On Summing Operators on JB*-triples

Antonio M. Peralta^{∗}

Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain. E-mail: aperalta@goliat.ugr.es

Abstract

In this paper we introduce 2-JB*-triple-summing operators on real and complex JB*-triples. These operators generalize 2-C*-summing operators on C*-algebras. We also obtain a Pietsch’s factorization theorem in the setting of 2-JB*-triple-summing operators on JB*- triples.

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

1 Introduction.

Let X, Y be Banach spaces, 0 < p < ∞, and T : X → Y a bounded linear operator. We say that T is p-summing if there is a constant C ≥ 0 such that for any finite sequence (x_{1}, …, x_{n}) of X we have

(∑_{k=1}^{n} ∥T (x_{k})∥^{p})^{\frac{1}{p}} ≤ C sup \{(∑_{k=1}^{n} |f(x_{k})|^{p})^{\frac{1}{p}} : f ∈ X^{∗}, ∥f∥ ≤ 1\} .

\underline{2}

In 1978, G. Pisier [20] introduced the following extension of the p-summing operators in the setting of C*-algebras. Let T be a bounded linear operator from a C*-algebra A to a Banach space Y, and 0 < p < ∞. We say that T is p-C*-summing if there exists a positive constant C such that for any finite sequence (a_{1}, …, a_{n}) of hermitian elements of A we have

∗Partially supported by D.G.I.C.Y.T. project no. PB 98-1371, and Junta de Andalucía grant FQM 0199

(∑_{i=1}^{n} ∥T (a_{i})∥^{p})^{\frac{1}{p}} ≤ C ∥ (∑_{i=1}^{n} |a_{i}|^{p})^{\frac{1}{p}} ∥,  (1)

where, for x in A, the “modulus” is defined by |x|^{2} := \frac{1}{2}(xx^{∗} + x^{∗}x). The smallest constant C for which (1) holds is denoted C_{p}(T). It is well known that every p-summing operator from a C*-algebra to a Banach space is p-C*-summing but the converse is false in general (compare [20, Remark 1.2]).

In [20] G. Pisier proved a Pietsch’s factorization theorem for p-C*-summing operators. Indeed, if T : A → Y is a p-C*-summing operator from a C*- algebra to a complex Banach space then there is a norm-one positive linear functional φ in A^{∗} such that

∥T (x)∥ ≤ C_{p}(T) (φ(|x|^{p}))^{\frac{1}{p}}

for every hermitian element x in A.

Complex JB*-triples were introduced by W. Kaup in the study of Bounded Symmetric Domains in complex Banach spaces ([15], [14]). The class of complex JB*-triples includes all C*-algebras and all JB*-algebras.

The aim of this paper is the study of summing operators on real and complex JB*-triples. In Section 2 we introduce the natural definition of p-JB*-summing operators in the setting of JB*-algebras. We obtain a Pietsch’s factorization theorem for p-JB*-summing operators. Section 3 deals with the definition and study of 2-JB*-triple-summing operators in the setting of complex JB*-triples. Operators which generalize 2-C*-summing and 2-JB*- summing operators on C*-algebras and JB*-algebras, respectively. For the most general class of 2-JB*-triple-summing operators, we obtain a Pietsch’s factorization theorem in the setting of JB*-triples (Theorems 3.5 and 3.6). It is worth mentioning that in the proof of this result, the so-called ”Little Grothendieck’s inequality” for JB*-triples [18] play a very important role. In the last section we establish analogous results in the setting of real JB*- triples. We also discuss the relations between 2-summing and 2-JB*-triple-summing operators.

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, J_{X} the natural embedding of X in its bidual 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).

2 Summing Operators on JB*-algebras

Let A be a JB*-algebra. Given x ∈ A, the modulus |x| is defined by |x|^{2} := x ∘ x^{∗} for all x ∈ A. Given a norm-one positive linear functional ψ ∈ A^{∗}, the mapping (x, y) ↦ (x/y)_{ψ} := ψ(x ∘ y^{∗}) is a positive sesquilinear form on A. If we denote N_{ψ} := {x ∈ A : ψ(x ∘ x^{∗}) = 0}, then the quotient A/N_{ψ} can be completed to a Hilbert space, which is denoted by H_{ψ}. The natural quotient map of A on H_{ψ} is denoted by J_{ψ}. Inspired by the definition of p-C*-summing operators, we introduced the following concept of p-JB*-summing operator in the setting of JB*-algebras.

Definition 2.1. Let 0 < p < ∞. A bounded linear operator T from a JB*- algebra A to a Banach space Y is said to be p-JB*-summing if there exists a positive constant C such that for any finite sequence (a_{1}, …, a_{n}) of hermitian elements of A we have

(∑_{i=1}^{n} ∥T (a_{i})∥^{p})^{\frac{1}{p}} ≤ C ∥ (∑_{i=1}^{n} |a_{i}|^{p})^{\frac{1}{p}} ∥.  (2)

The smallest constant C for which (2) holds is denoted C_{p}(T).

Remark 2.2. Let A be a C*-algebra. Then A is a JB*-algebra with respect x ∘ y := \frac{1}{2}(xy + yx). In this case, it is easy to see that a bounded linear operator from A to a Banach space is p-C*-summing if and only if it is p-JB*-summing.

The next result is an extension of Pietsch’s factorization theorem ([9]) in the JB*-algebra setting which is a verbatim extension of Pisier’s analogous result for C*-algebras [20, Proposition 1.1].

