ON THE AXIOMATIC DEFINITION OF REAL JB*-TRIPLES

ANTONIO M. PERALTA

Abstract. In the last twenty years, a theory of real Jordan triples has been developed. In 1994 T. Dang and B. Russo introduced the concept of J*B-triple. These J*B-triples include real C*-algebras and complex JB*-triples. However, concerning J*B-triples, an important problem was left open. Indeed, the question was whether the complexification of a J*B-triple is a complex JB*-triple in some norm extending the original norm. T. Dang and B. Russo solved this problem for commutative J*B-triples.

In this paper we characterize those J*B-triples with a unitary element whose complexifications are complex JB*-triples in some norm extending the original one. We actually find a necessary and sufficient new axiom to characterize those J*B-triples with a unitary element which are J*B-algebras in the sense of [1] or real JB*-triples in the sense of [4].

1. Introduction

We recall that a real (respectively, complex) Banach Jordan triple is a real (respectively, complex) Banach space U with a continuous trilinear (respectively, bilinear in the outer variables and conjugate linear in the middle one) product

U × U × U → U
(xyz) ↦ {x, y, z}

satisfying (1) {x, y, z} = {z, y, x};

(2) 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, x, y, z in U, where L(x, y)z := {x, y, z}.

2000 Mathematics Subject Classification. Primary 17C65, 46L05, and 46L70. Partially supported by D.G.I.C.Y.T. project no. PB 98-1371, and Junta de

Andalucía grant FQM 0199 .

A Banach Jordan triple U is commutative or abelian if

{{x, y, z} , u, v} = {x, y, {z, u, v}} = {x, {y, z, u} , v}

for all x, y, z, u, v ∈ U. An element u ∈ U is said to be unitary if L(u, u) coincides with the identity map on U.

A complex Jordan Banach triple E is said to be a (complex) JB*- triple if

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

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

We recall that a bounded linear operator T on a complex Banach space is said to be hermitian if ∥ exp(iαT)∥ = 1 for all real α.

Complex JB*-triples were introduced by W. Kaup in the study of Bounded Symmetric Domains in complex Banach spaces (see [5], [6]).

During the 80’s and early 90’s three notions of JB*-triples over the real field were introduced by H. Upmeier [9], T. Dang and B. Russo [3] and J. M. Isidro, W. Kaup and A. Rodríguez [4], respectively.

In 1994 T. Dang and B. Russo [3, Definition 1.3] gave the following axiomatic definition of a JB*-triple over the real field. A J*B-triple is a real Banach space E equipped with a structure of a real Banach Jordan triple which satisfies

(J*B 1) ∥{x, x, x}∥ = ∥x∥^{3} for all x in E; (J*B 2) ∥{x, y, z}∥ ≤ ∥x∥ ∥y∥ ∥z∥ for all x, y, z in E; (J*B 3) σ_{BL(E)}^{ℂ}(L(x, x)) ⊂ [0, +∞) for all x ∈ E; (J*B 4) σ_{BL(E)}^{ℂ}(L(x, y) − L(y, x)) ⊂ iℝ for all x, y ∈ E.

Every closed subtriple of a J*B-triple is a J*B-triple (c.f. [3, Remark 1.5]). The class of J*B-triples includes real C*-algebras and complex JB*-triples. Moreover, in [3, Proposition 1.4] it is shown that complex JB*-triples are precisely those complex Banach Jordan triples whose underlying real Banach space is a J*B-triple. In [3] the following important problems concerning J*B-triples were left open

(P 1) Is the complexification of a J*B-triple a complex JB*-triple in

some norm extending the original norm? (P 2) Is the bidual of a J*B-triple a J*B-triple with a separately

weak*-continuous triple product?

Problem (P 1) was affirmatively solved by T. Dang and B. Russo in the particular case of a commutative J*B-triple [3, Theorem 3.11]. They proved that the complexification of every commutative J*B-triple E is a complex JB*-triple in some norm extending the norm on E.

In this paper we study both problems in the general case of J*Btriples with a unitary element. We characterize those J*B-triples having a unitary element whose complexifications are complex JB*-triples in some norm extending the original norm. As a consequence we solve (P 2) for the class of J*B-triples with a unitary element where problem (P 1) has an affirmative solution.

In 1997 real JB*-triples were introduced by J. M. Isidro, W. Kaup, and A. Rodríguez [4], as norm-closed real subtriples of complex JB*- triples. Isidro, Kaup, and Rodríguez [4, Proposition 2.2] proved that given a real JB*-triple E, then there exists a unique complex JB*-triple structure on its complexification with a norm extending the norm on E.

