DERIVATIONS ON REAL AND COMPLEX JB*-TRIPLES

TONY HO, JUAN MARTíNEZ MORENO,

ANTONIO M. PERALTA, AND BERNARD RUSSO

Abstract

We prove that the real JB*-triple of all bounded linear operators from one real Hilbert space to another of a different dimension has an outer derivation, provided it is infinite dimensional. We prove that an infinite dimensional real spin factor has an outer derivation. We show that the corresponding complexifications also have outer derivations. We prove that every other real or complex Cartan factor has the inner derivation property, that is, all derivations are inner.

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

1. Introduction

At the regional conference held at the University of California, Irvine in 1985 [24], Harald Upmeier posed three basic questions regarding derivations on JB*- triples:

Q1: Are derivations automatically bounded?

Q2: When are all bounded derivations inner?

Q3: Can bounded derivations be approximated by inner derivations?

These three questions had all been answered in the binary cases. Q1 was answered affirmatively by Sakai [17] for C^{∗}-algebras and by Upmeier [23] for JBalgebras. Q2 was answered by Sakai [18] and Kadison [12] for von Neumann algebras and by Upmeier [23] for JW -algebras. Q3 was answered by Upmeier [23] for JB-algebras and follows trivially from the Kadison-Sakai answer to Q2 in the case of C^{∗}-algebras.

In the ternary case, both Q1 and Q3 were answered by Barton and Friedman in [3] for complex JB^{∗}-triples. In this paper we consider Q2 for real and complex JBW^{∗}-triples and Q1 and Q3 for real JB^{∗}-triples. A real or complex JB^{∗}-triple is said to have the inner derivation property if every derivation on it is inner. By pure algebra, every finite dimensional JB^{∗}-triple has the inner derivation property. Our main results, Theorems 2, 3, and 4 and Corollaries 2 and 3 determine which of the infinite dimensional real or complex Cartan factors have the inner derivation property.

Second author partially supported by D.G.I.C.Y.T. project no. PB 98-1371, and Junta de Andalucía grant FQM 0199. Third author supported by Programa Nacional F.P.I., Ministry of Education and Science grant and D.G.I.C.Y.T., project no. PB98-1371

1991 Mathematics Subject Classification 17C65, 46K70, 46L05, 46L10, and 46L70..

2 t. ho, j. martínez moreno, a. m. peralta, and b. russo

2. Background

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.

A JB-algebra is a complete normed Jordan real algebra (say A) satisfying the following two additional conditions for a, b ∈ A

(i) ∥a^{2}∥ = ∥a∥^{2} (ii) ∥a∥ ≤ ∥a + b∥. It is due to Wright (see [25]) that the complexification of a JB-algebra is a

JB*-algebra under a unique norm extending the given norm on the JB-algebra. Conversely, the self-adjoint part of a JB*-algebra is a JB-algebra under the restricted norm.

If H is a complex Hilbert space then the real Banach space H(H) of all bounded hermitian operators on H is a JB-algebra with respect to the Jordan product

x ∘ y := \frac{1}{2}(xy + yx).

A uniformly (respectively, weakly) closed unital real subalgebra of H(H) is called a JC-algebra (respectively, JW-algebra) on H. A norm (respectively, weakly) closed (complex) Jordan*-subalgebra of a C*-algebra (respectively, von Neumann algebra) is called a JC*-algebra (respectively, JW*-algebra). For more details on JB- and JB*-algebras we refer the reader to [9].

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

(i) (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 J , where L(a, b)x := {a, b, x};

(ii) For all a ∈ J, the map L(a, a) from J to J is an hermitian operator with

non-negative spectrum; (iii) ∥{a, a, a}∥ = ∥a∥^{3} for all a in J . 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). Also, every JB*-algebra is a JB*-triple with respect to {a, b, c} = (a∘b^{∗})∘c+(c∘b^{∗})∘a−(a∘c)∘b^{∗}. Conversely, every JB*-triple with a unitary element u (that is, {u, u, z} = z for every z) is a unital JB*-algebra with product a ∘ b = {a, u, b}, involution a^{∗} = {u, a, u}, and unit u. We refer to [15], [16] and [5] for recent surveys on the theory of JB*-triples.

Following [11], we recall that a real JB*-triple is a norm-closed real subtriple of a complex JB*-triple. Given a real JB*-triple J there exists a unique complex JB*-triple structure on the complexification Ĵ = J ⊕ i J, and a unique conjugation (i.e., conjugate-linear isometry of period 2) τ on Ĵ such that J = Ĵ^{τ} := {x ∈ Ĵ: τ(x) = x}. From this point of view, the real JB*-triples are real forms of complex JB*-triples.

The class of real JB*-triples includes all JB-algebras [9], all real C*-algebras [8], and all J*B-algebras [2].

A triple derivation or simply a derivation δ on a real or complex JB*-triple U is a linear operator satisfying

δ {a, b, c} = {δa, b, c} + {a, δb, c} + {a, b, δc}

derivations on real and complex jb*-triples 3

for all a, b, c ∈ U.

If U is a real or complex JB*-triple, we can conclude from the Jordan identity that δ(a, b) := L(a, b) − L(b, a) is a derivation, for every a, b ∈ U. An inner triple derivation δ on U is a finite sum of derivations of the form δ(a, b) (a, b ∈ U), i.e.,

δ = ∑_{i=1}^{n} δ(a_{j}, b_{j}).  (2.1)

The degree of an inner derivation is the least number of terms in a representation of the form (2.1). Any derivation which is not inner is called outer.

Remark 1. Let E be a real JB*-triple and δ a derivation on E. Then δ can extended to a derivation δ̃, on the complexification of E, defined by δ̃(x + iy) := δ(x) + iδ(y).

It is due to T. Barton and Y. Friedman [3], that every derivation on a complex JB*-triple is automatically continuous, so, by the previous comment, every derivation on a real JB*-triple is also continuous.

3. Inner Derivation Property

We say that a real or complex JB*-triple U has the inner derivation property (IDP for short) if every derivation on U is inner.

By [14, Chapter 8] every finite dimensional real or complex JB*-triple has the IDP. The next proposition shows that a real JB*-triple has the IDP whenever its complexification satisfies this property.

