JB*-triple to the dual of another complex JB*-triple factors through a complex Hilbert space. Actually, this fact is also claimed in the Barton-Friedman paper (see [BF1, Corollary 3.2]), but their proof relies on their alleged [BF1, Theorem 1.4].

Lemma 1.1 Let E and F be complex JB*-triples. Then every bounded bilinear form V on E×F has a separately weak*-continuous extension to E^{∗∗}×F^{∗∗}.

Proof. Let V be a bounded bilinear form on E × F. Let F denote the unique bounded linear operator from E to F^{∗} which satisfies

V (x, y) =< F (x), y >

for every (x, y) ∈ E × F. By [CIL, Lemma 5], F factors through a Hilbert space, and hence is weakly compact. By [HP, Lemma 2.13.1], we have F^{∗∗}(E^{∗∗}) ⊂ F^{∗}. Then the bilinear form Ṽ on E^{∗∗} × F^{∗∗} given by

Ṽ(α, β) =< F^{∗∗}(α), β >

extends V and is weak*-continuous in the second variable. But Ṽ is also weak*-continuous in the first variable because, for (α, β) ∈ E^{∗∗} × F^{∗∗}, the equality

< F^{∗∗}(α), β >=< α, F^{∗}(β) >

holds. □

Unfortunately, as the next example shows, Assertion 2 above is not true.

Example 1.2 Take E and F equal to the complex ℓ_{2} space, and consider the bounded bilinear form on E × F defined by V (x, y) := (S(x)|σ(y)) where S is the bounded linear operator on ℓ_{2} whose associated matrix is

(\begin{matrix}\frac{1}{2} & 0 & … & 0 & … \\ 0 & \frac{2}{3} & … & 0 & … \\ … & … & … & … & … \\ 0 & 0 & … & \frac{n}{n + 1} & … \\ … & … & … & … & … \end{matrix}) ,

and σ is the conjugation on ℓ_{2} fixing the elements of the canonical basis. Then V does not attain its norm.

It is worth mentioning that, although the bilinear form V above does not attain its norm, it satisfies inequality 1.1 for every x, y ∈ ℓ_{2} and every normone elements φ, ψ ∈ ℓ_{2}^{∗}. Therefore it does not become a counterexample to the Barton-Friedman claim. In fact we do not know if Theorem 1.4 of [BF1] is true.

Now that we know that Assertion 2 is not true, we prove that it is “almost” true.

Lemma 1.3 Let E, F be complex JB*-triples. Then the set of bounded bilinear forms on E×F whose separately weak*-continuous extensions to E^{∗∗}×F^{∗∗} attain their norms is norm-dense in the space L(^{2}(E × F)) of all bounded bilinear forms on E × F.

Proof. Let V be in L(^{2}(E × F)). Denote by Ṽ the (unique) separately weak*-continuous extension of V to E^{∗∗}×F^{∗∗}. By the proof of Lemma 1.1, we can assure the existence of a bounded linear operator F_{V} : E → F^{∗} satisfying F_{V}^{∗∗}(E^{∗∗}) ⊂ F^{∗} and

Ṽ(α, β) =< F_{V}^{∗∗}(α), β >

for every (α, β) ∈ E^{∗∗} × F^{∗∗}. It follows that Ṽ attains its norm whenever F_{V}^{∗∗} does. Since the mapping V ↦ F_{V}, from L(^{2}(E × F)) into the Banach space of all bounded linear operators from E to F^{∗}, is a surjective isometry, the result follows from [L, Theorem 1]. □

An alternative proof of the above Lemma can be given taking as a key tool [A, Theorem 1].

Now note that, if X and Y are dual Banach spaces, and if U is a separately weak*-continuous bilinear form on X × Y which attains its norm, then U actually attains its norm at a couple of extreme points of the closed unit balls of X and Y (hence at a couple of complete tripotents in the case that X and Y are complex JB*-triples). Since the Barton-Friedman proof of their claim actually shows that the inequality (1.1) holds (for suitable norm-one functionals φ ∈ E^{∗} and ψ ∈ F^{∗}) whenever the separately weak*-continuous extension of V given by Lemma 1.1 attains its norm at a couple of complete tripotents, the next theorem follows from Lemma 1.3.

Theorem 1.4 Let E, F be complex JB*-triples. Then the set of all bounded bilinear forms V on E × F such that there exist norm-one functionals φ ∈ E^{∗} and ψ ∈ F^{∗} satisfying

|V (x, y)| ≤ (3 + 2 \sqrt{3}) ∥V ∥ ∥x∥_{φ} ∥y∥_{ψ}

for every (x, y) ∈ E × F, is norm dense in L(^{2}(E × F)).

Another alleged proof of the Barton-Friedman claim [BF1, Theorem 1.4] (with constant 3+2\sqrt{3} replaced with 4(1+2\sqrt{3})) appears in the Chu-Iochum-Loupias paper already quoted (see [CIL, Theorem 6]). Such a proof relies on the Barton-Friedman version of the so called “Little Grothendieck’s Theorem” for complex JB*-triples [BF1, Theorem 1.3]. However, the Barton-Friedman argument for this “Little Grothendieck’s Theorem” also has a gap (see [P]).

Several authors (the second author of the present paper among others) subsumed the gap in the proof of Theorem 1.3 of [BF1] just commented, and formulated daring claims like the following (see [R1, Proposition 1] and the proof of Lemma 4 of [CM]). For every complex JBW*-triple W, every complex Hilbert space H, and every weak*-continuous linear operator T : W → H, there exists a norm-one functional φ ∈ W_{∗} such that the inequality

∥T (x)∥ ≤ \sqrt{2} ∥T ∥ ∥x∥_{φ}  (1.2)

holds for all x ∈ W. As in the case of the Barton-Friedman big Grothendieck’s inequality, we do not know if the above claim is true. In any case, the next lemma is implicitly shown in the proof of Theorem 1.3 of [BF1].

Lemma 1.5 Let W be a complex JBW*-triple, H a complex Hilbert space, and T a weak*-continuous linear operator from W to H which attains its norm. Then T satisfies inequality (1.2) for a suitable norm-one functional φ ∈ W_{∗}.

We note that, for W and H as in the above lemma, weak*-continuous linear operators from W to H need not attain their norms (see the introduction of [P]). Now, from Lemma 1.5 and [Z] we obtain the following result.

Theorem 1.6 Let W be a complex JBW*-triple and H a complex Hilbert space. Then the set of weak*-continuous linear operators T from W to H such that there exists a norm-one functional φ ∈ W_{∗} satisfying

∥T (x)∥ ≤ \sqrt{2} ∥T ∥ ∥x∥_{φ}

for all x ∈ W, is norm dense in the space of all weak*-continuous linear operators from W to H.

2 Little Grothendieck‘s Theorem for JBW*- triples

In this section we prove appropriate versions of “Little Grothendieck’s inequality” for real and complex JBW*-triples. We begin by considering the complex case, where the key tools are the Barton-Friedman result collected in Lemma 1.5, and a fine principle on approximation of operators by operators attaining their norms, due to R. A. Poliquin and V. E. Zizler [PZ].

Theorem 2.1 Let K > \sqrt{2} and ε > 0. Then, for every complex JBW*- triple W, every complex Hilbert space H, and every weak*-continuous linear operator T : W → H, there exist norm-one functionals φ_{1}, φ_{2} ∈ W_{∗} such that the inequality

∥T (x)∥ ≤ K ∥T ∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}}

holds for all x ∈ W.

