# Local triple derivations on C∗-algebras and JB∗-triples

Maria Burgos<EMAIL_ADDRESS>Departamento de Matematicas, Facultad de
Ciencias Sociales y de la Educacion, Universidad de Cdiz, 11405, Jerez de la
Frontera, Spain.  ,  Francisco J. Fernández-Polo<EMAIL_ADDRESS>Departamento
de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071
Granada, Spain.  and  Antonio M. Peralta<EMAIL_ADDRESS>Departamento de
Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071
Granada, Spain.

(Date: February, 2013)

###### Abstract.

In a first result we prove that every continuous local triple derivation on a
JB∗-triple is a triple derivation. We also give an automatic continuity
result, that is, we show that local triple derivations on a JB∗-triple are
continuous even if not assumed a priori to be so. In particular every local
triple derivation on a C∗-algebra is a triple derivation. We also explore the
connections between (bounded local) triple derivations and generalised
(Jordan) derivations on a C∗-algebra.

###### Key words and phrases:

triple derivation; (continuous) local triple derivation; generalised
derivation; generalised Jordan derivation; C∗-algebra; real C∗-algebra;
JB∗-triple; real JB∗-triple; range tripotent; compact tripotent; non-
commutative Urysohn’s lemma.

###### 2000 Mathematics Subject Classification:

Primary 47B47, 46L57; Secondary 17C65, 47C15, 46L05, 46L08

Authors partially supported by the Spanish Ministry of Economy and
Competitiveness, D.G.I. project no. MTM2011-23843, and Junta de Andalucía
grants FQM0199 and FQM3737.

##  1\. Introduction

An (associative) _derivation_ from a C∗-algebra $A$ into a Banach $A$-bimodule
$X$ is a linear mapping $D:A\to X$ satisfying $D(ab)=D(a)b+aD(b),$ for every
$a,b$ in $A$. A linear mapping $T:A\to X$ is a _local derivation_ if for each
$a$ in $A$ there is a derivation $D_{a}$ from $A$ into $X$ with
$D_{a}(a)=T(a).$ Local derivations were introduced by R.V. Kadison [27] who
proved that every continuous local derivation from a von Neumann algebra $M$
(i.e. a C∗-algebra which is also a dual Banach space) into a dual Banach
$M$-bimodule is a derivation. Kadison’s theorem motivated a flourishing line
of research which culminates in 2001 with a definite contribution by B.E.
Johnson [26], who shows that every bounded local derivation from a C∗-algebra
$A$ into a Banach $A$-bimodule is a derivation. In the just quoted paper,
Johnson showed that the continuity hypothesis is, in fact, superfluous by
proving that every local derivation from a C∗-algebra $A$ into a Banach
$A$-bimodule is continuous.

Besides of holomorphic motivations coming from the classification of bounded
symmetric domains in arbitrary complex Banach spaces (cf. [28]), there are
additional reasons, from the point of view of Functional Analysis, to study a
strictly wider class of complex Banach spaces which comprises the categories
of C∗-algebras, JB∗-algebras and ternary rings of operators, and is known as
the category of JB∗-triples. For example, contrary to what happens for
C∗-algebras, the category of JB∗-triples is stable under contractive
projections (cf. [36], [29] and [20]). A JB∗-triple is a complex Banach space
$E$ equipped with a triple product $\left\\{.,.,.\right\\}:E\times E\times
E\to E$ which is conjugate linear in the middle variable and symmetric and
linear in the outer variables satisfying certain algebraic-analytic axioms
(see Subsection 1.1 for more details). Every C∗-algebra is a JB∗-triple when
equipped with the triple product given by

$\left\\{x,y,z\right\\}=\frac{1}{2}(xy^{*}z+zy^{*}x).$

A _triple derivation_ on a JB∗-triple $E$ is a linear mapping $\delta:E\to E$
satisfying that

$\delta\left\\{a,b,c\right\\}=\left\\{\delta(a),b,c\right\\}+\left\\{a,\delta(b),c\right\\}+\left\\{a,b,\delta(c)\right\\},$

for every $a,b,c\in E$. A _local triple derivation_ on $E$ is a linear map
$T:E\to E$ such that for each $a$ in $E$ there exists a triple derivation
$\delta_{a}$ on $E$ satisfying $T(a)=\delta_{a}(a).$ Inspired by the results
proved by R.V. Kadison and B.E. Johnson it is natural to consider the
following problems:

######  Problem 1.1.

Is every bounded local triple derivation on a JB∗-triple $E$ a triple
derivation?