Proposition 2.3. Let A be a JB*-algebra, Y a Banach space, and T : A → Y a p-JB*-summing operator. Then there exists a norm-one positive linear functional φ on A such that

∥T (x)∥ ≤ C_{p}(T) (φ(|x|^{p}))^{\frac{1}{p}}

for every hermitian element x in A.

Proof. Let K denote the set of all positive linear functionals on A with norm less or equal to 1. Then K is a convex σ(A^{∗}, A)-compact subset of A^{∗} and

∥a∥ = sup_{f∈K} |f(a)|  (3)

for every hermitian element a ∈ A [12].

Let us denote by C the set of all continuous functions on K of the form

F_{{a_{1},... ,a_{n}}}(f) := C_{p}(T)^{p} f(∑_{i=1}^{n} |a_{i}|^{p}) − ∑_{i=1}^{n} ∥T a_{i}∥^{p},

where (a_{1}, …, a_{n}) is a finite collection of hermitian elements in A. Then C is a convex cone in C(K). Moreover, since T is p-JB*-summing, (3) assures that C is disjoint from the open cone O := {Φ ∈ C(K) : max Φ < 0}. By the Hahn-Banach theorem there exists a positive measure λ on K such that

∫_{K} F_{{a_{1},... ,a_{n}}}(k)λ(dk) ≥ 0

for every finite collection of hermitian elements (a_{1}, …, a_{n}) ∈ A. We can suppose that λ is a probability measure on K. Finally taking φ(x) := ∫_{K} k(x)λ(dk) (x ∈ A), we finish the proof.

Remark 2.4. Let A, Y, and T be as in Proposition 2.3 above with p = 2. If x ∈ A, then x = a + ib with a^{∗} = a, b^{∗} = b, and hence |x|^{2} = a^{2} + b^{2}. Therefore

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

The next result is a weak* version of Pietsch factorization theorem for JB*-algebras (Proposition 2.3) and a extension of [21, Lemma 4.1] in the JBW*-algebra setting. We recall that a JBW*-algebra is a JB*-algebra which is also a dual Banach space [12].

Proposition 2.5. Let A be a JBW*-algebra, Y a Banach space, and T : A → Y^{∗} a p-JB*-summing operator which is also weak*- continuous. Then there exists a norm-one positive linear functional φ in A_{∗} such that

∥T (x)∥ ≤ C_{p}(T) (φ(|x|^{p}))^{\frac{1}{p}}

for every hermitian element x in A.

Proof. By [12, 4.4.17] there exists a central projection e ∈ A^{∗∗} such that

L_{e} : A → e ∘ A^{∗∗}
L_{e}(x) := e ∘ x

is an isomorphism and A_{∗} = L_{e}^{∗}(A^{∗}). By Proposition 2.3, there exists a norm-one positive linear functional ψ ∈ A^{∗} such that

∥T (x)∥ ≤ C_{p}(T) (ψ(|x|^{p}))^{\frac{1}{p}}

for every hermitian element x in A. Now we take φ := L_{e}^{∗}(ψ) ∈ A_{∗}, f ∈ S_{Y} and compute

< f, T (x) >=< T^{∗}(f), x >=< L_{e}T^{∗}(f), x >≤ ∥T^{∗∗}(e ∘ x)∥
≤ C_{p}(T) (ψ(|e ∘ x|^{p}))^{\frac{1}{p}} = C_{p}(T) (ψ(e ∘ |x|^{p}))^{\frac{1}{p}} = C_{p}(T) (φ(|x|^{p}))^{\frac{1}{p}} .

Finally taking supremum over f ∈ S_{Y} we finish the proof.

3 Summing Operators on JB*-triples

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 non negative spectrum for all a in E;

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

We recall that a bounded linear operator on a complex Banach space is said to be hermitian if ∥ exp(iλT)∥ = 1 for all λ ∈ ℝ.

It is worth mentioning that every C*-algebra is a (complex) JB*-triple with respect to \{a, b, c\} = \frac{1}{2}(ab^{∗}c + cb^{∗}a) and also every JB*-algebra with respect to {a, b, c} = (a ∘ b^{∗}) ∘ c + (c ∘ b^{∗}) ∘ a − (a ∘ c) ∘ b^{∗}. We refer to [23], [24] and [8] for recent surveys on the theory of JB*-triples. A JBW*-triple is a JB*-triple which is a dual Banach space. If E is a JB*-triple then E^{∗∗} is a JBW*-triple [11]. It is well known that every JBW*-triple has a unique predual and the triple product is separately weak*-continuous [3].

Let E be a JB*-triple and φ a norm-one functional in E^{∗}. By [1, Proposition 1.2] the map (x, y) ↦ φ {x, y, z} is a positive sequilinear form on E, where z is any norm-one element of E^{∗∗} verifying φ(z) = 1 (If E is a JBW*- triple and φ ∈ S_{E_{∗}}, then z can be chosen in S_{E}).

If we define N_{φ} := {x ∈ E : ∥x∥_{φ} = 0}, the completion H_{φ} of E/N_{φ} is a Hilbert space with respect to the norm ∥.∥_{φ}. Throughout the paper the natural quotient map of E on H_{φ} will be denoted by J_{φ}.

Let W be a JBW*-triple. The strong* topology of W, denoted by S^{∗}(W, W_{∗}), is the topology on W generated by the family {∥.∥_{φ} : φ ∈ S_{W_{∗}}}.

The following definition is the natural extension of the 2-summing operators in the setting of JB*-triples.