Proof. Without loss of generality we can suppose ∥T ∥ = 1. Take δ > 0 such that δ ≤ ε^{2} and \sqrt{2((1 + δ)^{2} + δ)} ≤ K. By [PZ, Corollary 2] there is a rank one weak*-continuous linear operator T_{1} : W → H such that ∥T_{1}∥ ≤ δ and T − T_{1} attains its norm. Since T_{1} is of rank one and weak*-continuous, it also attains its norm. By Lemma 1.5, there exist norm-one functionals φ_{1}, φ_{2} ∈ W_{∗} such that

∥T_{1}(x)∥ ≤ \sqrt{2} ∥T_{1}∥ ∥x∥_{φ_{1}},
∥(T − T_{1})(x)∥ ≤ \sqrt{2} ∥T − T_{1}∥ ∥x∥_{φ_{2}}

for all x ∈ W. Therefore for x ∈ W we have

∥T (x)∥ ≤ ∥(T − T_{1})(x)∥ + ∥T_{1}(x)∥
≤ \sqrt{2} ∥T − T_{1}∥ ∥x∥_{φ_{2}} + \sqrt{2} ∥T_{1}∥ ∥x∥_{φ_{1}}
≤ \sqrt{2} (1 + δ) ∥x∥_{φ_{2}} + \sqrt{2δ} \sqrt{δ} ∥x∥_{φ_{1}}
≤ \sqrt{2((1 + δ)^{2} + δ)} (∥x∥_{φ_{2}}^{2} + δ ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}}
≤ K (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} .

□

Given a complex JBW*-triple W and norm-one elements φ_{1}, φ_{2} ∈ W_{∗} we denote by ∥.∥_{φ_{1},φ_{2}} the prehilbert seminorm on W given by ∥x∥_{φ_{1},φ_{2}}^{2} := ∥x∥_{φ_{1}}^{2} + ∥x∥_{φ_{2}}^{2}. The next result follows straightforwardly from Theorem 2.1.

Corollary 2.2 Let W be a complex JBW*-triple and T a weak*-continuous linear operator from W to a complex Hilbert space. Then there exist normone functionals φ_{1}, φ_{2} ∈ W_{∗} such that, for every x ∈ W, we have

∥T (x)∥ ≤ 2∥T ∥∥x∥_{φ_{1},φ_{2}}.

We recall that a JB*-algebra is a complete normed Jordan complex algebra (say A) endowed with a conjugate-linear algebra involution * satisfying ∥U_{x}(x^{∗})∥ = ∥x∥^{3} for every x ∈ A. Here, for every Jordan algebra A, and every x ∈ A, U_{x} denotes the operator on A defined by U_{x}(y) := 2x∘(x∘y)−x^{2}∘y, for all y ∈ A. We note that every JB*-algebra can be regarded as a complex JB*-triple under the triple product given by

{x, y, z} := (x ∘ y^{∗}) ∘ z + (z ∘ y^{∗}) ∘ x − (x ∘ z) ∘ y^{∗}

(see [BKU] and [Y]). By a JBW*-algebra we mean a JB*-algebra which is a dual Banach space. Every JBW*-algebra A has a unit 1 [Y], so that the binary product of A can be rediscovered from the triple product by means of the equality x ∘ y = {x, 1, y}.

Theorem 2.3 Let M > 2. Then, for every JBW*-algebra A, every complex Hilbert space H, and every weak*-continuous linear operator T : A → H, there exists a norm-one positive functional ξ ∈ A_{∗} such that the inequality

∥T (x)∥ ≤ M ∥T ∥ (ξ(x ∘ x^{∗}))^{\frac{1}{2}}

holds for all x ∈ A.

Proof. Taking K := \sqrt{M} and ε := \sqrt{\frac{M−2}{2}} in Theorem 2.1, we find normone functionals φ_{1}, φ_{2} ∈ A_{∗} such that

∥T (x)∥ ≤ K ∥T ∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}}

for all x ∈ A. Let i = 1, 2. We choose e_{i} ∈ A with φ_{i}(e_{i}) = ∥e_{i}∥ = 1, and denote by ξ_{i} the mapping x ↦ φ_{i}(x ∘ e_{i}) from A to ℂ. Clearly ξ_{i} is a norm-one weak*-continuous linear functional on A. Moreover, from the identity

{x, x, e_{i}} + {x^{∗}, x^{∗}, e_{i}} = 2e_{i} ∘ (x ∘ x^{∗})

we obtain that ξ_{i} is positive and that the equality ∥x∥_{φ_{i}}^{2} +∥x^{∗}∥_{φ_{i}}^{2} = 2ξ_{i}(x∘x^{∗}) holds. Therefore we have ∥x∥_{φ_{i}}^{2} ≤ 2ξ_{i}(x ∘ x^{∗}) and hence

∥T (x)∥ ≤ \sqrt{2}K ∥T ∥ (ξ_{2}(x ∘ x^{∗}) + ε^{2} ξ_{1}(x ∘ x^{∗}))^{\frac{1}{2}} .

Finally, putting ξ := \frac{1}{1+ε^{2}} (ξ_{2} + ε^{2} ξ_{1}), ξ becomes a norm-one positive functional in A_{∗} and for x ∈ A we have

∥T (x)∥ ≤ \sqrt{2 (1 + ε^{2})}K ∥T ∥ (ξ(x ∘ x^{∗}))^{\frac{1}{2}} = M ∥T ∥ (ξ(x ∘ x^{∗}))^{\frac{1}{2}} .

□

We recall that the bidual of every JB*-algebra A is a JBW*-algebra containing A as a JB*-subalgebra.

Corollary 2.4 Let A be a JB*-algebra and T a bounded linear operator from A to a complex Hilbert space. Then there exists a norm-one positive functional ξ ∈ A^{∗} satisfying

∥T (x)∥ ≤ 2∥T ∥ (ξ(x ∘ x^{∗}))^{\frac{1}{2}}

for all x ∈ A.

Proof. By Theorem 2.3, for n ∈ ℕ there is a norm-one positive functional ξ_{n} ∈ A^{∗} satisfying

∥T (x)∥ ≤ (2 + \frac{1}{n})∥T ∥ (ξ_{n}(x ∘ x^{∗}))^{\frac{1}{2}}

for all x ∈ A. Take in A^{∗} a weak* cluster point η of the sequence ξ_{n}. Then η is a positive functional with ∥η∥ ≤ 1, and the inequality

∥T (x)∥ ≤ 2∥T ∥ (η(x ∘ x^{∗}))^{\frac{1}{2}}

holds for all x ∈ A. If η = 0, then T = 0 and nothing has to be proved. Otherwise take ξ := \frac{1}{∥η∥}η. □

For background about JB- and JBW-algebras the reader is referred to [HS]. We recall that JB-algebras (respectively, JBW-algebras) are nothing but the self-adjoint parts of JB*-algebras (respectively, JBW*-algebras) [W] (respectively, [E]).

Corollary 2.5 Let K > 2\sqrt{2}. Then, for every JBW-algebra A, every real Hilbert space H, and every weak*-continuous linear operator T : A → H, there exists a norm-one positive functional ξ ∈ A_{∗} such that

∥T (x)∥ ≤ K ∥T ∥ (ξ(x^{2}))^{\frac{1}{2}}

for all x ∈ A.

Proof. Let Â denote the JBW*-algebra whose self-adjoint part is equal to A, and Ĥ be the Hilbert space complexification of H. Consider the complexlinear operator T̂ : Â → Ĥ, which extends T. Clearly we have ∥T̂∥ ≤ \sqrt{2}∥T ∥. By Theorem 2.3 there exists a norm-one positive functional ξ ∈ Â_{∗} such that

∥T (x)∥ = ∥T̂(x)∥ ≤ \frac{K}{\sqrt{2}} ∥T̂∥ (ξ(x^{2}))^{\frac{1}{2}} ≤ K ∥T ∥ (ξ(x^{2}))^{\frac{1}{2}}

for all x ∈ A. Since ξ is positive, ξ|_{A} is in fact a norm-one positive functional in A_{∗}. □

The next result follows from the above corollary in the same way that Corollary 2.4 was derived from Theorem 2.3.

Corollary 2.6 [P, Theorem 3.2]

Let A be a JB-algebra, H a real Hilbert space, and T : A → H a bounded linear operator. Then there is a norm-one positive linear functional φ ∈ A^{∗} such that

∥T (x)∥ ≤ 2\sqrt{2}∥T ∥(φ(x^{2}))\frac{1}{2}

for all x ∈ A.