Let X be a complex Banach space. A conjugation on X is a conjugatelinear isometry on X of period 2. If τ is a conjugation on X, then X^{τ} will stand for the real Banach space of all τ-fixed elements of X. Real normed spaces which can be written as X^{τ}, for some conjugation τ on X, are called real forms of X.

If E is a complex JB*-triple, then conjugations on E preserve the triple product of E (c.f. [6, Proposition 5.5]), and hence the real forms of E are real JB*-triples. In [4] it is shown that actually every real JB*- triple can be regarded as a real form of a suitable complex JB*-triple.

The class of real JB*-triples includes all JB-algebras, all real C*- algebras, all J*B-algebras (compare [1]), and obviously all complex JB*-triples. In [4, Proposition 2.5] it is shown that every real JB*- triple is a J*B-triple but the converse is an open problem. Problem:

(P 1′) Is every J*B-triple a real JB*-triple?

Problem (P 1′) has been also stated in [4, page 318]. Actually problems (P 1) and (P 1′) are equivalent.

In this paper (see Theorem 2.5) we also find a necessary and sufficient fifth axiom for a J*B-triple E with a unitary element to be a real JB*-triple in the sense of [4]. This new axiom assures that the operator L(x, x) has non negative numerical range (see definition below). Actually, this new axiom characterizes those J*B-triples with a unitary element which are J*B-algebras in the sense of [1].

2. The results

Let X be a Banach space. Through the paper we denote by B_{X}, S_{X}, and X^{∗} the closed unit ball, the unit sphere, and the dual space, respectively, of X. I_{X} will denote the identity operator on X, and if Y is another Banach space, then BL(X, Y) stands for the Banach space of all bounded linear operators from X to Y. We usually write BL(X) instead of BL(X, X).

Let E be a J*B-triple with a unitary element u. Then E is a unital real Jordan Banach algebra with product x ∘ y := {x, u, y} and involution x^{∗} := {u, x, u} (compare [9, Proposition 19.13]). By (J*B 2) and (J*B 1) it follows that ∥x^{∗}∥ = ∥x∥ and ∥x∥^{3} = ∥U_{x}(x^{∗})∥ for all x ∈ E, where U_{x}(y) := 2x ∘ (x ∘ y) − x^{2} ∘ y. Throughout the paper we will denote by E_{sym} the set of all ∗-symmetric elements in E.

Let X be a Banach space, and u a norm-one element in X. The set of states of X relative to u, D(X, u), is defined as the non empty, convex, and weak*-compact subset of X^{∗} given by

D(X, u) := {Φ ∈ B_{X^{∗}} : Φ(u) = 1}.

For x ∈ X, the numerical range of x relative to u, V (X, u, x), is given by V (X, u, x) := {Φ(x) : Φ ∈ D(X, u)}. The numerical radius of x relative to u, v(X, u, x), is given by

v(X, u, x) := max{|λ| : λ ∈ V (X, u, x)}.

It is well known that a bounded linear operator T on a complex Banach space X is hermitian if and only if V (BL(X), I_{X}, T) ⊆ ℝ (compare [2, Corollary 10.13]).

It is well known that for an element a of a unital Banach algebra A with unit u the following assertions hold

(a) V (A, u, a) = V (BL(A), I_{A}, L_{a}) = V (BL(A), I_{A}, R_{a}), where L_{a} (respectively, R_{a}) is the map given by x ↦ ax (respectively x ↦ xa);

(b) If B is a subalgebra of A with {a, u} ⊂ B, then V (B, u, a) = V (A, u, a).

Remark 2.1. Let X be a complex Banach space and τ a conjugation on X. We define a conjugation τ̃ on BL(X) by τ̃(T) := τT τ. If T is a τ̃-invariant element of BL(X), then we have T (X^{τ}) ⊆ X^{τ}, and hence we can consider Λ(T) := T |_{X^{τ}} as a bounded linear operator on the real Banach space X^{τ}. Since the mapping Λ : BL(X)τ̃ → BL(X^{τ}) is a linear contraction sending I_{X} to I_{X^{τ}}, we get

V (BL(X^{τ}), I_{X^{τ}} , Λ(T)) ⊆ V (BL(X)τ̃, I_{X}, T)

for all T ∈ BL(X)τ̃. On the other hand, by the Hahn-Banach Theorem, we have

V (BL(X)τ̃, I_{X}, T) = V (BL(X)_{ℝ}, I_{X}, T)

for every T ∈ BL(X)τ̃. It follows

V (BL(X^{τ}), I_{X^{τ}} , Λ(T)) ⊆ ℜe V (BL(X), I_{X}, T)

for all T ∈ BL(X)τ̃.