Definition 3.1. Let E be a JB*-triple and Y a Banach space. An operator T : E → Y is said to be 2-JB*-triple-summing if there exists a positive constant C such that for every finite sequence (x_{1}, …, x_{n}) of elements in E we have

∑_{i=1}^{n} ∥T (x_{i})∥^{2} ≤ C ∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥.  (4)

The smallest constant C for which (4) holds is denoted C_{2}(T).

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 [5, Corollary 10.13]). If T is a bounded linear operator on X, then we have V (BL(X), I_{X}, T) = \overline{co} W (T) where

W (T) = {x^{∗}(T (x)) : (x, x^{∗}) ∈ Γ},

and Γ ⊆ {(x, x^{∗}) : x ∈ S_{X}, x^{∗} ∈ S_{X^{∗}}, x^{∗}(x) = 1} verifies that its projection onto the first coordinate is norm dense in S_{X} [4, Theorem 9.3]. Moreover, the numerical radius of T can be calculated as follows

v(BL(X), I_{X}, T) = sup{|x^{∗}(T (x))| : (x, x^{∗}) ∈ Γ}.

In particular if X = Y^{∗}, then by the Bishop-Phelps-Bollobás theorem, it follows that

v(BL(X), I_{X}, T) = sup{|x^{∗}(T (x))| : x ∈ S_{X}, x^{∗} ∈ S_{Y} , x^{∗}(x) = 1}.

Remark 3.2. Let A be a JB*-algebra with unit 1, Y a Banach space, and T : A → Y a 2-JB*-triple-summing operator (regarded A as a JB*-triple). We have

∑_{i=1}^{n} ∥T (x_{i})∥^{2} ≤ C_{2}(T) ∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥  (5)

for every finite sequence (x_{1}, …, x_{n}) of elements in A. Since S := ∑_{i=1}^{n} L(x_{i}, x_{i}) is an hermitian operator on A, Sinclair’s theorem [5, Theorem 11.17] assures that

∥S∥ = sup{|Φ(S(z))| : z ∈ S_{A}, Φ ∈ S_{A^{∗}}, Φ(z) = 1}.

It is worth mentioning that Φ(S(z)) ≥ 0 for such Φ and z. Let z ∈ S_{A} and Φ ∈ S_{A^{∗}} with Φ(z) = 1. Let us define Ψ(x) := Φ(x ∘ z), then we have Ψ ∈ S_{A^{∗}}, Ψ(1) = Φ(z) = 1, and

Ψ(L(x, x)(1)) = Φ(L(x, x)(1) ∘ z)

= \frac{1}{2}Φ(\{x, x, z\} + \{x^{∗}, x^{∗}, z\}) = \frac{1}{2}(∥x∥_{Φ} + ∥x^{∗}∥_{Φ})

≥ \frac{1}{2}∥x∥_{Φ} = \frac{1}{2}Φ(L(x, x)(z))

for all x ∈ A. Therefore Φ(S(z)) ≤ 2Ψ(S(1)) and hence

∥S∥ ≤ 2 sup{Ψ(S(z)) : Ψ ∈ S_{A^{∗}}, Φ(1) = 1}

= 2 sup{Ψ(∑_{i=1}^{n} |x_{i}|^{2}) : Ψ ∈ S_{A^{∗}}, Φ(1) = 1} = 2∥ ∑_{i=1}^{n} |x_{i}|^{2}∥.

It follows from (5) that T is 2-JB*-summing. This shows that every 2-JB*- triple-summing operator from a unital JB*-algebra to a Banach space is 2-JB*-summing. Conversely, the inequality

∥ ∑_{i=1}^{n} |x_{i}|^{2}∥ = ∥ ∑_{i=1}^{n} L(x_{i}, x_{i})(1)∥ ≤ ∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥,

shows that every 2-JB*-summing operator from a unital JB*-algebra is also 2-JB*-triple-summing.

In 1987 T. Barton and Y. Friedman [1, Corollary 3.1] established a Ringrose-type inequality for JB*-triples. However the Barton-Friedman proof of this inequality is based in [1, Theorem 1.3], result which has a gap (see [17] and [18]). Now we can follow the same ideas to prove this Ringrosetype inequality, but replacing [1, Theorem 1.3] by [18, Theorem 3]. Given a JB*-triple E and norm-one elements φ_{1}, φ_{2} ∈ E^{∗} we denote by ∥.∥_{φ_{1},φ_{2}} the prehilbert seminorm on E given by ∥x∥_{φ_{1},φ_{2}}^{2} := ∥x∥_{φ_{1}}^{2} + ∥x∥_{φ_{2}}^{2}.

Proposition 3.3. Let E and F be JB*-triples, T : E → F a bounded linear operator, and x_{1}, …, x_{n} in E. Then

∥ ∑_{i=1}^{n} L(T (x_{i}), T (x_{i}))∥ ≤ 2∥T ∥^{2}∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥.

Proof. Let us denote S := ∑_{i=1}^{n} L(T (x_{i}), T (x_{i})). Then S is an hermitian operator on F. By Sinclair’s theorem [5, Theorem 11.17]

∥S∥ = sup{|ψ(S(z))| : z ∈ S_{F}, ψ ∈ S_{F^{∗}}, ψ(z) = 1}.

Note that ψ(S(Z)) > 0 for every such ψ and z. Fix ε > 0 and choose ψ and z such that

∥S∥ ≤ ψ(S(z)) + ε.