Remark 2.7 1.− Since every C*-algebra becomes a JB*-algebra under the Jordan product x ∘ y := \frac{1}{2}(xy + yx), it follows from Theorem 2.3 that, given M > 2, a von Neumann algebra A, and a weak*-continuous linear operator T from A to a complex Hilbert space, there exists a norm-one positive functional φ ∈ A_{∗} satisfying

∥T (x)∥ ≤ M ∥T ∥ (φ(\frac{1}{2}(xx^{∗} + x^{∗}x)))^{\frac{1}{2}}

for all x ∈ A. A lightly better result can be derived from [H, Proposition 2.3].

2.− As is asserted in [CIL], Corollary 2.4 can be proved by translating verbatim Pisier’s arguments for the case of C*-álgebras [P2, Theorem 9.4]. We note that actually Corollary 2.4 contains Pisier’s result. Moreover, it is worth mentioning that our proof of Corollary 2.4 avoids any use of ultraproducts techniques.

Following [IKR], we define real JB*-triples as norm-closed real subtriples of complex JB*-triples. In [IKR] it is shown that every real JB*-triple E can be regarded as a real form of a complex JB*-triple. Indeed, given a real JB*-triple E there exists a unique complex JB*-triple structure on the complexification Ê = E ⊕ i E, and a unique conjugation (i.e., conjugatelinear isometry of period 2) τ on Ê such that E = Ê^{τ} := {x ∈ Ê : τ(x) = x}. The class of real JB*-triples includes all JB-algebras [HS], all real C*-algebras [G], and all J*B-algebras [Al].

By a real JBW*-triple we mean a real JB*-triple whose underlying Banach space is a dual Banach space. As in the complex case, the triple product of every real JBW*-triple is separately weak*-continuous [MP], and the bidual E^{∗∗} of a real JB*-triple E is a real JBW*-triple whose triple product extends the one of E [IKR]. Noticing that every real JBW*-triple is a real form of a complex JBW*-triple [IKR], it follows easily that, if W is a real JBW*- triple and if φ is a norm-one element in W_{∗}, then, for z ∈ W such that φ(z) = ∥z∥ = 1, the mapping x ↦ (φ \{x, x, z\})^{\frac{1}{2}} is a prehilbert seminorm on W (not depending on z). Such a seminorm will be denoted by ∥.∥_{φ}.

Now we proceed to deal with “Little Grothendieck’s inequality” for real JBW*-triples. We begin by showing the appropriate version of Lemma 1.5 for real JBW*-triples. Such a version is obtained by adapting the proof of a recent result of the first author for real JB*-triples (see [P]) to the setting of real JBW*-triples.

Lemma 2.8 Let M > 1 + 3\sqrt{2}. Then, for every real JBW*-triple W, every real Hilbert space H, and every weak*-continuous linear operator T : W → H which attains its norm, there exists a norm one functional φ ∈ W_{∗} such that

∥T (x)∥ ≤ M ∥T ∥ ∥x∥_{φ}

for all x ∈ W.

Proof. We follow with minors changes the line of proof of [P, Theorem 4.3]. Without loss of generality we can suppose ∥T ∥ = 1. Write

K = [2\sqrt{2}(\frac{M^{2}}{1 + 3\sqrt{2}} − (1 + \sqrt{2}))]^{\frac{1}{2}} > 2\sqrt{2}

and ρ = \frac{2\sqrt{2}}{1+\sqrt{2}}. By [IKR, Lemma 3.3], there exists a complete tripotent e ∈ W with 1 = ∥T (e)∥. Then denoting by ξ the linear functional on W given by ξ(x) := (T (x)|T (e)) for every x ∈ W, ξ belongs to W_{∗} and satisfies ∥ξ∥ = ξ(e) = 1. Moreover, when in the proof of [P, Theorem 4.3] Corollary 2.5 replaces [P, Theorem 3.2], we obtain the existence of a normone functional ψ ∈ W_{∗} with ψ(e) = 1 such that

∥T (x)∥ ≤ K∥x∥_{ψ} + (1 + \sqrt{2}) ∥x∥_{ξ}

for all x ∈ W. Setting φ := \frac{1}{1+ρ}(ξ + ρ ψ), φ is a norm-one functional in W_{∗} with φ(e) = 1, and we have

∥T (x)∥ ≤ \sqrt{(1 + \sqrt{2})^{2} + \frac{K^{2}}{ρ}} \sqrt{∥x∥_{ξ}^{2} + ρ ∥x∥_{ψ}^{2}}

= ([(1 + \sqrt{2})^{2} + \frac{K^{2}}{ρ}](1 + ρ))^{\frac{1}{2}} ∥x∥_{φ} = M ∥x∥_{φ}

for all x ∈ W. □

When in the proof of Theorem 2.1 Lemma 2.8 replaces Lemma 1.5, we arrive in the following result.

Theorem 2.9 Let K > 1+3\sqrt{2} and ε > 0. Then, for every real JBW*-triple W, every real Hilbert space H, and every weak*-continuous linear operator T : W → H, there exist norm-one functionals φ_{1}, φ_{2} ∈ W_{∗} such that the inequality

∥T (x)∥ ≤ K ∥T ∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}}

holds for all x ∈ W.

For norm-one elements φ_{1}, φ_{2} in the predual of a given real JBW*-triple W, we define the prehilbert seminorm ∥.∥_{φ_{1},φ_{2}} on W verbatim as in the complex case.

Corollary 2.10 Let W be a real JBW*-triple and T a weak*-continuous linear operator from W to a real Hilbert space. Then there exist norm-one functionals φ_{1}, φ_{2} ∈ W_{∗} such that, for every x ∈ W, we have

∥T (x)∥ ≤ 6∥T ∥∥x∥_{φ_{1},φ_{2}}.

3 Grothendieck’s Theorem for JBW*-triples

In this section we prove “Grothendieck’s inequality” for separately weak*- continuous bilinear forms defined on the cartesian product of two JBW*- triples.

Theorem 3.1 Let M > 4(1 + 2\sqrt{3}) and ε > 0. For every couple (V, W) of complex JBW*-triples and every separately weak*-continuous bilinear form V on V × W, there exist norm-one functionals φ_{1}, φ_{2} ∈ V_{∗}, and ψ_{1}, ψ_{2} ∈ W_{∗} satisfying

|V (x, y)| ≤ M ∥V ∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} (∥y∥_{ψ_{2}}^{2} + ε^{2} ∥y∥_{ψ_{1}}^{2})^{\frac{1}{2}}

for all (x, y) ∈ V × W.

Proof. We begin by noticing that a bilinear form U on V ×W is separately weak*-continuous if and only if there exists a weak*-to-weak-continuous linear operator F_{U} : V → W_{∗} such that the equality

U(x, y) =< F_{U}(x), y >

holds for every (x, y) ∈ V × W.

Put T := F_{V} : V → W_{∗} in the sense of the above paragraph. By [CIL, Lemma 5] there exist a Hilbert space H and bounded linear operators S : V → H, R : H → W_{∗} satisfying T = R S and ∥R∥ ∥S∥ ≤ 2(1 + 2\sqrt{3}) ∥T ∥. Notice that in fact we can enjoy such a factorization in such a way that R is injective. Indeed, take H′ equals to the orthogonal complement of Ker(R) in H, R′ := R|_{H′} and S′ := π_{H′} S, where π_{H′} is the orthogonal projection from H onto H′, to have T = R′ S′ with R′ injective and ∥R′∥ ∥S′∥ ≤ 2(1 + 2\sqrt{3}) ∥T ∥.

Next we show that S is weak*-continuous. By [DS, Corollary V.5.5] it is enough to prove that S is weak*-continuous on bounded subsets of V. Let x_{λ} be a bounded net in V weak*-convergent to zero. Take a weak cluster point h of S(x_{λ}) in H. Then R(h) is a weak cluster point of T (x_{λ}) = R S(x_{λ}) in W_{∗}. Moreover, since T is weak*-to-weak-continuous, we have T (x_{λ}) → 0 weakly. It follows R(h) = 0 and hence h = 0 by the injectivity of R. Now, zero is the unique weak cluster point in H of the bounded net S(x_{λ}), and therefore we have S(x_{λ}) → 0 weakly.