Proposition 1. Let E be a real JB*-triple. Suppose that the complexification Ê of E has the IDP. Then E has the IDP. Moreover, if M is a bound of the degree of all inner derivations of Ê, then 2M is a bound of the degree of all inner derivations of E.

Proof. Suppose that E is a real JB*-triple such that Ê has the IDP. Let δ be a derivation of E. We denote by δ̂ the derivation on Ê, extending δ to Ê. Since Ê has the IDP, then δ̂ is an inner derivation of degree n, i.e.,

δ̂ = ∑_{k=1}^{n} δ(a_{k}, b_{k})

where a_{k}, b_{k} ∈ Ê. Since Ê = E ⊕ iE, it follows that a_{k} = a_{k,1} + ia_{k,2} and b_{k} = b_{k,1} + ib_{k,2} for suitable a_{k,l}, b_{k,l} ∈ E, l = 1, 2 and k = 1, …, n.

Consider now x ∈ E. We can compute

δ(a_{k}, b_{k})x = δ(a_{k,1} + ia_{k,2}, b_{k,1} + ib_{k,2})x =
{a_{k,1} + ia_{k,2}, b_{k,1} + ib_{k,2}, x} − {b_{k,1} + ib_{k,2}, a_{k,1} + ia_{k,2}, x} =
{a_{k,1}, b_{k,1}, x} + {a_{k,2}, b_{k,2}, x} + i({a_{k,2}, b_{k,1}, x} − {a_{k,1}, b_{k,2}, x})−
− {b_{k,1}, a_{k,1}, x} − {b_{k,2}, a_{k,2}, x} − i({b_{k,2}, a_{k,1}, x} − {b_{k,1}, a_{k,2}, x}) =
= δ(a_{k,1}, b_{k,1})(x) + δ(a_{k,2}, b_{k,2})(x)+

+i({a_{k,2}, b_{k,1}, x} − {a_{k,1}, b_{k,2}, x} − {b_{k,2}, a_{k,1}, x} + {b_{k,1}, a_{k,2}, x}).

Therefore,

E ∋ δ(x) = δ̂(x) = ∑_{k=1}^{n} δ(a_{k,1} + ia_{k,2}, b_{k,1} + ib_{k,2})x =

= (∑_{k=1}^{n}(δ(a_{k,1}, b_{k,1}) + δ(a_{k,2}, b_{k,2}))+

+i ∑_{k=1}^{n}(L(a_{k,2}, b_{k,1}) − L(a_{k,1}, b_{k,2}) − L(b_{k,2}, a_{k,1}) + L(b_{k,1}, a_{k,2}))(x)

Since the elements a_{k,l}, b_{k,l} ∈ E, we have

(∑_{k=1}^{n}(δ(a_{k,1}, b_{k,1}) + δ(a_{k,2}, b_{k,2})))(E) ⊂ E

and

(i ∑_{k=1}^{n}(L(a_{k,2}, b_{k,1}) − L(a_{k,1}, b_{k,2}) − L(b_{k,2}, a_{k,1}) + L(b_{k,1}, a_{k,1}))(E) ⊂ iE.

Therefore,

(∑_{k=1}^{n}(L(a_{k,2}, b_{k,1}) − L(a_{k,1}, b_{k,2}) − L(b_{k,2}, a_{k,1}) + L(b_{k,1}, a_{k,1}))(x) = 0

for all x ∈ E. Thus,

δ(x) = δ̂(x) = ∑_{k=1}^{n}(δ(a_{k,1}, b_{k,1}) + δ(a_{k,2}, b_{k,2}))(x)

for all x ∈ E, proving that δ is an inner derivation with degree ≤ 2n.

From the above proposition, it is easy to see that if E is a real JB*-triple which does not satisfy the IDP, then its complexification also does not satisfy the IDP.

3.1 Reversible unital JB*-algebras We recall that the (complex) type 1 Cartan factor can be defined as the complex Banach space BL(H, K), of all bounded linear operators between two complex Hilbert spaces H and K, with triple product given by

\{a, b, c\} = \frac{1}{2}(ab^{∗}c + cb^{∗}a).

Next we give a brief description of the (complex) Cartan factors of type 2 and 3. Let H be a complex Hilbert space equipped with a conjugation (conjugate-linear isometry of period 2) j : H → H; then for any z ∈ B(H) we can define its transpose as z^{t} := j z^{∗} j. The type 2 Cartan factor coincides with the Banach space of all t-skew symmetric elements in B(H) (z^{t} = −z), and the type 3 Cartan factor is defined as the Banach space of all t-symmetric elements of B(H) (z^{t} = z). The

derivations on real and complex jb*-triples 5

triple product of these Cartan factors is the restriction of the triple product in B(H).

We recall that a JC-algebra (or a JC*-algebra) A is said to be reversible if x_{1}x_{2} … x_{n} + x_{n}x_{n−1} … x_{1} ∈ A, for all n ∈ N and x_{1}, …, x_{n} ∈ A.

Proposition 2. Cartan factors of type 1 with dim H = dim K, Cartan factors of type 2 with dim H even, or infinite, and all Cartan factors of type 3 are reversible JW*-algebras.

Proof. Let C^{3} be a type 3 Cartan factor. Since x^{t} = x for all x ∈ C^{3}, we have (x_{1}...x_{n} + x_{n}...x_{1})^{t} = x_{n}...x_{1} + x_{1}...x_{n} ∈ C^{3}.

Let C^{2} be a type 2 Cartan factor with dim H even or infinite. Then C^{2} contains a distinguished unitary

u = ( \begin{matrix}0 & 1 & 0 & 0 & 0 & . & . \\ −1 & 0 & 0 & 0 & 0 & . & . \\ 0 & 0 & 0 & 1 & 0 & . & . \\ 0 & 0 & −1 & 0 & 0 & . & . \\ 0 & 0 & 0 & 0 & 0 & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \end{matrix} ) .