The mapping J_{ψ}T : E → H_{ψ} is a bounded linear operator. Let m ∈ ℕ. By [18, Theorem 3], there are norm-one functionals φ_{1,m}, φ_{2,m} ∈ E^{∗} such that

∥J_{ψ}T (x)∥ ≤ (\sqrt{2} + \frac{1}{m})∥T ∥ (∥x∥_{φ_{1,m}}^{2} + \frac{1}{m}∥x∥_{φ_{2,m}}^{2})^{\frac{1}{2}} , i. e.

ψ \{T (x), T (x), z\} ≤ (\sqrt{2} + \frac{1}{m})^{2}∥T ∥^{2} (φ_{1,m} \{x, x, e_{1,m}\} + \frac{1}{m}φ_{2,m} \{x, x, e_{2,m}\})

for all x ∈ E, where e_{1,m}, e_{2,m} ∈ S_{E^{∗∗}} verify φ_{i,m}(e_{i,m}) = 1 for i ∈ {1, 2}. Therefore

∥S∥ − ε ≤ ψ(S(z)) ≤ (\sqrt{2} + \frac{1}{m})^{2}∥T ∥^{2} (φ_{1,m}(∑_{i=1}^{n} L(x_{i}, x_{i})e_{1,m})+

\frac{1}{m}φ_{2,m}(∑_{i=1}^{n} L(x_{i}, x_{i})e_{2,m})) ≤ (\sqrt{2} + \frac{1}{m})^{2}(1 + \frac{1}{m})∥T ∥^{2} ∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥.

Finally, letting ε → 0, m → ∞, we get

∥ ∑_{i=1}^{n} L(T (x_{i}), T (x_{i}))∥ ≤ 2∥T ∥^{2}∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥.

From the above proposition, we immediately obtain the following corollary.

Corollary 3.4. Let E and F be JB*-triples, Y a Banach space, T : F → Y a 2-JB*-triple-summing operator, and R : E → F a bounded linear operator. Then T R : E → Y is a 2-JB*-triple-summing operator.

Now we deal with the following characterization of 2-JB*-triple-summing operators from a JBW*-triple to a complex Banach space which generalizes Pietsch’s factorization theorem for C*-algebras [21, Theorem 3.2].

Theorem 3.5. Let T be a weak*-continuous linear operator from a JBW*- triple W with values in a Banach space Y^{∗}. The following assertions are equivalent.

1. T is 2-JB*-triple-summing.

2. There are norm-one functionals φ_{1}, φ_{2} in W_{∗} and a positive constant

C(T) such that

∥T (x)∥ ≤ C(T)∥x∥_{φ_{1},φ_{2}}

for all x ∈ W.

Proof. 1 ⇒ 2.− By [6, Proposition 2] there is a (unital) JBW*-algebra A and a contractive projection P : A → W. Actually, by [23, Theorem D.20], P can be supposed weak*-continuous. Since, by Corollary 3.4 and Remark 3.2, it follows that T P : A → Y^{∗} is a 2-JB*-summing operator, which is also weak*-continuous, we conclude, by Theorem 2.5, that there exists a norm-one positive functional ψ ∈ A_{∗} such that

∥T P (α)∥ ≤ \sqrt{2 C_{2}(T)}(ψ(α ∘ α^{∗}))^{\frac{1}{2}}

for all α ∈ A.

It is worth mentioning that, by the same arguments given in the proof of [22, Corollary 1], the natural quotient map J_{ψ} is weak*-continuous. Therefore, we can apply [18, Theorem 3] to the restriction J_{ψ}|_{W} : W → H_{ψ} to get norm-one functionals φ_{1}, φ_{2} ∈ W_{∗} such that

ψ(x ∘ x^{∗}) ≤ 4 ∥x∥_{φ_{1},φ_{2}}^{2},

and hence

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

for all x ∈ W.

2 ⇒ 1.− Let (x_{1}, …, x_{n}) be a finite sequence of elements of W, and e_{1}, e_{2} ∈ S_{W} such that φ_{i}(e_{i}) = 1 for i ∈ {1, 2}. Then we have

∑_{i=1}^{n} ∥T (x_{i})∥^{2} ≤ C(T)^{2} ∑_{i=1}^{n} ∥x_{i}∥_{φ_{1},φ_{2}}^{2}

= C(T)^{2} (φ_{1}(∑_{i=1}^{n} L(x_{i}, x_{i})(e_{1})) + φ_{2}(∑_{i=1}^{n} L(x_{i}, x_{i})(e_{2})))

≤ 2C(T)^{2}∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥.

Inequality which shows that T is 2-JB*-triple-summing.

Let E be a complex JB*-triple and Φ ∈ D(BL(E), I_{E}). Since for every x ∈ E, the operator L(x, x) is hermitian and has non-negative spectrum, it follows from [5, Lemma 38.3] that the mapping (x, y) → Φ(L(x, y)) from E × E to ℂ becomes a positive sesquilinear form on E. Then we define the prehilbert seminorm ∥|.∥|_{Φ} on E by ∥|x∥|_{Φ}^{2} := Φ(L(x, x)).

Our next result is the natural extension of Pietsch’s factorization theorem in the setting of JB*-triples.

Theorem 3.6. Let T be a bounded operator from a JB*-triple E with values in a Banach space Y. The following assertions are equivalent.

1. T is 2-JB*-triple-summing.

2. There is a state Ψ ∈ D(BL(E), I_{E}) and a positive constant C(T) such

that

∥T (x)∥ ≤ C(T)∥|x∥|_{Ψ}

for every x ∈ E.