By [8, Lemma 1.6] we know that if we denote by H the real Banach space of all hermitian operators on X which lie in BL(X)τ̃, then for every Φ ∈ D(BL(X), I_{X}), there exists Ψ ∈ D(BL(X^{τ}, I_{X^{τ}}) such that Φ(T) = Ψ(Λ(T)) for every T in H. As a consequence

V (BL(X), I_{X}, T) = V (BL(X^{τ}), I_{X^{τ}} , Λ(T))

for every T ∈ H.

Let E be a J*B-triple with a unitary element. The next proposition allows us to calculate the norm of a symmetric element x ∈ E from the numerical radius of the map L(x, x).

Proposition 2.2. Let E be a J*B-triple with a unitary element u. Then for every x ∈ E_{sym} the following assertions hold

V (BL(E), I_{E}, L(x, x)) ⊂ [0, +∞),

and

∥x∥^{2} = ∥x^{2}∥ = v(E, u, x^{2}) = v(BL(E), I_{E}, L(x, x)).

Moreover

∥2x ∘ x^{∗}∥ = v(E, u, L_{2x∘x^{∗}}) = v(BL(E), I_{E}, L(x, x) + L(x^{∗}, x^{∗}))

for all x ∈ E.

Proof. Let x ∈ E_{sym}. We denote by C(x) := \overline{Span}^{∥.∥}\{x^{n} : n = 0, 1, 2, …} the closed subalgebra of E generated by x and u. Since x^{∗} = x we conclude that C(x) is a commutative closed subtriple of E. By [3, Remark 1.5], it follows that C(x) is a commutative J*Btriple with a unitary element. Now Theorem 3.11 in [3] assures us that there exists a commutative complex JB*-triple B(x) and a conjugation τ on B(x) such that C(x) = B(x)^{τ}. Since B(x) is a commutative complex JB*-triple with a unitary element u, we can conclude that B(x) is a commutative C*-algebra with product z ∘ y = {z, u, y} and involution z^{∗} = {u, z, u}. Moreover, the conjugation τ on B(x) satisfies τ(z^{∗}) = (τ(z))^{∗} for all z ∈ B(x). By Remark 2.1, the identity L(x, x) = L_{x^{2}}, and since C(x) is a subalgebra of E containing {x, u}, it follows that

V (BL(B(x)), I_{B(x)}, L(x, x)) = V (BL(C(x)), I_{C(x)}, L(x, x))
= V (BL(C(x)), I_{C(x)}, L_{x^{2}}) = V (C(x), u, x^{2}) = V (E, u, x^{2})
(1)  = V (BL(E), I_{E}, L(x, x)).

Since B(x) is a complex JB*-triple, and hence L(x, x) is hermitian and has non-negative spectrum, it follows from (1) and [2, Lemma 38.3] that

V (BL(E), I_{E}, L(x, x)) = V (BL(B(x)), I_{B(x)}, L(x, x)) ⊂ [0, +∞).

Since B(x) is a commutative C*-algebra and x ∈ B(x) is an hermitian element (x^{∗} = {u, x, u} = x), Sinclair’s theorem (c.f. [2, Theorem 10.17]) and equality (1) assure that

∥L(x, x)∥ = ∥x∥^{2} = ∥x^{2}∥ = v(B(x), u, x^{2})
= v(BL(B(x)), I_{B(x)}, L_{x^{2}}) = v(BL(B(x)), I_{B(x)}, L(x, x))
= v(BL(E), I_{E}, L(x, x)) = v(E, u, x^{2}).

Finally, let x ∈ E. From the equality L(x, x) + L(x^{∗}, x^{∗}) = L_{2x∘x^{∗}} and following the arguments given above, we conclude that

∥2x ∘ x^{∗}∥ = v(B(x ∘ x^{∗}), u, 2x ∘ x^{∗}) = v(BL(B(x ∘ x^{∗})), I_{B(x∘x^{∗})}, L_{2x∘x^{∗}})

= v(BL(C(x ∘ x^{∗})), I_{C(x∘x^{∗})}, L_{2x∘x^{∗}}) = v(C(x ∘ x^{∗})), u, 2x ∘ x^{∗})

= v(E, u, 2x ∘ x^{∗}) = v(BL(E), I_{E}, L(x, x) + L(x^{∗}, x^{∗})).