In this case we can provide a new C*-algebra structure to B(H) with product a·b = au^{∗}b and involution a^{♯} = ua^{∗}u in which C^{2} becomes a JC*-subalgebra under a ∘ b = (a · b + b · a)/2. With this Jordan product, C^{2} is reversible since

(x_{1}u^{∗}x_{2} ⋯ u^{∗}x_{n} + x_{n}u^{∗} ⋯ x_{2}u^{∗}x_{1})^{t} =
(−1)^{n+(n−1)}(x_{n}u^{∗}x_{n−1} ⋯ x_{2}u^{∗}x_{1} + x_{1}u^{∗}x_{2} ⋯ x_{n−1}u^{∗}x_{n}) =
−(x_{1}u^{∗}x_{2} ⋯ u^{∗}x_{n} + x_{n}u^{∗} ⋯ x_{2}u^{∗}x_{1}).

We recall that if A is an algebra, then a derivation D of A is a linear mapping D : A → A satisfying D(ab) = D(a)b + aD(b), for all a, b ∈ A. If A is a Jordan algebra, an inner algebra derivation of A is a finite sum of commutators of the form [L_{a}, L_{b}] for some a, b ∈ A, where L_{a}x := a ∘ x. For an inner algebra derivation D, the degree of D is the least natural number n satisfying D = ∑_{i=1}^{n}[L_{a_{i}}, L_{b_{i}}].

Lemma 1. Let Z be a JB*-algebra, with unit u, regarded as a complex JB*- triple. If δ is a triple derivation of Z, then L_{δ(u)} is an inner triple derivation of Z of degree 1.

Proof. Simply note that for every triple derivation δ of Z, we have

δu = δ{u, u, u} = {δu, u, u} + {u, δu, u} + {u, u, δu} =
= 2{δu, u, u} + {u, δu, u} = 2δu ∘ u + (δu)^{∗}

and hence,

(δu)^{∗} = −δu.

Now considering

L_{δu}z = δu ∘ z = \frac{1}{2}(δu ∘ z − (−δu) ∘ z) =

\frac{1}{2}(δu ∘ z − (δu)^{∗} ∘ z) = \frac{1}{2}(\{δu, u, z\} − \{u, δu, z\}),

it follows that L_{δu} is an inner triple derivation of degree one.

Lemma 2. [3, p. 263] Let Z be a unital JB*-algebra and D an algebra derivation of Z which commutes with the involution of Z. Then D is a triple derivation of Z.

Conversely, if Z is a JB*-triple with a unitary element u and δ is a triple derivation of Z, then δ − L_{δu} is an algebra derivation of Z which commutes with the involution on Z. In particular if δ is an inner derivation of degree one, i. e., δ = δ(x, y), then

δ − L_{δ(u)} = \frac{1}{2}([L_{x+x^{∗}}, L_{y+y^{∗}}] + [L_{−i(x−x^{∗})}, L_{−i(y−y^{∗})}]).

Proof. The first statement is clear. To prove the second one, let δ be a triple derivation of Z. It is easy to check that

(δ − L_{δu})(x ∘ y) = δ{x, u, y} − {δu, u, {x, u, y}} =
= {δx, u, y} + {x, δu, y} + {x, u, δy} − {δu, u, {x, u, y}} =
= {δx, u, y} + {x, δu, y} + {x, u, δy}−
−{{δu, u, x}, u, y} + {x, {u, δu, u}, y} − {x, u, {δu, u, y}} =
= δx ∘ y + {x, δu, y} + x ∘ δy−
−(δu ∘ x) ∘ y + {x, (δu)^{∗} , y} − x ∘ (δu ∘ y) =

(applying (δu)^{∗} = −δu)

= (δ − L_{δu})(x) ∘ y + {x, δu, y} + x ∘ (δ − L_{δu})(y) − {x, δu, y} =
= (δ − L_{δu})(x) ∘ y + x ∘ (δ − L_{δu})(y).

Thus δ − L_{δu} is an algebra derivation.

The verification of the last formula is left to the reader.

By [22, Theorem 13] (see also [1, p. 255]), each JW-algebra A admits a decomposition into weakly closed ideals of the form

A = I_{fin} ⊕ I_{∞} ⊕ II_{1} ⊕ II_{∞} ⊕ III.

See [22] and [1] for the meaning of these symbols. A JW-algebra A is called properly non-modular if its modular part I_{fin} ⊕ II_{1} vanishes.

In 1980, H. Upmeier showed that each algebra derivation on a properly nonmodular JW-algebra is the sum of six commutators of the form [L_{a}, L_{b}] ([23, Theorem 3.8]), and each algebra derivation on a reversible JW-algebra of type I_{fin} is the sum of five commutators ([23, Theorem 3.9]).

The proof of the following theorem is implicitly contained in [23], and we include it here for completeness.

Theorem 1. Let A be a reversible JW-algebra of type II_{1}. Then each derivation of A is a sum of at most 140 commutators of the form [L_{a}, L_{b}].

Proof. Let A be a reversible JW-algebra of type II_{1}. We will denote by U(A) its complex enveloping von Neumann algebra (the smallest von Neumann algebra containing A). By [1, Theorem 8], U(A)^{+} (that is, U(A) with the Jordan product w_{1} ∘ w_{2} = (w_{1}w_{2} + w_{2}w_{1})/2) is also of type II_{1}. So following the proof of [23, Theorem 3.10], it follows that each derivation of A has the form D(x) = ad(w)(x) := [w, x] (x ∈ A), where w = −w^{∗} ∈ U(A). Moreover since U(A)^{+} is of type II_{1}, w is the sum of ten commutators in U(A) (see [7, Theorem 2.3]), so that, each derivation of A has the form

D = ∑_{j=1}^{10} ad([w_{1,j}, w_{2,j}]).

Now since A = R(A)_{∫⊣}, [19], where R(A) is the real enveloping algebra of A, we have, by [20, Lemma 6.1] and [21, Lemma 2.3,Theorem 2.4],

U(A) = R(A) + ⟩ R(A).

Hence, every element w_{l,j} is the sum, w_{l,j} = u_{l,j} + iv_{l,j}, where u_{l,j}, v_{l,j} ∈ R(A). Since for every u_{l}, v_{l} ∈ R(A), the equalities

[u_{1} + iv_{1}, u_{2} + iv_{2}] = [u_{1}, u_{2}] − [v_{1}, v_{2}] + i([u_{1}, v_{2}] + [v_{1}, u_{2}]),
[u_{1} + iu_{2}, x] = [u_{1}, x] + i[u_{2}, x]

hold for all x ∈ A, and since D maps A in A, we have

∑_{j=1}^{10}[[u_{1,j}, v_{2,j}] + [v_{1,j}, u_{2,j}], x] = 0

for all x ∈ A. Thus

D = ad(w) = ∑_{j=1}^{10} ad([u_{1,j}, u_{2,j}] − [v_{1,j}, v_{2,j}]) = ∑_{j=1}^{20} ad([z_{1,j}, z_{2,j}])

where z_{i,j} ∈ R(A) and w = ∑_{j=1}^{20}[z_{1,j}, z_{2,j}].