Now that we know that the operator S is weak*-continuous, we apply Theorem 2.1 with K = \sqrt{\frac{M}{2(1+2\sqrt{3})}} > \sqrt{2} to find norm-one functionals φ_{1}, φ_{2} ∈ V_{∗}, and ψ_{1}, ψ_{2} ∈ W_{∗} satisfying

∥S(x)∥ ≤ K ∥S∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} and
∥R^{∗}(y)∥ ≤ K ∥R^{∗}∥ (∥y∥_{ψ_{2}}^{2} + ε^{2} ∥y∥_{ψ_{1}}^{2})^{\frac{1}{2}}

for all x ∈ V and y ∈ W. Therefore

|V (x, y)| = | < T (x), y > | = | < S(x), R^{∗}(y) > |

≤ \frac{M}{2(1 + 2\sqrt{2})} ∥R∥ ∥S∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} (∥y∥_{ψ_{2}}^{2} + ε^{2} ∥y∥_{ψ_{1}}^{2})^{\frac{1}{2}}

≤ M ∥V ∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} (∥y∥_{ψ_{2}}^{2} + ε^{2} ∥y∥_{ψ_{1}}^{2})^{\frac{1}{2}} ,

for all (x, y) ∈ V × W. □

In the same way that Theorem 2.3 was derived from Theorem 2.1, we can obtain from Theorem 3.1 that, given M > 8 (1 + 2\sqrt{3}), JBW*-algebras A, B, and a separately weak*-continuous bilinear form V on A × B, there exist norm-one positive functionals φ ∈ A_{∗} and ψ ∈ B_{∗} satisfying

|V (x, y)| ≤ M∥V ∥ (φ(x ∘ x^{∗}))^{\frac{1}{2}} (ψ(y ∘ y^{∗}))^{\frac{1}{2}}

for all (x, y) ∈ A × B. As a relevant particular case we obtain the following result.

Corollary 3.2 Let M > 8(1 + 2\sqrt{3}). For every couple (A, B) of von Neumann algebras and every separately weak*-continuous bilinear form V on A × B, there exist norm-one positive functionals φ ∈ A_{∗} and ψ ∈ B_{∗} satisfying

|V (x, y)| ≤ M∥V ∥ (φ(\frac{1}{2}(xx^{∗} + x^{∗}x)))^{\frac{1}{2}} (ψ(\frac{1}{2}(yy^{∗} + y^{∗}y)))^{\frac{1}{2}}

for all (x, y) ∈ A × B.

A refined version of the above corollary can be found in [H, Proposition 2.3].

Now we proceed to deal with Grothendieck’s Theorem for real JBW*- triples. The following lemma generalizes [CIL, Lemma 5] to the real case.

Lemma 3.3 Let E and F be real JB*-triples and T : E → F^{∗} a bounded linear operator. Then T has a factorization T = R S through a real Hilbert space with ∥R∥ ∥S∥ ≤ 4(1 + 2\sqrt{3}) ∥T ∥

Proof.

Let us consider the JB*-complexifications Ê and F̂ of E and F, respectively, and denote by T̂ : Ê → F̂^{∗} the complex linear extension of T, so that we easily check that ∥T̂∥ ≤ 2∥T ∥. As we have mentioned before, T̂ has a factorization T̂ = R̂Ŝ through a complex Hilbert space H, with ∥R̂∥ ∥Ŝ∥ ≤ 2(1 + 2\sqrt{3}) ∥T̂∥.

Since T̂ is the complex linear extension of T, the inclusion T̂(E) ⊆ F^{∗} holds. Put H := \overline{Ŝ(E)}, the closure of Ŝ(E) in H. Then H is a real Hilbert space and we have R̂(H) ⊆ \overline{R̂(Ŝ(E))} = \overline{T̂(E)} ⊆ F^{∗}.

Finally we define the bounded linear operators S := Ŝ|_{E} : E → H and R := R̂|_{H} : H → F^{∗}. It is easy to see that T = R S and

∥R∥ ∥S∥ ≤ ∥R̂∥ ∥Ŝ∥ ≤ 2(1 + 2\sqrt{3}) ∥T̂∥ ≤ 4(1 + 2\sqrt{3}) ∥T ∥.

□

When in the proof of Theorem 3.1 Lemma 3.3 and Theorem 2.9 replace [CIL, Lemma 5] and Theorem 2.1, respectively, we obtain the following theorem.

Theorem 3.4 Let M > 4(1 + 2\sqrt{3}) (1 + 3\sqrt{2})^{2} and ε > 0. For every couple (V, W) of real JBW*-triples and every separately weak*-continuous bilinear form U on V ×W, there exist norm-one functionals φ_{1}, φ_{2} ∈ V_{∗}, and ψ_{1}, ψ_{2} ∈ W_{∗} satisfying

|U(x, y)| ≤ M ∥U∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} (∥y∥_{ψ_{2}}^{2} + ε^{2} ∥y∥_{ψ_{1}}^{2})^{\frac{1}{2}}

for all (x, y) ∈ V × W.

Thanks to Lemma 3.3, Lemma 1.1 remains true when real JB*-triples replace complex ones. Then Theorems 3.4 and 3.1 give rise to the real and complex cases, respectively, of the result which follows.

Corollary 3.5 Let M > 4(1 + 2\sqrt{3}) (1 + 3\sqrt{2})^{2} (respectively, M > 4(1 + 2\sqrt{3})) and ε > 0. Then for every couple (E, F) of real (respectively, complex) JB*-triples and every bounded bilinear form U on E×F there exist norm-one functionals φ_{1}, φ_{2} ∈ E^{∗} and ψ_{1}, ψ_{2} ∈ F^{∗} satisfying

|U(x, y)| ≤ M ∥U∥ (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} (∥y∥_{ψ_{2}}^{2} + ε^{2} ∥y∥_{ψ_{1}}^{2})^{\frac{1}{2}}

for all (x, y) ∈ E × F.

Remark 3.6 In the complex case of the above corollary, the interval of variation of the constant M can be enlarged by arguing as follows. Let M > 3 + 2\sqrt{3}, ε > 0, E and F be complex JB*-triples, and U a norm-one bounded bilinear form on E × F. Consider the separately weak*-continuous bilinear form Ũ on E^{∗∗} × F^{∗∗} which extends U, and take a weak*-to-weak continuous linear operator T : E^{∗∗} → F^{∗} satisfying

Ũ(α, β) =< T (α), β >

for all (α, β) ∈ E^{∗∗} × F^{∗∗}. Choose δ > 0 such that δ ≤ ε^{2} and (3 + 2\sqrt{3})(1 + δ) ≤ M. By [PZ, Corollary 2] there is a rank one weak*-to-weak continuous linear operator T_{1} : E^{∗∗} → F^{∗} such that ∥T_{1}∥ ≤ δ and T_{2} := T − T_{1} attains its norm. Since T_{1} is of rank one and weak*-continuous, it also attains its norm. For i = 1, 2, consider the separately weak*-continuous bilinear form Ũ_{i} on E^{∗∗} × F^{∗∗} defined by

Ũ_{i}(α, β) =< T_{i}(α), β >,

and put U_{i} = Ũ_{i}|_{E×F}, so that U_{i} is a bounded bilinear form on E × F whose separately weak*-continuous extension to E^{∗∗} × F^{∗∗} attains its norm. By the proof of [BF1, Theorem 1.4], there exist norm-one functionals φ_{1}, φ_{2} ∈ E^{∗} and ψ_{1}, ψ_{2} ∈ F^{∗} such that

|U_{i}(x, y)| ≤ (3 + 2\sqrt{3}) ∥U_{i}∥ ∥x∥_{φ_{i}}∥y∥_{ψ_{i}},

for all (x, y) ∈ E × F and i = 1, 2.