□

The following corollary shows that the symmetric part of a J*B-triple with a unitary element is a JB-algebra.

Corollary 2.3. Let E be a J*B-triple with a unitary element u. Then E_{sym} is a JB-algebra.

Proof. By Proposition 2.2 the equality

(2)  ∥x∥^{2} = v(BL(E), I_{E}, L(x, x)) = v(E, u, x^{2})

holds for all x ∈ E_{sym}.

Now let x, y ∈ E_{sym} and ϕ ∈ S_{E^{∗}} with ϕ(u) = 1. By Proposition 2.2, we also have V (E, u, z^{2}) = V (BL(E), I_{E}, L(z, z) = L_{z^{2}}) ⊂ [0, +∞), for all z ∈ E_{sym}. In particular ϕ(z^{2}) ≥ 0 for all z ∈ E_{sym}. Thus, it follows that

∥x^{2} + y^{2}∥ ≥ ϕ(x^{2} + y^{2}) ≥ ϕ(x^{2}).

Finally, taking supreme on {ϕ ∈ S_{E^{∗}} : ϕ(u) = 1}, we conclude by (2) that

∥x^{2} + y^{2}∥ ≥ ∥x^{2}∥,

which shows that E_{sym} is a JB-algebra. □

Remark 2.4. Let E be a J*B-triple. Suppose that the complexification of E admits a complex JB*-triple structure with a norm extending the original norm of E. Then E is clearly a real JB*-triple. Let us denote by Ê := E ⊕ iE the complexification of E, by τ the canonical conjugation on Ê satisfying E = Ê^{τ} and let x ∈ E. Since Ê is a complex JB*-triple, it follows that L(x, x) is an hermitian operator with non-negative spectrum, and hence, by [2, Lemma 38.3],

V (BL(Ê), IÊ, L(x, x)) ⊂ [0, +∞).

By Remark 2.1 we also have

V (BL(E), I_{E}, L(x, x)) = V (BL(Ê), IÊ, L(x, x)) ⊂ [0, +∞).

Thus it seems natural to add the axiom

(3)  V (BL(E), I_{E}, L(x, x)) ⊂ [0, +∞) for all x ∈ E

to the structure of the J*B-triple E, in order to show that E is also a real JB*-triple.

We can now state our main result which shows that a J*B-triple E with a unitary element is a real JB*-triple if and only if E satisfies axiom (3). Previously, we recall (c.f. [1]) that a J*B-algebra is a real Jordan algebra A with unit and an involution ^{∗} equipped with a complete algebra norm such that ∥U_{x}(x^{∗})∥ = ∥x∥^{3} and ∥x^{∗}x∥ ≤ ∥x^{∗}x + y^{∗}y∥ for all x, y ∈ A.

Theorem 2.5. Let E be a J*B-triple with a unitary element u. The following assertions are equivalent

(a) V (BL(E), I_{E}, L(x, x)) ⊂ [0, +∞) for all x ∈ E;

(b) E is a J*B-algebra with product x∘y := {x, u, y} and involution x^{∗} := {u, x, u};

(c) E is a real JB*-triple;

(d) The complexification of E is a complex JB*-triple in some norm extending the original norm on E.

Proof. (a) ⇒ (b) As we have seen before, E is a real Jordan Banach algebra with product x ∘ y := {x, u, y} and involution x^{∗} := {u, x, u}. We have also seen that ∥U_{x}(x^{∗})∥ = ∥x∥^{3}, for all x ∈ E. Hence only the inequality

∥x^{∗} ∘ x∥ ≤ ∥x^{∗} ∘ x + y^{∗} ∘ y∥

has to be shown for all x, y ∈ E.

By Proposition 2.2, it follows that

(4) ∥2z ∘ z^{∗}∥ = v(E, u, 2z ∘ z^{∗}) = v(BL(E), I_{E}, L(z, z) + L(z^{∗}, z^{∗}))

for all z ∈ E.

Let ϕ ∈ S_{E^{∗}} with ϕ(u) = 1. By hypothesis, V (BL(E), I_{E}, L(x, x)) is contained in [0, +∞) for all x ∈ E, then we can conclude by the identity L(x, x) + L(x^{∗}, x^{∗}) = L_{2x∘x^{∗}} that ϕ(x ∘ x∗) ≥ 0 for all x ∈ E. Therefore, given x, y ∈ E we have

∥x ∘ x^{∗} + y ∘ y^{∗}∥ ≥ ϕ(x ∘ x^{∗} + y ∘ y^{∗}) ≥ ϕ(x ∘ x^{∗}).