Our next goal is to prove that every element [z_{1,j}, z_{2,j}] is a finite sum of commutators of elements in A.

Let z_{1,j}, z_{2,j} ∈ R(A), and l ∈ {1, 2}. We denote by z_{l,j}^{s} (respectively, z_{l,j}^{a}) the symmetric part (respectively, the skew-symmetric part) of z_{l,j}. Since for every j, [z_{1,j}^{a}, z_{2,j}^{s}] and [z_{1,j}^{s}, z_{2,j}^{a}] are symmetric elements and w^{∗} = −w, we deduce that

w = ∑_{j=1}^{20}[z_{1,j}^{s}, z_{2,j}^{s}] + [z_{1,j}^{a}, z_{2,j}^{a}].

Now, since A = R(A)_{∫⊣}, we have z_{1,j}^{s}, z_{2,j}^{s} ∈ A. So it is enough to show that every commutator [z_{1,j}^{a}, z_{2,j}^{a}] is a finite sum of commutators of elements in A.

8 t. ho, j. martínez moreno, a. m. peralta, and b. russo

By [4, p. 121], R(A) is isomorphic to the matrix algebra M_{2}(B), where B is a suitable real associative ∗-algebra.

Following the proof of Lemma 3.11 in [23], it follows that each commutator of

skew-symmetric elements in M_{2}(B) has the form ( \begin{matrix}a & −c^{∗} \\ c & b \end{matrix} ), with

a + b = [a_{1}, a_{2}] + [b_{1}, b_{2}] + [c_{1}, c_{2}] + [d_{1}, d_{2}],

where a_{j}, b_{j}, and c_{j} are skew-symmetric elements in B while d_{1} and d_{2} are symmetric elements in B.

On the other hand, since for a, b, c, α_{j} and β_{j} ∈ B, with a^{∗} = −a, b^{∗} = −b, α_{j}^{∗} = α_{j} and β_{j}^{∗} = −β_{j}, the following identities hold:

( \begin{matrix}0 & −c^{∗} \\ c & 0 \end{matrix} ) = [( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & c^{∗} \\ c & 0 \end{matrix} )],

2 ( \begin{matrix}a − b & 0 \\ 0 & b − a \end{matrix} ) = [( \begin{matrix}0 & a − b \\ b − a & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )],

( \begin{matrix}[α_{1}, α_{2}] & 0 \\ 0 & [α_{1}, α_{2}] \end{matrix} ) = [( \begin{matrix}α_{1} & 0 \\ 0 & α_{1} \end{matrix} ) , ( \begin{matrix}α_{2} & 0 \\ 0 & α_{2} \end{matrix} )],

( \begin{matrix}[β_{1}, β_{2}] & 0 \\ 0 & [β_{1}, β_{2}] \end{matrix} ) = [( \begin{matrix}0 & −β_{2} \\ β_{2} & 0 \end{matrix} ) , ( \begin{matrix}0 & −β_{1} \\ β_{1} & 0 \end{matrix} )], and

( \begin{matrix}a & −c^{∗} \\ c & b \end{matrix} ) = ( \begin{matrix}0 & −c^{∗} \\ c & 0 \end{matrix} ) + \frac{1}{2} ( \begin{matrix}a − b & 0 \\ 0 & b − a \end{matrix} ) + \frac{1}{2} ( \begin{matrix}a + b & 0 \\ 0 & a + b \end{matrix} ) ,

it may be concluded that each commutator [z_{1,j}^{a}, z_{2,j}^{a}], is the sum of six commutators of elements in A. Therefore, we have proved that

w = ∑_{j=1}^{140}[x_{1,j}, x_{2,j}]

where x_{l,j} ∈ A, for all l, j, which proves that

D = ∑_{j=1}^{140} ad([x_{1,j}, x_{2,j}]) = ∑_{j=1}^{140}[L_{x_{1,j}}, L_{x_{2,j}}].

Recall that a derivation on a JB-algebra is automatically continuous and a JBalgebra has an approximate unit [9, 3.5.4]. Thus a derivation leaves each closed ideal invariant. By combining the above theorem with the comments preceding it, we have the following corollary.

Corollary 1. Each derivation on a reversible JW-algebra is a sum of at most 151 commutators of the form [L_{a}, L_{b}].

The next theorem is the main result of this section.

Theorem 2. Cartan factors of type 1 with dim H = dim K, Cartan factors of type 2 with dim H even, or infinite, and all Cartan factors of type 3 have the IDP. Moreover, every derivation of the above Cartan factors has degree at most 153.

derivations on real and complex jb*-triples 9

Proof. By Proposition 2, such factors are unital reversible JW*-algebras. So it is enough to prove the statement for a unital reversible JW*-algebra Z.

It is well known that Z decomposes in the form Z = X + iX, where X is the symmetric part of Z, and hence X is a reversible JW-algebra.

If δ is a triple derivation of Z, then by Lemma 2, δ − L_{δu} is a derivation of the JB*-algebra Z which commutes with the involution, hence its restriction to X is a derivation of X. From the identity:

(δ − L_{δu})(z) = (δ − L_{δu})(x + iy) = (δ − L_{δu}) |_{X} (x) + i(δ − L_{δu}) |_{X} (y),

it follows that (δ − L_{δu}) |_{X} determines (δ − L_{δu}). Now, Corollary 1 gives (except summing the 0 commutator)

(δ − L_{δu})(z) = ∑_{j=1}^{152}[L_{a_{j}}, L_{b_{j}}](x) + i ∑_{j=1}^{152}[L_{a_{j}}, L_{b_{j}}](y) =

= ∑_{j=1}^{152}[L_{a_{j}}, L_{b_{j}}](x + iy) = ∑_{j=1}^{152}[L_{a_{j}}, L_{b_{j}}](z).