Therefore

|U(x, y)| ≤ |U_{2}(x, y)| + |U_{1}(x, y)|
≤ (3 + 2\sqrt{3})(∥U_{2}∥ ∥x∥_{φ_{2}}∥y∥_{ψ_{2}} + ∥U_{1}∥ ∥x∥_{φ_{1}}∥y∥_{ψ_{1}})
≤ (3 + 2\sqrt{3})((1 + δ) ∥x∥_{φ_{2}}∥y∥_{ψ_{2}} + δ ∥x∥_{φ_{1}}∥y∥_{ψ_{1}})
≤ (3 + 2\sqrt{3})(1 + δ) (∥x∥_{φ_{2}}∥y∥_{ψ_{2}} + δ ∥x∥_{φ_{1}}∥y∥_{ψ_{1}})
≤ (3 + 2\sqrt{3})(1 + δ) \sqrt{∥x∥_{φ_{2}}^{2} + δ∥x∥_{φ_{1}}^{2}}\sqrt{∥y∥_{ψ_{2}}^{2} + δ∥y∥_{ψ_{1}}^{2}}
≤ M (∥x∥_{φ_{2}}^{2} + ε^{2} ∥x∥_{φ_{1}}^{2})^{\frac{1}{2}} (∥y∥_{ψ_{2}}^{2} + ε^{2} ∥y∥_{ψ_{1}}^{2})^{\frac{1}{2}}

for all (x, y) ∈ E × F.

We do not know if the value ε = 0 is allowed in Theorems 3.1 and 3.4. In any case, as the next result shows, the value ε = 0 is allowed for a “big quantity” of separately weak*-continuous bilinear forms.

Theorem 3.7 Let M > 4(1 + 2\sqrt{3}) (1 + 3\sqrt{2})^{2} (respectively, M > 4(1 + 2\sqrt{3})) and V, W be real (respectively, complex) JBW*-triples. Then the set of all separately weak*-continuous bilinear forms U on V ×W such that there exist norm-one functionals φ ∈ V_{∗} and ψ ∈ W_{∗} satisfying

|U(x, y)| ≤ M ∥U∥ ∥x∥_{φ} ∥y∥_{ψ}

for all (x, y) ∈ V × W, is norm dense in the set of all separately weak*- continuous bilinear forms on V × W.

Proof. Let U a non zero separately weak*-continuous bilinear form on V ×W. By the proof of Theorem 3.4 (respectively, Theorem 3.1) there exists a real (respectively, complex) Hilbert space H such that for all (x, y) ∈ V ×W we have

U(x, y) :=< F (x), G(y) >,

where F : V → H and G : W → H^{∗} are weak*-continuous linear operators satisfying ∥F ∥ ∥G∥ ≤ L ∥U∥ with L = 4(1 + 2\sqrt{3}) (respectively, L = 2(1 + 2\sqrt{3})).

By [Z], there are sequences {F_{n} : V → H} and {G_{n} : W → H^{∗}} of weak*- continuous linear operators, converging in norm to F and G, respectively, and such that F_{n} and G_{n} attain their norms for every n. Then, putting

U_{n}(x, y) :=< F_{n}(x), G_{n}(y) > ((n, x, y) ∈ ℕ × V × W),

{U_{n}} becomes a sequence of separately weak*-continuous bilinear forms on V × W, converging in norm to U. Take \sqrt{\frac{M}{L}} > K > 1 + 3\sqrt{2} (respectively,

\sqrt{\frac{M}{L}} > K > \sqrt{2}). Applying Lemma 2.8 (respectively, Lemma 1.5), for n ∈ ℕ we find norm-one functionals φ_{n} ∈ V_{∗} and ψ_{n} ∈ W_{∗} satisfying

∥F_{n}(x)∥ ≤ K ∥F_{n}∥ ∥x∥_{φ_{n}} and
∥G_{n}(y)∥ ≤ K ∥G_{n}∥ ∥y∥_{ψ_{n}}

for all (x, y) ∈ V × W.

Set

δ = \frac{\frac{M}{K^{2}} − L}{1 + L} \frac{∥U∥}{2} > 0,

and take m ∈ ℕ such that the inequalities

| ∥F_{n}∥ ∥G_{n}∥ − ∥F ∥ ∥G∥ | < δ,
| ∥U_{n}∥ − ∥U∥ | < δ, and

∥U_{n}∥ ≥ \frac{∥U∥}{2}

hold for every n ≥ m.

Now for n ≥ m and (x, y) ∈ V × W we have

|U_{n}(x, y)| ≤ K^{2} ∥F_{n}∥ ∥G_{n}∥ ∥x∥_{φ_{n}}∥y∥_{ψ_{n}}

≤ K^{2} (∥F ∥ ∥G∥ + δ) ∥x∥_{φ_{n}}∥y∥_{ψ_{n}}
≤ K^{2} (L ∥U∥ + δ) ∥x∥_{φ_{n}}∥y∥_{ψ_{n}}

≤ K^{2} (L ∥U_{n}∥ + δ (1 + L)) ∥x∥_{φ_{n}}∥y∥_{ψ_{n}}

= K^{2} (L ∥U_{n}∥ + (\frac{M}{K^{2}} − L)\frac{∥U∥}{2}) ∥x∥_{φ_{n}}∥y∥_{ψ_{n}}

≤ M ∥U_{n}∥ ∥x∥_{φ_{n}}∥y∥_{ψ_{n}}.

□

As we noticed before Corollary 3.5, Lemma 1.1 remains true in the real setting. Then, given real or complex JB*-triples E, F, the mapping sending each element U ∈ L(^{2}(E × F)) to its unique separately weak*-continuous bilinear extension Ũ to E^{∗∗} × F^{∗∗} is an isometry from L(^{2}(E × F)) onto the Banach space of all separately weak*-continuous bilinear forms on E^{∗∗} ×F^{∗∗}. Therefore we obtain the following corollary.

Corollary 3.8 Let M > 4(1 + 2\sqrt{3}) (1 + 3\sqrt{2})^{2} (respectively, M > 4(1 + 2\sqrt{3})) and E, F be real (respectively, complex) JB*-triples. Then the set of all bounded bilinear forms U on E × F such that there exist norm-one functionals φ ∈ E^{∗} and ψ ∈ F^{∗} satisfying

|U(x, y)| ≤ M ∥U∥ ∥x∥_{φ} ∥y∥_{ψ}

for all (x, y) ∈ E × F, is norm dense in L(^{2}(E × F)).

We note that Theorem 1.4 is finer than the complex case of the above corollary. However, since Theorem 1.4 depends on the proof of [BF1, Theorem 1.4], it is much more difficult.

Remark 3.9 We do not know if the value ε = 0 is allowed in Theorems 2.1 and 2.9 (respectively, in Theorems 3.1 and 3.4) for some value of the constant K (respectively, M). Concerning this question, it is worth mentioning that the following three assertions are equivalent:

1. There is a universal constant G such that, for every real (respectively, complex) JBW*-triple W and every couple (φ_{1}, φ_{2}) of norm-one functionals in W_{∗} × W_{∗}, we can find a norm-one functional φ ∈ W_{∗} satisfying

∥x∥_{φ_{i}} ≤ G∥x∥_{φ}

for every x ∈ W and i = 1, 2.

2. There is a universal constant Ĝ such that for every couple of real (respectively, complex) JBW*-triples (V, W) and every separately weak*- continuous bilinear form U on V × W, there are norm-one functionals φ ∈ V_{∗}, and ψ ∈ W_{∗} satisfying

|U(x, y)| ≤ Ĝ ∥U∥ ∥x∥_{φ}∥y∥_{ψ}

for all (x, y) ∈ V × W.

3. There is a universal constant G̃ such that for every real (respectively, complex) JBW*-triple W and every weak*-continuous linear operator T from W to a real (respectively, complex) Hilbert space, there exists a norm-one functional φ ∈ W_{∗} satisfying

∥T (x)∥ ≤ G̃ ∥T ∥ ∥x∥_{φ}

for all x ∈ W.