3. There are norm-one functionals φ_{1}, φ_{2} in E^{∗} and a positive constant

C(T)′ such that

∥T (x)∥ ≤ C(T)′∥x∥_{φ_{1},φ_{2}}

for all x ∈ E.

Proof. 1 ⇒ 2.− Let K denote the set of states of BL(E) relative to I_{E}. Then K is a non empty, convex, and weak*-compact subset of BL(E)^{∗}. Moreover, by Sinclair’s theorem [5, Theorem 11.17],

∥T ∥ = sup_{Φ∈K} |Φ(T)|  (6)

for every hermitian operator T on X.

Let us denote by C the set of all continuous functions on K of the form

F_{{x_{1},... ,x_{n}}}(Φ) := C_{2}(T)Φ(∑_{i=1}^{n} L(x_{i}, x_{i})) − ∑_{i=1}^{n} ∥T (x_{i})∥^{2}

where n ∈ ℕ and {x_{1}, …, x_{n}} ⊂ E. Since for every {x_{1}, …, x_{n}} ⊂ E, the map ∑_{i=1}^{n} L(x_{i}, x_{i}) is an hermitian operator on E and T is 2-JB*-triple-summing, (6) assures that C is disjoint from the open cone O := {φ ∈ C(K) : max φ < 0}. Therefore, by the Hahn-Banach theorem there is a probability measure μ on K such that

∫_{K} F_{{x_{1},... ,x_{n}}}(k) μ(dk) ≥ 0

for every finite collection of elements {x_{1}, …, x_{n}} ∈ E. Finally taking Ψ(T) := ∫_{K} T (k) μ(dk) we obtain 2.

2 ⇒ 3.− Let Ψ the state given in 2. The map ∥|.∥|_{Ψ} is a pre-Hilbert seminorm on E. Denoting N := {x ∈ E : ∥|x∥|_{Ψ} = 0}, then the quotient E/N can be completed to a Hilbert space H. Let us denote by Q the natural quotient map from E to H. By [18, Corollary 1] (see also [19, Corollary 1.11]) there are norm-one functionals φ_{1}, φ_{2} ∈ E^{∗} such that the inequality

∥Q(x)∥ = ∥|x∥|_{Ψ} ≤ 2∥x∥_{φ_{1},φ_{2}}

holds for every x ∈ E. Then it follows that

∥T (x)∥ ≤ 2C(T)∥x∥_{φ_{1},φ_{2}}

for every x ∈ E.

The implication 3 ⇒ 1.− follows as (2 ⇒ 1) in Theorem 3.5.

Let T : E → Y be a 2-JB*-triple-summing operator from a JB*-triple to a Banach space. By Theorem 3.6 above, there are norm-one functionals φ_{1}, φ_{2} in E^{∗} and a positive constant C(T)′ such that

∥T (x)∥ ≤ C(T)′∥x∥_{φ_{1},φ_{2}}  (7)

for all x ∈ E. Let α ∈ E^{∗∗}. Since by [2, Theorem 3.2], the strong*-topology of E^{∗∗} is compatible with the duality, it follows that there is a net (x_{λ}) ⊆ E converging to α in the strong*-topology and hence also in the weak*-topology of E^{∗∗}. Since the seminorm ∥.∥_{φ_{1},φ_{2}} is strong*-continuous, by (7) and the weak*-lower semicontinuity of the norm we have

∥T^{∗∗}(α)∥ ≤ C(T)′∥α∥_{φ_{1},φ_{2}}.

Therefore, by Theorem 3.6 we conclude that T^{∗∗} is 2-JB*-triple-summing. We have thus proved the following lemma.

Lemma 3.7. Let T : E → Y be a 2-JB*-triple-summing operator from a JB*-triple to a Banach space. Then there are norm-one functionals φ_{1}, φ_{2} in E^{∗} and a positive constant C(T)′ such that

∥T^{∗∗}(α)∥ ≤ C(T)′∥α∥_{φ_{1},φ_{2}}

for all α ∈ E^{∗∗}. In particular T^{∗∗} is 2-JB*-triple-summing.

Remark 3.8. Let A be a JB*-algebra. By [12, Proposition 3.5.4] A has an increasing approximate identity of hermitian elements, i. e., there is a net (u_{λ})_{Λ} ⊆ A where Λ is a directed set, u_{λ}^{∗} = u_{λ}, ∥u_{λ}∥ ≤ 1, and ∥u_{λ}∘x−x∥ → 0 for every x ∈ A. Then

∥L(x, x)(u_{λ}) − |x|^{2}∥ = ∥|x|^{2} ∘ u_{λ} + (u_{λ} ∘ x^{∗}) ∘ x − (u_{λ} ∘ x) ∘ x^{∗} − |x|^{2}∥ → 0

and hence

∥ ∑_{i=1}^{n} |x_{i}|^{2}∥ = lim_{λ} ∥ ∑_{i=1}^{n} L(x_{i}, x_{i})(u_{λ})∥ ≤ ∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥

for every finite sequence (x_{1}, …, x_{n}) ∈ A. It follows that every 2-JB*- summing operator from A to a Banach space is 2-JB*-triple-summing (regarded A as a JB*-triple). Conversely if T : A → Y is a 2-JB*-triple-summing operator then, by Lemma 3.7, T^{∗∗} : A^{∗∗} → Y^{∗∗} is a 2-JB*-triple-summing operator. Since A^{∗∗} is a unital JBW*-algebra, it follows, by Remark 3.2, that T^{∗∗} (and hence T) is a 2-JB*-summing operator.