Now applying the identity:

[L_{a}, L_{b}] + [L_{c}, L_{d}] = 2(δ̃ − L_{δ̃u}),

for all a, b, c and d in X, where

δ̃ = δ(\frac{a + ic}{2}, \frac{b + id}{2}),

we obtain

(δ − L_{δu})(z) = ∑_{j=1}^{152}[L_{a_{j}}, L_{b_{j}}](z) = 2 ∑_{j=1}^{76}(δ(c_{j}, d_{j}) − L_{δ(c_{j},d_{j})u})(z).

Finally applying Lemma 1 it follows that

δ = 2 ∑_{j=1}^{76}(δ(c_{j}, d_{j}) − L_{δ(c_{j},d_{j})u} ) + L_{δu}

is an inner derivation with degree at most 153.

Following [13], we define a real Cartan factor to be a real form of a complex Cartan factor. Combining the above theorem and Proposition 1, we obtain the following result for real Cartan factors.

Corollary 2. If E is either a real form of a type 1 Cartan factor with dim H = dim K, or a real form of a Cartan factor of type 2 with dim H even, or infinite, or a real form of a Cartan factor of type 3, then every derivation on E is inner with degree at most 306.

3.2 Real or Complex spin factors In this subsection we prove that no infinite dimensional real spin factor satisfies the IDP. So by Proposition 1, it may be concluded that no complex spin factor satisfies the IDP.

10 t. ho, j. martínez moreno, a. m. peralta, and b. russo

We recall that a complex spin Cartan factor is a JB*-triple which can be equipped with a complete inner product (.|.) and a conjugation ∗ such that the triple product satisfies

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

and the norm is given by

∥x∥^{2} := (x|x) + ((x|x)^{2} − |(x|x^{∗})|^{2})^{\frac{1}{2}} .

By a real spin factor we mean any real form of a complex spin factor. By [13, Theorem 4.1], we know that every real spin factor E is an l_{1}-sum,

E = X_{1} ⊕^{ℓ_{1}} X_{2}

where X_{1} and X_{2} are closed subspaces of a real Hilbert space X, satisfying X_{2} = X_{1}^{⊥}, and the triple product on E is given by

\{x, y, z\} = (x|y)z + (z|y)x − (x|\overline{z})\overline{y},

where (.|.) is the inner product of X and the map x ↦ \overline{x} is given by \overline{x} = (x_{1}, −x_{2}), for all x = (x_{1}, x_{2}) ∈ E.

Our goal is to build a derivation which is not inner in the case of an infinite dimensional real spin factor E = X_{1} ⊕^{ℓ_{1}} X_{2}. Without loss of generality we can assume that X_{1} is also infinite dimensional.

First we suppose that E is separable. Let {e_{n} : n ∈ N} be a countable orthonormal basis of X_{1}. Since \overline{e_{n}} = e_{n}, it is easy to check that \{e_{n}, e_{n}, e_{n}\} = e_{n} and ∥δ(e_{2k−1}, e_{2k})∥ ≤ 2, hence the operator

δ_{0} := ∑_{k=1}^{∞} \frac{1}{2^{k}} δ(e_{2k−1}, e_{2k})

is a well-defined derivation on E. Our goal is to show that δ_{0} is not inner. Suppose that δ_{0} is inner; then

δ_{0} = ∑_{j=1}^{P} δ(a_{j}, b_{j})

for suitable a_{j}, b_{j} ∈ E, with, a_{j} = a_{j,1} + a_{j,2} and b_{j} = b_{j,1} + b_{j,2} where a_{j,i} and b_{j,i} are in X_{i} (j = 1, …, P,i = 1, 2). Hence

δ_{0} = ∑_{j=1}^{P} δ(a_{j}, b_{j}) =

= ∑_{j=1}^{P} δ(a_{j,1}, b_{j,1}) + δ(a_{j,1}, b_{j,2}) + δ(a_{j,2}, b_{j,1}) + δ(a_{j,2}, b_{j,2}).

It is easy to check that for all x_{1} ∈ X_{1},

δ(a_{j,2}, b_{j,2})(x_{1}) = δ(a_{j,1}, b_{j,2})(x_{1}) = δ(a_{j,2}, b_{j,1})(x_{1}) = 0

and δ_{0}(X_{2}) = 0. Therefore

δ_{0}(x_{1}) = ∑_{j=1}^{P} δ(a_{j,1}, b_{j,1})(x_{1})

derivations on real and complex jb*-triples 11

for all x_{1} ∈ X_{1}.

Now we define K as the linear span of {a_{j,1}, b_{j,1} : j = 1 … P }. Let x_{1} ∈ K^{⊥}∩X_{1}; then

0 = ∑_{j=1}^{P} δ(a_{j,1}, b_{j,1})(x_{1}) = δ_{0}(x_{1}) = ∑_{k=1}^{∞} \frac{1}{2^{k}} δ(e_{2k−1}, e_{2k})(x_{1}) =

= ∑_{k=1}^{∞} \frac{1}{2^{k}} (\{e_{2k−1}, e_{2k}, x_{1}\} − \{e_{2k}, e_{2k−1}, x_{1}\}) =

= ∑_{k=1}^{∞} \frac{1}{2^{k}} ((e_{2k−1}|e_{2k})x_{1} + (x_{1}|e_{2k})e_{2k−1} − (e_{2k−1}|x_{1})e_{2k}−

−(e_{2k}|e_{2k−1})x_{1} − (x_{1}|e_{2k−1})e_{2k} + (e_{2k}|x_{1})e_{2k−1}) =

= ∑_{k=1}^{∞} \frac{1}{2^{k−1}} ((x_{1}|e_{2k})e_{2k−1} − (e_{2k−1}|x_{1})e_{2k}).

Thus (x_{1}|e_{2k}) = (e_{2k−1}|x_{1}) = 0 for all k ∈ N, so x_{1} = 0 since {e_{n}} is a basis of X_{1}. Therefore K^{⊥} ∩ X_{1} = 0, and hence X_{1} = K is finite dimensional, which is impossible.