The implication 1 ⇒ 2 follows from Theorems 3.1 and 3.4.

Assume that Assertion 2 above is true. Let W be a real (respectively, complex) JBW*-triple, H a real (respectively, complex) Hilbert space, and T : W → H a weak*-continuous linear operator. Consider the separately weak*-continuous bilinear form U on W ×H given by U(x, y) := (T (x)|y) (respectively, U(x, y) := (T (x)|σ(y)), where σ is a conjugation on H). Regarding H as a JBW*-triple under the triple product \{x, y, z\} := \frac{1}{2}((x|y)z + (z|y)x), and applying the assumption, we find norm-one functionals φ ∈ W_{∗} and ψ ∈ H_{∗} satisfying

|U(x, y)| ≤ Ĝ ∥U∥ ∥x∥_{φ}∥y∥_{ψ}
≤ Ĝ ∥T ∥ ∥x∥_{φ}∥y∥

for all (x, y) ∈ W × H. Taking y = T (x) (respectively, y = σ(T (x))) we obtain

∥T (x)∥ ≤ Ĝ∥T ∥∥x∥_{φ}

for all x ∈ W. In this way Assertion 3 holds.

Finally let us assume that Assertion 3 is true. Let W be a real (respectively, complex) JBW*-triple and φ_{1}, φ_{2} norm-one functionals in W_{∗}. Since ∥.∥_{φ_{1},φ_{2}} comes from a suitable separately weak*-continuous positive sesquilinear form < ., . > on W by means of the equality ∥x∥_{φ_{1},φ_{2}}^{2} =< x, x >, it follows from the proof of [R1, Corollary] that there exists a weak*-continuous linear operator T from W to a real (respectively, complex) Hilbert space satisfying ∥x∥_{φ_{1},φ_{2}} = ∥T (x)∥ for all x ∈ W (which implies ∥T ∥ ≤ \sqrt{2}). Now applying the assumption we find a norm one functional φ ∈ W_{∗} such that

∥x∥_{φ_{1},φ_{2}} = ∥T (x)∥ ≤ G̃∥T ∥∥x∥_{φ} ≤ \sqrt{2}G̃∥x∥_{φ}

for all x ∈ W. As a consequence, for i = 1, 2 we have

∥x∥_{φ_{i}} ≤ \sqrt{2}G̃∥x∥_{φ}

for all x ∈ W.

4 Some Applications

We define the strong*-topology S^{∗}(W, W_{∗}) of a given real or complex JBW*- triple W as the topology on W generated by the family of seminorms {∥.∥_{φ} : φ ∈ W_{∗}, ∥φ∥ = 1}. In the complex case, the above notion has been introduced by T. J. Barton and Y. Friedman in [BF2]. When a JBW*-algebra A is regarded as a complex JBW*-triple, S^{∗}(A, A_{∗}) coincides with the so-called “algebra-strong* topology” of A, namely the topology on A generated by the family of seminorms of the form x ↦ \sqrt{ξ(x ∘ x^{∗})} when ξ is any positive functional in A_{∗} [R1, Proposition 3]. As a consequence, when a von Neumann algebra M is regarded as a complex JBW*-triple, S^{∗}(M, M_{∗}) coincides with the familiar strong*-topology of M (compare [S, Definition 1.8.7]).

We note that, if W is a complex JBW*-triple, then, denoting by W_{ℝ} the realification of W (i.e., the real JBW*-triple obtained from W by restriction of scalar to ℝ), we have S^{∗}(W, W_{∗}) = S^{∗}(W_{ℝ}, (W_{ℝ})_{∗}). Indeed, the mapping φ ↦ ℜe φ identifies W_{∗} with (W_{ℝ})_{∗}, and, when φ has norm one, the equality ∥x∥_{φ} = ∥x∥_{ℜe φ} holds for every x ∈ W.

Proposition 4.1 Let W be a real (respectively, complex) JBW*-triple. The following topologies coincide in W:

1. The strong*-topology of W.

2. The topology on W generated by the family of seminorms of the form x ↦ \sqrt{< x, x >}, where < ., . > is any separately weak*-continuous positive sesquilinear form on W.

3. The topology on W generated by the family of seminorms x ↦ ∥T (x)∥, when T runs over all weak*-continuous linear operators from W to arbitrary real (respectively, complex) Hilbert spaces.

Proof. Let us denote by τ_{1}, τ_{2}, and τ_{3} the topologies arising in paragraphs 1, 2, and 3, respectively. The inequality τ_{1} ≥ τ_{3} follows from Corollary 2.10 (respectively, Corollary 2.2). Since the proof of [R1, Corollary 1] shows that for every separately weak*-continuous positive sesquilinear form < ., . > on W there exists a weak*-continuous linear operator T from W to a real (respectively, complex) Hilbert space satisfying \sqrt{< x, x >} = ∥T (x)∥ for all x ∈ W, we have τ_{3} ≥ τ_{2}. Finally, since for every norm-one functional φ ∈ W_{∗} there is a separately weak*-continuous positive sesquilinear form < ., . > satisfying ∥x∥_{φ} = \sqrt{< x, x >} for all x ∈ W, the inequality τ_{2} ≥ τ_{1} follows. □

For every Banach space X, B_{X} will stand for the closed unit ball of X. For every dual Banach space X (with a fixed predual denoted by X_{∗}), we denote by m(X, X_{∗}) the Mackey topology on X relative to its duality with X_{∗}.

Corollary 4.2 Let W be a real or complex JBW*-triple. Then the strong*- topology of W is compatible with the duality (W, W_{∗}).

Proof. We apply the characterization of S^{∗}(W, W_{∗}) given by paragraph 3 in Proposition 4.1. Clearly S^{∗}(W, W_{∗}) is stronger than the weak*-topology σ(W, W_{∗}) of W. On the other hand, if T is a weak*-continuous linear operator from W to a Hilbert space H, and if we put T = S^{∗} for a suitable bounded linear operator S : H_{∗} → W_{∗}, then S(B_{H_{∗}}) is an absolutely convex and weakly compact subset of W_{∗} and we have ∥T (x)∥ = sup | < x, S(B_{H_{∗}}) > |. This shows that S^{∗}(W, W_{∗}) is weaker than m(W, W_{∗}). □

The complex case of the above corollary is due to T. J. Barton and Y. Friedman [BF2]. The complex case of Proposition 4.1 is claimed in [R1, Corollary 2] (see also [R2, Proposition D.17]), but the proof relies on [R1, Proposition 1], which subsumes a gap from [BF1] (see the comments before Lemma 1.5). Now that we have saved [R1, Corollary 2], all subsequent results in [R1] concerning the strong*-topology of complex JBW*-triples are valid. Moreover, keeping in mind Proposition 4.1 and Corollary 4.2, some of those results remain true for real JBW*-triples with verbatim proof. For instance, the following assertions hold:

1. Linear mappings between real JBW*-triples are strong*-continuous if and only if they are weak*-continuous (compare [R1, Corollary 3]).

2. If W is a real JBW*-triple, and if V is a weak*-closed subtriple, then the inequality S^{∗}(W, W_{∗})|_{V} ≤ S^{∗}(V, V_{∗}) holds, and in fact S^{∗}(W, W_{∗})|_{V} and S^{∗}(V, V_{∗}) coincide on bounded subsets of V (compare [R1, Proposition 2]).

It follows from the first part of Assertion 2 above and a new application of Proposition 4.1 that, if W is a real JBW*-triple, and if V is a weak*- complemented subtriple of W, then we have S^{∗}(W, W_{∗})|_{V} = S^{∗}(V, V_{∗}). Since every real JBW*-triple V is weak*-complemented in the realification of a complex JBW*-triple W (see V as a real form of its JB*-complexification), and S^{∗}(W, W_{∗}) = S^{∗}(W_{ℝ}, (W_{ℝ})_{∗}), the results [R1, Theorem] and [R2, Theorem D.21] for complex JBW*-triples can be transferred to the real setting, providing the following result.