4 Summing Operators on real JB*-triples

Real JB*-triples were defined by J. M. Isidro, W. Kaup, and A. Rodríguez [13], as norm-closed real subtriples of complex JB*-triples. In [13], it is shown that given a real JB*-triple E, then there exists a unique complex JB*-triple structure on its complexification Ê = E ⊕ iE and a unique conjugation (conjugate-linear isometry of period 2) τ on Ê such that E = Ê^{τ} := {z ∈ Ê : τ(z) = z}. All JB-algebras, all real C*-algebras and obviously all complex JB*-triples are examples of real JB*-triples. 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 [16], and the bidual E^{∗∗} of a real JB*-triple E is a real JBW*-triple whose triple product extends the one of E [13]. Noticing that every real JBW*-triple is a real form of a complex JBW*-triple [13], 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}. The strong* topology on W, denoted by S^{∗}(W, W_{∗}), is the topology on W generated by the family {∥.∥_{φ} : φ ∈ S_{W_{∗}}}.

As in the complex case, we say that a linear operator T from a real JB*-triple E to a real Banach space Y is 2-JB*-triple-summing if there exists a positive constant C such that for every finite sequence (x_{1}, …, x_{n}) of elements in E we have

∑_{i=1}^{n} ∥T (x_{i})∥^{2} ≤ C ∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥.  (8)

The smallest constant C for which (8) holds is again denoted C_{2}(T).

Let T : E → F be a bounded linear operator between real JB*-triples and let M > \sqrt{2}. Let us consider T̂ : Ê → F̂ the natural complex linear extension of T. By Proposition 3.3

∥ ∑_{k=1}^{n} L(T̂(z_{k}), T̂(z_{k}))∥ ≤ 2M^{2}∥T̂∥^{2}∥ ∑_{k=1}^{n} L(z_{k}, z_{k})∥,

for every finite sequence (z_{1}, …, z_{n}) ⊆ Ê. In particular, the inequality

∥ ∑_{k=1}^{n} L(T (x_{k}), T (x_{k}))∥ ≤ 8M^{2}∥T ∥^{2}∥ ∑_{k=1}^{n} L(x_{k}, x_{k})∥,

holds for every finite sequence (x_{1}, …, x_{n}) ⊆ E. We deduce, as in the complex case, the following result.

Corollary 4.1. Let E and F be real JB*-triples, Y a real Banach space, T : F → Y a 2-JB*-triple-summing operator, and R : E → F a bounded linear operator. Then T R : E → Y is a 2-JB*-triple-summing operator.

Remark 4.2. Let E be a real JB*-triple, Y a real Banach space, and T : E → Y a 2-JB*-triple-summing operator. We denote by Ỹ the complex Banach space Y ⊕ iY equipped with the norm

∥x + iy∥_{c} := sup{∥αx − βy∥ : α + iβ ∈ ℂ with |α + iβ| = 1}.

Then T can be extended to a complex linear operator T̂ : Ê → Ỹ. We claim that T̂ is 2-JB*-triple-summing. Indeed, given (x_{1} + iy_{1}, …, x_{n} + iy_{n}) ⊆ Ê we have

∑_{k=1}^{n} ∥T̂(x_{k} + iy_{k})∥^{2} ≤ 2 ∑_{k=1}^{n} ∥T (x_{k})∥^{2} + ∥T (y_{k})∥^{2}  (9)

≤ C_{2}(T) ∥2 ∑_{k=1}^{n} L(x_{k}, x_{k}) + L(y_{k}, y_{k})∥

Now, since S := ∑_{k=1}^{n} L(x_{k}, x_{k}) + L(y_{k}, y_{k}) is an hermitian operator on Ê it follows, by Sinclair’s theorem, that

∥2S∥ = sup{2Φ(S(z)) : Φ ∈ SÊ^{∗}, z ∈ SÊ, Φ(z) = 1}

= sup{2 ∑_{k=1}^{n} ∥x_{k}∥_{Φ}^{2} + ∥y_{k}∥_{Φ}^{2} : Φ ∈ SÊ^{∗}, z ∈ SÊ, Φ(z) = 1}

= sup{∑_{k=1}^{n} ∥x_{k} + iy_{k}∥_{Φ}^{2} + ∥τ(x_{k} + iy_{k})∥_{Φ}^{2} : Φ ∈ SÊ^{∗}, z ∈ SÊ, Φ(z) = 1}

≤ ∥ ∑_{k=1}^{n} L(x_{k} + iy_{k}, x_{k} + iy_{k})∥ + ∥ ∑_{k=1}^{n} L(τ(x_{k} + iy_{k}), τ(x_{k} + iy_{k}))∥

= 2∥ ∑_{k=1}^{n} L(x_{k} + iy_{k}, x_{k} + iy_{k})∥.

Therefore, we conclude by (9) that T̂ is 2-JB*-triple-summing and C_{2}(T̂) ≤ 2C_{2}(T).

Let E be a real JB*-triple. Following [19], we known that given Φ ∈ D(BL(E), I_{E}) then the mapping (x, y) → Φ(L(x, y)) from E × E to ℝ is a positive symmetric bilinear form on E, and hence ∥|x∥|_{Φ}^{2} := Φ(L(x, x)) defines a prehilbert seminorm on E.

With the help of the previous remark, we can now obtain the following Pietsch’s factorization theorem in the setting of real JB*-triples.

Theorem 4.3. Let T be a linear operator from a real JB*-triple E with values in a real Banach space Y. The following assertions are equivalent.