This proves that δ_{0} is not an inner derivation. Suppose now that dim X_{1} > ℵ_{0} and let {e_{n}}_{N} be a countable set of orthonormal vectors in X_{1}. Let us denote by H the real separable Hilbert space generated by {e_{n}}_{N}, and by δ_{0} the derivation on E given by

δ_{0} := ∑_{k=1}^{∞} \frac{1}{2^{k}} δ(e_{2k−1}, e_{2k}).

Since δ_{0}(H) ⊆ H, it follows that δ_{0}|_{H} is a derivation of the real spin factor H, which is not inner by the previous case. Actually we claim that δ_{0} is not an inner derivation on E. Suppose, contrary to our claim, that δ_{0} is inner on E; then

δ_{0} = ∑_{j=1}^{P} δ(a_{j}, b_{j})

with a_{j}, b_{j} ∈ E. Since

E = (H ⊕^{ℓ_{2}} H^{⊥}) ⊕^{ℓ_{1}} X_{2},

the elements, a_{j} and b_{j}, can be expressed as a_{j} = h_{j} + x_{j,3} and b_{j} = k_{j} + y_{j,3} where h_{j} and k_{j} are in H and x_{j,3}, y_{j,3} ∈ H^{⊥} ⊕^{ℓ_{1}} X_{2} (j = 1, …, P). Thus

δ_{0} = ∑_{j=1}^{P} δ(a_{j}, b_{j}) =

= ∑_{j=1}^{P} δ(h_{j}, k_{j}) + δ(h_{j}, y_{j,3}) + δ(x_{j,3}, k_{j}) + δ(x_{j,3}, y_{j,3}).

It is easy to check that

δ(h_{j}, y_{j,3})h = −(h_{j}|h)\overline{y_{j,3}} − (h|h_{j})y_{j,3} ∈ H^{⊥} ⊕^{ℓ_{1}} X_{2},

δ(x_{j,3}, k_{j})h = (h|k_{j})x_{j,3} + (k_{j}|h)\overline{x_{j,3}} ∈ H^{⊥} ⊕^{ℓ_{1}} X_{2}

and

δ(x_{j,3}, y_{j,3})(h) = 0,

for all h ∈ H. From the last identity we have

δ_{0}(h) = ∑_{j=1}^{P} δ(h_{j}, k_{j})(h) + ∑_{j=1}^{P}(δ(h_{j}, y_{j,3}) + δ(x_{j,3}, k_{j}))(h),

for all h ∈ H. Since δ_{0}(H) ⊆ H and ∑_{j=1}^{P}(δ(h_{j}, y_{j,3})+δ(x_{j,3}, k_{j}))(H) ⊆ H^{⊥}⊕^{ℓ_{1}} X_{2}, we have that

δ_{0}(h) = ∑_{j=1}^{P} δ(h_{j}, k_{j})(h)

for all h ∈ H. Therefore δ_{0}|_{H} is an inner derivation on H which is impossible, hence δ_{0} is not an inner derivation on E.

We have thus proved the following theorem.

Theorem 3. Every infinite dimensional real or complex spin factor has a derivation which is not inner, i. e., none of infinite dimensional real or complex spin factors has the IDP.

3.3 Non-square type 1 As in the case of a real or complex spin factor, we are going to build an outer derivation in every real form of an infinite dimensional and non-square (dim H ≠ dim K) type 1 Cartan factor. Again using Proposition 1, we will conclude that no complex infinite dimensional non-square type 1 Cartan factor satisfies the IDP.

By [13, Theorem 4.1], we know that the real forms of a complex type 1 Cartan factor are precisely the real Banach space BL(X, Y) of all bounded linear operators between two real Hilbert spaces X and Y or the real Banach space BL(P, Q) of all bounded linear operators between two Hilbert spaces P, Q over the quaternion field. Thus it is enough to prove that BL(X, Y), with +∞ = dim(X) > dim(Y), possesses an outer derivation. We will divide the proof in several steps. In a first step we suppose that Y = R. In this case BL(X, R) is isometrically isomorphic, as a real JB*-triple, to X equipped with the triple product

\{x, y, z\} = \frac{1}{2}((x|y)z + (z|y)x)

for all x, y, z ∈ X.

Let δ a derivation on X, then

δ {x, y, z} = {δx, y, z} + {x, δy, z} + {x, y, δz} (∗)

for all x, y, z ∈ X. Now from the expression of the triple product, we have

δ \{x, y, z\} = \frac{1}{2}((x|y)δz + (z|y)δx),

\{δx, y, z\} = \frac{1}{2}((δx|y)z + (z|y)δx)),

\{x, δy, z\} = \frac{1}{2}((x|δy)z + (z|δy)x)),

\{x, y, δz\} = \frac{1}{2}((x|y)δz + (δz|y)x)).

Thus it follows from (∗) that

\frac{1}{2}(((δx|y) + (x|δy))z + ((z|δy) + (δz|y))x) = 0

for all x, y, z ∈ X. In particular we have

(x|δy) = −(δx|y)

for all x, y ∈ X, i. e., δ^{∗} = −δ. Therefore every derivation on X, regarded as the real type 1 Cartan factor BL(X, R), is a skew-symmetric operator on X. Conversely:

Lemma 3. If X is a real Hilbert space, regarded as the real Cartan factor BL(X, R), then the derivations on X coincide with the skew-symmetric operators on X.

Proof. Suppose that T is a skew-symmetric operator on X. The identities

T \{x, y, z\} = \frac{1}{2}((x|y)T z + (z|y)T x),

\{T x, y, z\} = \frac{1}{2}((T x|y)z + (z|y)T x)),

\{x, T y, z\} = −\frac{1}{2}((T x|y)z + (T z|y)x)),

\{x, y, T z\} = \frac{1}{2}((x|y)T z + (T z|y)x)),

show that T is a derivation on X.

The next proposition characterizes the inner derivations on X.

Proposition 3. The inner derivations on X, regarded as the real Cartan factor BL(X, R), coincide with the finite rank operators on X which are skewsymmetric.