Theorem 4.3 Let W be a real JBW*-triple. Then the triple product of W is jointly S^{∗}(W, W_{∗})-continuous on bounded subsets of W, and the topologies m(W, W_{∗}) and S^{∗}(W, W_{∗}) coincide on bounded subsets of W.

Our concluding goal in this paper is to establish, in the setting of real JB*- triples, a result on weakly compact operators originally due to H. Jarchow [J] in the context of C*-algebras, and later extended to complex JB*-triples by C-H. Chu and B. Iochum [CI]. This could be made by transferring the complex results to the real setting by a complexification method. However, we prefer to do it in a more intrinsic way, by deriving the result from the second assertion in Theorem 4.3 according to some ideas outlined in [R2, pp. 142-143].

Proposition 4.4 Let X be a dual Banach space (with a fixed predual X_{∗}). Then the Mackey topology m(X, X_{∗}) coincides with the topology on X generated by the family of semi-norms x ↦ ∥T (x)∥, where T is any weak*- continuous linear operator from X to a reflexive Banach space.

Proof. Let us denote by τ the second topology arising in the statement. As in the proof of Corollary 4.2, if T is a weak*-continuous linear operator from X to a reflexive Banach space, then there exists an absolutely convex and weakly compact subset D of X_{∗} such that the equality

∥T (x)∥ = sup | < x, D > |

holds for every x ∈ X. This shows that τ ≤ m(X, X_{∗}).

Let D be an absolutely convex and weakly compact subset of X_{∗}. Consider the Banach space ℓ_{1}(D) and the bounded linear operator

F : ℓ_{1}(D) → X_{∗}

given by

F ({λ_{φ}}_{φ∈D}) := ∑_{φ∈D} λ_{φ}φ.

Then we have F (B_{ℓ_{1}(D)}) = D, and hence F is weakly compact. By [DFJP] there exists a reflexive Banach space Y together with bounded linear operators S : ℓ_{1}(D) → Y, R : Y → X_{∗} such that F = R S. Then, for x ∈ X, we have

sup | < x, D > | = sup | < x, F (B_{ℓ_{1}(D)}) > |

= sup | < x, R(S(B_{ℓ_{1}(D)})) > | ≤ ∥S∥ sup | < x, R(B_{Y}) > |
= ∥S∥ ∥R^{∗}(x)∥.

Since D is an arbitrary absolutely convex and weakly compact subset of X_{∗}, and R^{∗} is a weak*-continuous linear operator from X to the reflexive Banach space Y^{∗}, the inequality m(X, X_{∗}) ≤ τ follows.□

Let X be a dual Banach space (with a fixed predual X_{∗}). In agreement with Proposition 4.1, we define the strong*-topology of X, denoted by S^{∗}(X, X_{∗}), as the topology on X generated by the family of semi-norms x ↦ ∥T (x)∥, where T is any weak*-continuous linear operator from X to a Hilbert space.

Proposition 4.5 Let X be a dual Banach space (with a fixed predual X_{∗}). Then the following assertions are equivalent:

1. The topologies m(X, X_{∗}) and S^{∗}(X, X_{∗}) coincide on bounded subsets of X.

2. For every weak*-continuous linear operator F from X to a reflexive Banach space, there exists a weak*-continuous linear operator G from X to a Hilbert space satisfying ∥F (x)∥ ≤ ∥G(x)∥ + ∥x∥ for all x ∈ X.

3. For every weak*-continuous linear operator F from X to a reflexive Banach space, there exist a weak*-continuous linear operator G from X to a Hilbert space and a mapping N : (0, ∞) → (0, ∞) satisfying

∥F (x)∥ ≤ N(ε) ∥G(x)∥ + ε ∥x∥

for all x ∈ X and ε > 0.

Proof. 1 ⇒ 2.− Let F be a weak*-continuous linear operator from X to a reflexive Banach space. Then, by Proposition 4.4

O := {y ∈ B_{X} : ∥F (y)∥ ≤ 1}

is a m(X, X_{∗})|_{B_{X}}-neighborhood of zero in B_{X}. By assumption, there exist Hilbert spaces H_{1}, …, H_{n} and weak*-continuous linear operators G_{i} : X → H_{i} (i : 1, …, n) such that

O ⊇ ∩_{i=1}^{n}{y ∈ B_{X} : ∥G_{i}(y)∥ ≤ 1}.

Now set H := (⨁_{i=1}^{n} H_{i})_{ℓ_{2}}, and consider the weak*-continuous linear operator G : X → H defined by G(x) := (G_{1}(x), …, G_{n}(x)). Notice that

{y ∈ B_{X} : ∥G(y)∥ ≤ 1} ⊆ ∩_{i=1}^{n}{y ∈ B_{X} : ∥G_{i}(y)∥ ≤ 1} ⊆ O.

Finally, if x ∈ X \ \{0\}, then \frac{1}{∥x∥+∥G(x)∥} x lies in \{y ∈ B_{X} : ∥G(y)∥ ≤ 1\} ⊆ O, and hence ∥F (\frac{1}{∥x∥+∥G(x)∥} x)∥ ≤ 1.

2 ⇒ 3.− Let F be a weak*-continuous linear operator from X to a reflexive Banach space. By assumption, for every n ∈ ℕ there exists a Hilbert space H_{n} and a weak*-continuous linear operator G_{n} from X to H_{n} such that ∥nF (x)∥ ≤ ∥G_{n}(x)∥ + ∥x∥ for all x ∈ X. Now set H := (⨁_{n∈ℕ} H_{n})_{ℓ_{2}}, and consider the bounded linear operator G : X → H defined by G(x) := \{\frac{1}{n∥G_{n}∥} G_{n}(x)\} and the mapping N : ε → ∥G_{n(ε)}∥ (where n(ε) denotes the smallest natural number satisfying n > \frac{1}{ε}). Then G is weak*-continuous. Indeed, given y = {h_{n}} ∈ H, we can take for n ∈ ℕ α_{n} in X_{∗} satisfying (G_{n}(x)|h_{n}) =< x, α_{n} > for every x ∈ X, so that we have

∑_{n∈ℕ}∥\frac{α_{n}}{n∥G_{n}∥}∥ ≤ ∑_{n∈ℕ} \frac{∥h_{n}∥}{n} ≤ \sqrt{∑_{n∈ℕ}∥h_{n}∥^{2}} \sqrt{∑_{n∈ℕ} \frac{1}{n^{2}}} < ∞,

and hence α := ∑_{n∈ℕ} \frac{α_{n}}{n ∥G_{n}∥} is an element of X_{∗} satisfying (G(x)|h) =< x, α > for all x ∈ X. Moreover, for all ε > 0 and x ∈ X we have

∥F (x)∥ ≤ \frac{1}{n(ε)}∥G_{n(ε)}(x)∥ + \frac{1}{n(ε)}∥x∥

≤ ∥G_{n(ε)}∥ ∥G(x)∥ + \frac{1}{n(ε)}∥x∥ ≤ N(ε)∥G(x)∥ + ε∥x∥.

3 ⇒ 1.− Let x_{λ} be a net in B_{X} converging to zero in the topology S^{∗}(X, X_{∗}). Let F be a weak*-continuous linear operator from X to a reflexive Banach space, and ε > 0. By assumption, there exist a weak*-continuous linear operator G from X to a Hilbert space and a mapping N : (0, ∞) → (0, ∞) satisfying

∥F (x)∥ ≤ N(\frac{ε}{2}) ∥G(x)∥ + \frac{ε}{2} ∥x∥

for all x ∈ X. Take λ_{0} such that ∥G(x_{λ})∥ ≤ \frac{ε}{2 N( \frac{ε}{2} )} whenever λ ≥ λ_{0}. Then we have ∥F (x_{λ})∥ ≤ ε for all λ ≥ λ_{0}. By Proposition 4.4, x_{λ} m(X, X_{∗})- converges to zero. □

We can now state the following characterization of weakly compact operators on JB*-triples.