1. T is 2-JB*-triple-summing.

2. There is a state Ψ ∈ D(BL(E), I_{E}) and a positive constant C(T) such

that

∥T (x)∥ ≤ C(T)∥|x∥|_{Ψ}

for every x ∈ E.

3. There are norm-one functionals φ_{1}, φ_{2} in E^{∗} and a positive constant

C(T)′ such that

∥T (x)∥ ≤ C(T)′∥x∥_{φ_{1},φ_{2}}

for all x ∈ E.

Proof. 1 ⇒ 2.− By Remark 4.2 above, we see that T can be extended to a complex linear operator T̂ : Ê → Ỹ which is also 2-JB*-triple summing, where Ỹ denotes the complexification of Y defined in Remark 4.2. Now by Theorem 3.6 there exists a state Φ ∈ D(BL(Ê), IÊ) and a positive constant C(T̂) such that

∥T (x)∥ ≤ C(T̂)∥|x∥|_{Φ} ≤ 2\sqrt{2C_{2}(T)}∥|x∥|_{Φ}

for every x ∈ E. By [19, Corollary 1.7] there exists Ψ ∈ D(BL(E), I_{E}) such that

∥|x∥|_{Φ} = ∥|x∥|_{Ψ}

for all x ∈ E. Therefore

∥T (x)∥ ≤ 2\sqrt{2C_{2}(T)}∥|x∥|_{Ψ}

for every x ∈ E.

The rest of the proof runs as in Theorem 3.6.

The next lemma can be derived form Theorem 4.3 above as Lemma 3.7 was derived from Theorem 3.6.

Lemma 4.4. Let T : E → Y be a 2-JB*-triple-summing operator from a real JB*-triple to a Banach space. Then there are norm-one functionals φ_{1}, φ_{2} in E^{∗} and a positive constant C(T) such that

∥T^{∗∗}(α)∥ ≤ C(T)∥α∥_{φ_{1},φ_{2}}

for all α ∈ E^{∗∗}. In particular T^{∗∗} is 2-JB*-triple-summing.

Our last goal is to obtain a weak*-version of Theorem 4.3 above. The next remark play a fundamental role in the proof of such result.

Remark 4.5. Let T : W → Y^{∗} be a 2-JB*-triple-summing and weak*- continuous operator form a real JBW*-triple to a dual Banach space. Let us denote by Ŵ and τ the complexification of W and the canonical conjugation τ on Ŵ, respectively. We define

ϕ : Ŵ^{∗} → Ŵ^{∗}

by

ϕ(f)(z) = \overline{f(τ(z))}.

From [13] we can assure that ϕ is a conjugation (conjugate-linear isometry of period 2) on Ŵ^{∗}. Furthermore the map

(Ŵ^{∗})^{ϕ} := {f ∈ Ŵ^{∗} : ϕ(f) = f} → (Ŵ^{τ})^{∗}
f ↦ f|_{W}

is an isometric bijection. In the same way, the predual W_{∗} of W can be identified with (Ŵ_{∗})^{ϕ} := {f ∈ Ŵ_{∗} : ϕ(f) = f}. The construction can be realized one more time to get a conjugation ϕ̂ on Ŵ^{∗∗} such that

W^{∗∗} ≅ (Ŵ^{∗∗})ϕ̂.

Since T is weak*-continuous, there is a bounded linear operator R : W_{∗} → Y such that R^{∗} = T. Let Ỹ denote the complexification of Y defined in Remark 4.2 and R̃ : Ŵ_{∗} → Ỹ the complex linear extension of R. Then T̃ := (R̃)^{∗} : Ŵ → (Ỹ)^{∗} is a weak*-continuous operator extending T to Ŵ and verifying ∥T̃∥ = ∥R̃∥ ≤ 2∥R∥ = 2∥T ∥. Now we can repeat the same arguments given in Remark 4.2 to assure that T̃ is 2-JB*-triple-summing (and C_{2}(T̃) ≤ 2C_{2}(T)).

We can now state the weak*-version of Theorem 4.3.

Theorem 4.6. Let T be a weak*-continuous linear operator from a real JBW*-triple W with values in a real Banach space Y^{∗}. The following assertions are equivalent.

1. T is 2-JB*-triple-summing.

2. There are norm-one functionals φ_{1}, φ_{2} in W_{∗} and a positive constant

C(T) such that

∥T (x)∥ ≤ C(T)∥x∥_{φ_{1},φ_{2}}

for all x ∈ W.

Proof. 1 ⇒ 2.− By Remark 4.5 above, we see that T can be extended to a weak*-continuous operator T̃ : Ŵ → (Ỹ)^{∗} which is also 2-JB*-triple summing, where Ỹ denotes the complexification of Y defined in Remark 4.2. Now by Theorem 3.5 there are norm-one functionals ψ_{1}, ψ_{2} in Ŵ_{∗} and a positive constant C(T̃) such that

∥T (x)∥ ≤ C(T̂) ∥x∥_{ψ_{1},ψ_{2}} ≤ 2\sqrt{2C_{2}(T)} ∥x∥_{ψ_{1},ψ_{2}}  (10)

for all x ∈ W.