Proof. Let

δ = ∑_{j=1}^{P} δ(a_{j}, b_{j})

be an inner derivation on X. Since

δ(a_{j}, b_{j})(x) = \frac{1}{2}((x|b_{j})a_{j} − (x|a_{j})b_{j}),

it follows that δ is a finite rank operator. The other implication follows from Lemma 3

Remark 2. Since for every infinite dimensional Hilbert space X there exists 14 t. ho, j. martínez moreno, a. m. peralta, and b. russo

a skew-symmetric operator T on X satisfying that T^{2} = −Id, we conclude from Lemma 3 and Proposition 3 that T is an outer derivation on X. It follows that BL(X, R) does not satisfy the IDP.

Our next goal is to build derivations on BL(X, Y) from derivations on X = BL(X, R).

Lemma 4. Let δ be a derivation on a real Hilbert space X (regarded as a real type 1 Cartan factor) and let Y be another real Hilbert space. Then the operator

δ̃: BL(X, Y) → BL(X, Y)
δ̃a = aδ

is a derivation on BL(X, Y).

Proof. Since δ is a derivation on X, δ^{∗} = −δ (see Lemma 3). Given a, b, c ∈ BL(X, Y), we have

{δ̃a, b, c} + {a, δ̃b, c} + {a, b, δ̃c} =

= \frac{1}{2}(aδb^{∗}c + cb^{∗}aδ + aδ^{∗}b^{∗}c + cδ^{∗}b^{∗}a + ab^{∗}cδ + cδb^{∗}a) =

= \frac{1}{2}(aδb^{∗}c + cb^{∗}aδ − aδb^{∗}c − cδb^{∗}a + ab^{∗}cδ + cδb^{∗}a) =

= \frac{1}{2}(cb^{∗}aδ + ab^{∗}cδ) = \{a, b, c\} δ = δ̃\{a, b, c\} ,

which proves that δ̃ is a derivation.

At this moment, we need the following identification. Let us fix a norm one element y_{0} ∈ Y. In the sequel we will identify each h ∈ X, with the operator

f_{h} : X → Y
f_{h}(x) := (x|h)y_{0} (x ∈ X).

In this way X can be regarded with the subspace of BL(X, Y) formed by all operators of the form f_{h} with h ∈ X. Using this identification it is easy to check that if δ and δ̃ are as in Lemma 4, then δ̃(X) ⊆ X. In fact

δ̃(f_{h})(x) = f_{h}(δx) = (δx|h)y_{0} =
= (x|δ^{∗}h)y_{0} = (x| − δh)y_{0} = f_{−δh}(x).

The next lemma is the key tool of the main result of this subsection.

Lemma 5. Let δ and δ̃ be as in Lemma 4, and suppose that δ̃ is an inner derivation. Then δ has rank less or equal than the hilbertian dimension of Y.

Proof. Since δ̃ is an inner derivation on BL(X, Y), δ̃ is the sum

δ̃ = ∑_{j=1}^{P} δ(a_{j}, b_{j})

for suitable a_{j}, b_{j} ∈ BL(X, Y). As we have seen previously for each h ∈ X, δ̃f_{h} = f_{−δh} ∈ X. On the other hand,

f_{−δh} = δ̃(f_{h}) = ∑_{j=1}^{P} δ(a_{j}, b_{j})(f_{h}) =

= ∑_{j=1}^{P} \frac{1}{2}(a_{j}b_{j}^{∗}f_{h} + f_{h}b_{j}^{∗}a_{j} − b_{j}a_{j}^{∗}f_{h} − f_{h}a_{j}^{∗}b_{j}) =

= (∑_{j=1}^{P} \frac{1}{2}(a_{j}b_{j}^{∗} − b_{j}a_{j}^{∗}))f_{h} + f_{h}(∑_{j=1}^{P} \frac{1}{2}(b_{j}^{∗}a_{j} − a_{j}^{∗}b_{j})) =

= Rf_{h} + f_{h}T,

where

R = ∑_{j=1}^{P} \frac{1}{2}(a_{j}b_{j}^{∗} − b_{j}a_{j}^{∗}) : Y → Y

T = ∑_{j=1}^{P} \frac{1}{2}(b_{j}^{∗}a_{j} − a_{j}^{∗}b_{j}) : X → X

are two skew-symmetric operators. Moreover

f_{h}T (x) = (T x|h)y_{0} = (x| − T h)y_{0} = f_{−T h}(x)

for all x ∈ X, so that f_{h}T = f_{−T h}, and

Rf_{h} = δ̃f_{h} − f_{h}T = f_{−δh−T h} = f_{h′} ∈ X.

Therefore, for all x, h ∈ X, the equality

Rf_{h}(x) = (x|h)R(y_{0}) = (x|h′)y_{0},

holds. Thus, we have R(y_{0}) = λy_{0} for a suitable λ ∈ R. Since R is a skew-symmetric operator and λ is a real eigenvalue of R, λ = 0.

In this way, since Rf_{h} = 0, we have

f_{−δh} = δ̃(f_{h}) = f_{h}T = f_{−T h}

for all h ∈ X, and hence T = δ.

Since each b_{j}^{∗}a_{j} and each a_{j}^{∗}b_{j} are operators which factorize through Y, they have rank at most the hilbertian dimension of Y. Therefore so does

δ = T = ∑_{j=1}^{P} \frac{1}{2}(b_{j}^{∗}a_{j} − a_{j}^{∗}b_{j}).

Theorem 4. Let X be an infinite dimensional real Hilbert space, and Y a real Hilbert space with hilbertian dimension less than the hilbertian dimension of X. Then BL(X, Y) does not satisfy the IDP.

Proof. We recall that, since X is infinite dimensional, there exists a bounded linear operator T on X, such that T^{2} = −Id_{X} and T^{∗} = −T. Hence T has rank 16 t. ho, j. martínez moreno, a. m. peralta, and b. russo

equal to the hilbertian dimension of X. Since T^{∗} = −T, Lemma 3 assures that T is a derivation on X. Moreover, by Lemma 4, the operator T̃ given by T̃a = aT (a ∈ BL(X, Y)), is a derivation on BL(X, Y). If T̃ is an inner derivation, then Lemma 5, assures that T has rank at most the hilbertian dimension of Y, which is impossible since dim(X) > dim(Y).

Again combining the above Theorem and Proposition 1, we obtain the following corollary.

Corollary 3. The complex infinite dimensional non-square type 1 Cartan factors and their real forms do not satisfy the IDP.