Theorem 4.6 Let E be a real (respectively, complex) JB*-triple, X a real (respectively, complex) Banach space, and T : E → X a bounded linear operator. The following assertions are equivalent:

1. T is weakly compact.

2. There exist a bounded linear operator G from E to a real (respectively, complex) Hilbert space and a function N : (0, +∞) → (0, +∞) such that

∥T (x)∥ ≤ N(ε)∥G(x)∥ + ε∥x∥

for all x ∈ E and ε > 0.

3. There exist norm one functionals φ_{1}, φ_{2} ∈ E^{∗} and a function N : (0, +∞) → (0, +∞) such that

∥T (x)∥ ≤ N(ε) ∥x∥_{φ_{1},φ_{2}} + ε∥x∥

for all x ∈ E and ε > 0.

Proof. The implication 2 ⇒ 3 follows from Corollary 2.10 (respectively, Corollary 2.2). The implication 3 ⇒ 2 holds because, for norm-one functionals φ_{1}, φ_{2} ∈ E^{∗}, ∥.∥_{φ_{1},φ_{2}} is a prehilbert seminorm on E, and hence there exists a bounded linear operator G from E to a Hilbert space satisfying ∥G(x)∥ = ∥x∥_{φ_{1},φ_{2}} for all x ∈ E. On the other hand, the implication 2 ⇒ 1 is known to be true, even if E is an arbitrary Banach space (see for instance [J, Theorem 20.7.3]). To conclude the proof, let us show that 1 implies 2. Assume that Assertion 1 holds. Then, by [DFJP], there exist a reflexive Banach space Y and bounded linear operators F : E → Y and S : Y → X such that T = S F and ∥S∥ ≤ 1. By Theorem 4.3 and Proposition 4.5, there exist a weak*-continuous linear operator G̃ from E^{∗∗} to a Hilbert space and a mapping N : (0, ∞) → (0, ∞) satisfying

∥F^{∗∗}(α)∥ ≤ N(ε) ∥G̃(α)∥ + ε ∥α∥

for all α ∈ E^{∗∗} and ε > 0. By putting G := G̃|_{E}, the inequality in Assertion 2 follows. □

The complex case of the above theorem is established in [CI, Theorem 11], with ∥.∥_{φ_{1},φ_{2}} in Assertion 3 replaced with ∥.∥_{φ} for a single norm-one functional φ ∈ E^{∗}. As we have noticed in similar occasions, we do not know if such a replacement is correct.

References

[A] Acosta, M. D.: On bilinear mappings attaining their norms, Studia Math. 131 (2), 155-165 (1998).

[Al] Alvermann, K.: Real normed Jordan algebras with involution, Arch. Math. 47, 135-150 (1986).

[BF1] Barton, T. and Friedman Y.: Grothendieck’s inequality for JB*-triples and applications, J. London Math. Soc. (2) 36, 513-523 (1987).

[BF2] Barton, T. and Friedman Y.: Bounded derivations of JB*-triples, Quart. J. Math. Oxford 41, 255-268 (1990).

[BT] Barton, T. and Timoney, R. M.: Weak*-continuity of Jordan triple products and its applications, Math. Scand. 59, 177-191 (1986).

[BKU] Braun, R. B., Kaup, W., and Upmeier, H.: A holomorphic characterization of Jordan-C*-algebras, Math. Z. 161, 277-290 (1978).

[CI] Chu, C-H. and Iochum, B.: Weakly compact operators on Jordan triples, Math. Ann. 281, 451-458 (1988).

[CIL] Chu, C-H., Iochum, B., and Loupias, G.: Grothendieck’s theorem and factorization of operators in Jordan triples, Math. Ann. 284, 41-53 (1989).

[CM] Chu, C-H., Mellon, P.: The Dunford-Pettis property in JB*-triples, J. London Math. Soc. (2) 55, 515-526 (1997).

[DFJP] Davis, W. J., Figiel, T., Johnson W. B., and Pelczynski, A.: Factoring weakly compact operators, J. Funct. Anal. 17, 311-327 (1974).

[Di] Dineen, S.: The second dual of a JB*-triple system, In: Complex analysis, functional analysis and approximation theory (ed. by J. Múgica), 67-69, (North-Holland Math. Stud. 125), North-Holland, Amsterdam-New York, 1986.

[DS] Dunford, N. and Schwartz, J. T.: Linear operators, Part I, Interscience, New York 1958.

[E] Edwards, C. M.: On Jordan W*-algebras, Bull. Sci. Math. (2) 104, 393-403 (1980).

[G] Goodearl, K. R.: Notes on real and complex C*-algebras, Shiva Publ. 1982.

[Gro] Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. Sao Paolo 8, 1-79 (1956).

[H] Haagerup, U.: The Grothendieck inequality for bilinear forms on C*- algebras, Avd. Math. 56, 93-116 (1985).

[HS] Hanche-Olsen, H. and Størmer, E.: Jordan operator algebras, Monographs and Studies in Mathematics 21, Pitman, London-Boston-Melbourne 1984.

[HP] Hille, E. and Phillips, R. S.: Functional analysis and semigroups, A.M.S. Colloquium Publications Vol. XXXI, 1957.

[IKR] Isidro, J. M., Kaup, W., and Rodríguez, A.: On real forms of JB*- triples, Manuscripta Math. 86, 311-335 (1995).

[J] Jarchow, H.: Locally Convex Spaces, Teubner, Stuttgart, 1981.

[K1] Kaup, W.: Algebraic characterization of symmetric complex Banach manifolds, Math. Ann. 228, 39-64 (1977).

[K2] Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183, 503-529 (1983).

[KU] Kaup, W. and Upmeier, H.: Jordan algebras and symmetric Siegel domains in Banach spaces, Math. Z. 157, 179-200, (1977).

[L] Lindenstrauss, J.: On operators which attain their norm, Israel J. Math. 1, 139-148 (1963).

[MP] Martínez, J. and Peralta A.M.: Separate weak*-continuity of the triple product in dual real JB*-triples, Math. Z., to appear.

[P] Peralta, A. M.: Little Grothendieck’s theorem for real JB*-triples, Math. Z., to appear.

[P1] Pisier, G.: Grothendieck’s theorem for non commutative C*-algebras with an appendix on Grothendieck’s constant, J. Funct. Anal. 29, 397-415 (1978).

[P2] Pisier, G.: Factorization of linear operators and geometry of Banach spaces, Publ. Amer. Math. Soc. CBMS 60, Am. Math. Soc. 1986.

[P3] Pisier, G.: Factorization of operators through L_{p∞} or L_{p1} and noncommutative generalizations, Math. Ann. 276, 105-136 (1986).

[PZ] Poliquin, R. A. and Zizler, V. E.: Optimization of convex functions on w*-compact sets, Manuscripta Math. 68, 249-270 (1990).

[R1] Rodríguez A.: On the strong* topology of a JBW*-triple, Quart. J. Math. Oxford (2) 42, 99-103 (1989).

[R2] Rodríguez A.: Jordan structures in Analysis. In Jordan algebras: Proc. Oberwolfach Conf., August 9-15, 1992 (ed. by W. Kaup, K. McCrimmon and H. Petersson), 97-186. Walter de Gruyter, Berlin, 1994.

[S] Sakai, S.: C*-algebras and W*-algebras, Springer-Verlag, Berlin 1971.

[U] Upmeier, H.: Symmetric Banach Manifolds and Jordan C*-algebras, Mathematics Studies 104, (Notas de Matemática, ed. by L. Nachbin) North Holland 1985.

[W] Wright, J. D. M.: Jordan C*-algebras, Michigan Math. J. 24, 291-302 (1977).

[Y] Youngson, M. A.: Non unital Banach Jordan algebras and C*-triple systems, Proc. Edinburg Math. Soc. 24, 19-31 (1981).

[Z] Zizler, V.: On some extremal problems in Banach spaces, Math. Scand. 32, 214-224 (1973).

A. M. Peralta and A. Rodríguez Palacios Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada 18071 Granada, Spain aperalta@goliat.ugr.es and apalacio@goliat.ugr.es