Let e_{1}, e_{2} ∈ SŴ with ψ_{1}(e_{1}) = ψ_{2}(e_{2}) = 1. The map (x, y) ↦ (x|y) := ℜe(ψ_{1} {x, y, e_{1}}+ψ_{2} {x, y, e_{2}}) is a positive bilinear form on W. If we denote N := {x ∈ W : (x|x) = 0}, the quotient W/N can be completed to a a Hilbert space, which is denoted by H. The natural quotient map of W on H will be denoted by J_{ψ_{1},ψ_{2}}. We note that, by the same arguments given in the proof of [22, Corollary 1], it may be concluded that J_{ψ_{1},ψ_{2}} is weak*- continuous. Now By [18, Theorem 5] it follows that there exist norm-one functionals φ_{1}, φ_{2} ∈ S_{W_{∗}} such that

∥J_{ψ_{1},ψ_{2}}(x)∥^{2} = ℜe(ψ_{1} {x, x, e_{1}} + ψ_{2} {x, x, e_{2}}) = ∥x∥_{ψ_{1},ψ_{2}}^{2} ≤ 6^{2} ∥x∥_{φ_{1},φ_{2}}^{2}

for all x ∈ W. Therefore, by (10), we conclude that

∥T (x)∥ ≤ 12\sqrt{2C_{2}(T)} ∥x∥_{φ_{1},φ_{2}}

for all x ∈ W.

The implication 2 ⇒ 1.− follows as in Theorem 3.5.

Remark 4.7. Let T : E → Y be a 2-summing operator from a real or complex JB*-triple to a Banach space. Let φ ∈ S_{E^{∗}} and z ∈ S_{E^{∗∗}} satisfying φ(z) = 1. By [2, Proof of Theorem 3.2] we have

|φ(x)| ≤ ∥x∥_{φ} = (φ(L(x, x)z))^{\frac{1}{2}}

for all x in E, and hence

∑_{k=1}^{n} ∥T (x_{k})∥^{2} ≤ C_{2}(T)^{2} sup {∑_{k=1}^{n} f(L(x, x)z) : f ∈ S_{E^{∗}}, z ∈ S_{E}, f(z) = 1}

≤ C_{2}(T)^{2}∥ ∑_{i=1}^{n} L(x_{i}, x_{i})∥,

for every finite sequence (x_{1}, …, x_{n}) ⊆ E. Therefore every 2-summing operator from a real or complex JB*-triple to a Banach space is 2-JB*-triple-summing.

Corollary 4.8. Let T be a 2-summing operator from a real or complex JB*- triple E to a Banach space. Then there are norm-one functionals φ_{1}, φ_{2} in E^{∗} and a positive constant C(T) such that

∥T (x)∥ ≤ C(T)∥x∥_{φ_{1},φ_{2}}

for all x ∈ E.

Let X and Y be Banach spaces. We recall that an operator T : X → Y is said to be of cotype q (2 ≤ q < ∞), if there is a constant C such that for any {x_{1}, …, x_{n}} ⊆ X the inequality

( ∑_{j=1}^{n} ∥T (x_{j}∥^{q})^{\frac{1}{q}} ≤ C ( ∫_{D} ∥ ∑_{j=1}^{n} ε_{j}x_{j}∥^{2}d(μ))^{\frac{1}{2}}

holds, where ε_{j} ∈ {−1, 1}; D = {−1, 1}^{ℕ} and μ is the uniform probability measure on D. A Banach space X is said to be of cotype q if I_{X} is of cotype q. By [21, page 120], we know that if X is a Banach space of cotype q then I_{X} is (q, 1)−summing, i. e., there is a constant C such that, for all finite sequences (x_{i}) in X, we have

( ∑_{i=1}^{n} ∥x_{i}∥^{q})^{\frac{1}{q}} ≤ C sup\{∑_{i=1}^{n} |ξ(x_{i})| : ξ ∈ X^{∗}, ∥ξ∥ ≤ 1\}.

In general it can not be expected that if Y is a Banach space of cotype 2 then I_{Y} could be 2-summing. However, as we are showing in what follows, if Y is a Banach space of cotype 2 then every bounded linear operator from a real or complex JB*-triple to Y is always 2-JB*-triple-summing. Indeed, in [7, Theorem 12] C-H. Chu, B. Iochum and G. Loupias show that if T : E → Y is a bounded linear operator from a JB*-triple to a Banach space of cotype 2, then there are norm-one functionals φ_{1}, φ_{2} in E^{∗} and a positive constant C(Y) (depending only on Y) such that

∥T (x)∥ ≤ C(Y)∥T ∥ ∥x∥_{φ_{1},φ_{2}}  (11)

for all x ∈ E. Therefore, we conclude by Theorem 3.6 that T is 2-JB*-triple-summing. We have thus proved the following corollary.

Corollary 4.9. Every bounded linear operator from a real or complex JB*- triple to a Banach space of cotype 2 is 2-JB*-triple-summing.

Remark 4.10. It is worth mentioning that in [7, Theorem 12] the authors affirm that if T : E → Y is a bounded linear operator from a JB*-triple to a Banach space of cotype 2, then there exists a norm-one functional φ in E^{∗} and a positive constant C(Y) (depending only on Y) such that

∥T (x)∥ ≤ C(Y)∥T ∥ ∥x∥_{φ}

for all x ∈ E. In the proof of this theorem, the result [7, Proposition 4] ([1, Theorem 1.3]) play a fundamental role. Since, as we have mentioned before, the proof of the last result contains some subtle difficulties (compare [17, 18]), the original setting of [7, Theorem 12] is only a conjecture. However, when in the Chu-Iochum-Loupias proof, [18, Theorem 3] (see also [19, Corollary 1.11]) replaces [7, Proposition 4] we obtain the statement in (11).

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