By virtue of the previous results we know that there exist real and complex JB*-triples having outer derivations. Therefore it is natural to ask if any derivation can be approximated (in a convenient topology) by inner derivations. Upmeier [23] proved that there exists a unital JB-algebra X, and a derivation D on X which can not be approximated in norm by inner algebra derivations. Let X̂ denotes the complexification of X, and D̂ the complex linear extension of D to X̂, then X̂ is a unital JB*-algebra with unit u, and hence a JB*-triple, and D̂ is a triple derivation, since D̂ is an algebra derivation which commutes with the involution (see Lemma 2). We claim that D̂ cannot be approximated in norm by inner triple derivations. Otherwise for ε > 0 there would exist an inner triple derivation

δ = ∑_{j}^{P} δ(e_{j}, f_{j})

such that

∥D̂ − δ∥ < ε.

Now by Lemma 2

δ − L_{δ(u)} = ∑_{j}^{P} δ(e_{j}, f_{j}) − L_{δ(e_{j} ,f_{j})(u)} =

= \frac{1}{2}∑_{j}^{P} [L_{a_{j}}, L_{c_{j}}] + [L_{b_{j}}, L_{d_{j}}]

where e_{j} = \frac{1}{2}(a_{j} + ib_{j}), f_{j} = \frac{1}{2}(c_{j} + id_{j}) with a_{j}, b_{j}, c_{j}, d_{j} in X. Therefore δ − L_{δ(u)} is an inner derivation on X such that

∥D − (δ − L_{δ(u)})∥ = ∥D − L_{D(u)} − (δ − L_{δ(u)})∥ ≤
≤ ∥D̂ − δ∥ + ∥L_{D(u)} − L_{δ(u)}∥ ≤
≤ ∥D̂ − δ∥ + ∥L_{D(u)−δ(u)}∥ ≤
≤ ∥D̂ − δ∥ + ∥(D̂ − δ)(u)∥ ≤ 2ε

which is imposible, since D cannot be approximated in norm by inner derivation.

On the other hand, D is also a derivation on the real JB*-triple X. If D could be approximated in norm by inner triple derivations on X, then for every ε > 0 there exists

δ = ∑_{j}^{P} δ(e_{j}, f_{j})

with e_{j}, f_{j} ∈ X such that ∥D − δ∥ ≤ ε. In this case, δ = ∑_{j}^{P} δ(e_{j}, f_{j}), is an inner derivation on X̂ and

∥(D̂ − δ)∥ ≤ 2ε.

This is impossible.

Upmeier [23], also proved that every algebra derivation on a JB-algebra can be approximated in the strong operator topology by inner derivations. In [3, Theorem 4.6], T. Barton and Y. Friedman proved that the set of all inner derivations on a JB*-triple is dense in the set of all derivations with respect to the strong operator topology. This result can be extended to real JB*-triples.

Theorem 5. The set of all inner derivations on a real JB*-triple is dense in the set of all derivations with respect to the strong operator topology.

Proof. Let E be a real JB*-triple and δ a derivation on E. We consider

δ̂: Ê → Ê
δ̂(x + iy) := δ(x) + iδ(y)

the natural extension of δ to Ê. Since Ê is a complex JB*-triple, by [3, Theorem 4.6], it follows that for every x_{1}, …, x_{n} ∈ E ⊂ Ê and every ε > 0, there exists an inner derivation

δ_{1} = ∑_{j=1}^{P} δ(a_{j}, b_{j})

on Ê such that ∥δ̂(x_{l}) − δ_{1}(x_{l})∥ ≤ ε for all l = 1, …, n.

Since a_{j} = a_{j,1} + ia_{j,2} and b_{j} = b_{j,1} + ib_{j,2}, where a_{j,k} and b_{j,k} are in E, it is easy to check that

δ_{1}(x_{l}) = ∑_{j=1}^{P}(δ(a_{j,1}, b_{j,1}) + δ(a_{j,2}, b_{j,2})+

+i(L(a_{j,2}, b_{j,1}) + L(b_{j,1}, a_{j,2}) − L(a_{j,1}, b_{j,2}) − L(b_{j,2}, a_{j,1})))x_{l}.

Since a_{j,k}, b_{j,k} and x_{l} are elements in E, it follows that

∑_{j=1}^{P} i(L(a_{j,2}, b_{j,1}) + L(b_{j,1}, a_{j,2}) − L(a_{j,1}, b_{j,2}) − L(b_{j,2}, a_{j,1}))x_{l} ∈ i E.

Thus

∥δ(x_{l}) − ∑_{j=1}^{P}(δ(a_{j,1}, b_{j,1}) + δ(a_{j,2}, b_{j,2})(x_{l})∥ ≤

≤ ∥δ(x_{l}) − ∑_{j=1}^{P}(δ(a_{j,1}, b_{j,1}) + δ(a_{j,2}, b_{j,2})(x_{l})−

18 t. ho, j. martínez moreno, a. m. peralta, and b. russo

−i ∑_{j=1}^{P}(L(a_{j,2}, b_{j,1}) + L(b_{j,1}, a_{j,2}) − L(a_{j,1}, b_{j,2}) − L(b_{j,2}, a_{j,1}))(x_{l})∥ =

= ∥δ̂(x_{l}) − ∑_{j=1}^{P}(δ(a_{j,1}, b_{j,1}) + δ(a_{j,2}, b_{j,2})+

+i(L(a_{j,2}, b_{j,1}) + L(b_{j,1}, a_{j,2}) − L(a_{j,1}, b_{j,2}) − L(b_{j,2}, a_{j,1}))) x_{l}∥ =

= ∥δ̂(x_{l}) − δ_{1}(x_{l})∥ ≤ ε

for all l = 1, …, n.

Problem: If we could obtain a universal bound for the degree of all derivation in a type 2 Cartan factor with dim H odd, we could try to determine all JBW*- triples of type I satisfying the IDP following the techniques contained in T. Ho’s dissertation [10].

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J. Martínez and A. M. Peralta Dept. Análisis Matemático Ftad. de Ciencias Universidad de Granada 18071 Granada, Spain jmmoreno@goliat.ugr.es and aperalta@goliat.ugr.es

T. Ho and B. Russo

Department of Mathematics University of California Irvine, California 92697-3875 U. S. A. brusso@math.uci.edu