Finally, taking supreme on {ϕ ∈ S_{E^{∗}} : ϕ(u) = 1}, we conclude by (4) that

∥x^{∗} ∘ x∥ ≤ ∥x^{∗} ∘ x + y^{∗} ∘ y∥.

(b) ⇒ (c) By [1, Theorem 4.4] it follows that the norm on E can be extended to the complexification Ê = E ⊕ iE of E such that (Ê, ∘,^{∗} , ∥.∥) is a JB*-algebra. Actually [1, Theorem 4.4] shows that E is a real form of its complexification. Since every JB*-algebra is a complex JB*-triple ([9, Proposition 20.35]), we conclude that E is a real JB*-triple.

The implications (c) ⇒ (d) and (d) ⇒ (a) follow from [4, Proposition 2.2] and Remark 2.4, respectively.

□

The next definition is motivated by the above theorem.

Definition 2.6. An axiomatic JB*-triple is a J*B-triple E satisfying the following fifth axiom (J*B 5) V (BL(E), I_{E}, L(x, x)) ⊂ [0, +∞) for all x ∈ E.

Clearly every real JB*-triple is an axiomatic JB*-triple and every axiomatic JB*-triple is a J*B-triple. By Theorem 2.5 we know that if E is a Banach Jordan triple with a unitary element then problem (P 1) has an affirmative solution for E if and only if E is a real JB*-triple if and only if E is an axiomatic JB*-triple.

It seems natural to ask if we have found an axiomatic definition of real JB*-triples, but for the moment, we do not know if every axiomatic JB*-triple is a real JB*-triple.

Now we deal with problem (P 2). By [4, Lemma 4.2 and Theorem 4.4] we know that the bidual of a real JB*-triple is a real JB*-triple with a separate weak*-continuous triple product. Thus the following corollary is derived from Theorem 2.5.

Corollary 2.7. Let E be a J*B-triple with a unitary element. Suppose that V (BL(E), I_{E}, L(x, x)) ⊂ [0, +∞) for all x ∈ E, i.e., E is an axiomatic JB*-triple with a unitary element. Then the bidual of E is again a J*B-triple with a unitary element and separately weak*- continuous triple product.

If E is a real JB*-triple which is also a Banach dual space, then E has a unique predual and the triple product of E is separately weak*- continuous (c.f. [7]). Now applying Theorem 2.5, we obtain the following corollary.

Corollary 2.8. Let E be an axiomatic JB*-triple with a unitary element. Suppose that E is a Banach dual space. Then E has a unique predual and the triple product of E is separately weak*-continuous.

We leave open the following problems (P 3) Is the complexification of an axiomatic JB*-triple a complex

JB*-triple in some norm extending the original norm? This problem is equivalent to ask whether an axiomatic JB*-triple is a real JB*-triple.

(P 4) Is the bidual of an axiomatic JB*-triple an axiomatic JB*-triple

with a separately weak*-continuous triple product? (P 5) Let E be an axiomatic JB*-triple which is also a Banach dual

space. Does E have a unique predual? Is the triple product on E separately weak*-continuous?

We have shown that problems (P 3), (P 4), and (P 5) have an affirmative solution whenever E has a unitary element.

References

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

[2] Bonsall, F. F. and Duncan, J.: Complete Normed Algebras, Springer-Verlag, New York 1973.

[3] Dang, T. and Russo, B.: Real Banach Jordan triples, Proc. Amer. Math. Soc. 122, 135-145 (1994).

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

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

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

[7] Martínez, J. and Peralta A.M.: Separate weak*-continuity of the triple product in dual real JB*-triples, Math. Z., 234, 635-646 (2000).

[8] Peralta, A. M. and Rodríguez, A.: Grothendieck’s inequalities revisited, In Recent Progress in Functional Analysis: Proceedings of the International Functional Analysis Meeting on the Occasion of the 70th Birthday of Professor Manuel Valdivia, Valencia, Spain, July 3-7, 2000, (ed. by K.D. Bierstedt, J. Bonet, M. Maestre, J. Schmets), to appear. North-Holland 2001.

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

Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.

E-mail address: aperalta@goliat.ugr.es