######  Problem 1.2.

Is every local triple derivation on a JB∗-triple $E$ a triple derivation?

######  Problem 1.3.

Is every local triple derivation on a JB∗-triple $E$ continuous?

We note that the above problems also make sense in the setting of C∗-algebras
and are interesting questions in its own right.

######  Problem 1.4.

Is every (bounded) local triple derivation on a C∗-algebra $B$ a triple
derivation?

M. Mackey gave in [31, Theorem 5.11] a positive answer to the above Problem
(1.1) under the additional hypothesis of $E$ being a JBW∗-triple (i.e. a
JB∗-triple which is also a dual Banach space). Problems (1.1), (1.2) and (1.3)
already appear in [31] and in [14]. Mackey’s theorem can be considered an
appropriate version of Kadison’s theorem in the setting of JB∗-triples.
Problem 1.4 was solved by J. Garcs and the authors of this note for bounded
local triple derivations on unital C∗-algebras in [14]. The above problems
have remained open in their full generality until now.

In this paper we completely solve Problems (1.1) and (1.3) (and consequently
Problems (1.2) and (1.4)). In Section 2 we prove that every continuous local
triple derivation on a JB∗-triple is a triple derivation (see Theorem 2.4).
Our strategy consists in studying the behavior of the bitranspose, $T^{**}$,
on compact and range tripotents in $E^{**}$. It follows from our first main
result that Problems (1.3) and (1.2) are equivalent. In a subsequent result,
we explore an automatic continuity result for local triple derivations. T.
Barton and Y. Friedman proved in [8] that every triple derivation on a
JB∗-triple is automatically continuous, we shall adapt their arguments to show
that the same statement remains valid for local triple derivations (see
Theorem 2.8). In Section 3, we particularize our results to the setting of
C∗-algebras and explore the connections between (local) triple derivations and
generalised derivations on a C∗-algebra. Among the consequences, we establish
appropriate generalizations, to general C∗-algebras, of results due to V.
Shu’lman [35] and J. Li and Z. Pan [30] in the context of unital C∗-algebras.

###  1.1. Preliminaries

A JB∗-triple is a complex Banach space $E$ together with a continuous triple
product $\left\\{.,.,.\right\\}:E\times E\times E\to E,$ which is conjugate
linear in the middle variable and symmetric bilinear in the outer variables
satisfying the following axioms:

  1. $(a)$

(Jordan Identity)

(1)
$\left\\{a,b,\left\\{x,y,z\right\\}\right\\}=\left\\{\left\\{a,b,x\right\\},y,z\right\\}-\left\\{x,\left\\{b,a,y\right\\},z\right\\}+\left\\{x,y,\left\\{a,b,z\right\\}\right\\},$

for all $a,b,x,y,z$ in $E$;

  2. $(b)$

If $L(a,b)$ denotes the operator on $E$ given by
$L(a,b)x=\left\\{a,b,x\right\\},$ the mapping $L(a,a)$ is an hermitian
operator with non-negative spectrum;

  3. $(c)$

$\|\left\\{a,a,a\right\\}\|=\|a\|^{3}$, for every $a\in E$.

Given $a,b\in E$, the symbol $Q(a,b)$ will denote the conjugate linear
operator defined by $Q(a,b)(x)=\left\\{a,x,b\right\\}$. We shall write $Q(a)$
instead of $Q(a,a)$.

Every C∗-algebra is a JB∗-triple via the triple product given by

(2) $2\left\\{x,y,z\right\\}=xy^{*}z+zy^{*}x,$

and every JB∗-algebra is a JB∗-triple under the triple product

(3) $\left\\{x,y,z\right\\}=(x\circ y^{*})\circ z+(z\circ y^{*})\circ
x-(x\circ z)\circ y^{*}.$

Every element $e$ in a JB∗-triple $E$ satisfying $\left\\{e,e,e\right\\}=e$ is
called tripotent. When a C∗-algebra, $A$, is regarded as a JB∗-triple, the set
of tripotents of $A$ is precisely the set of all partial isometries in $A$.

Associated with each tripotent $e$ in a JB∗-triple $E$, there is a
decomposition of $E$ (called _Peirce decomposition_) in the form:

$E=E_{0}(e)\oplus E_{1}(e)\oplus E_{2}(e),$

where $E_{k}(e)=\\{x\in E:L(e,e)x=\frac{k}{2}x\\}$ for $k=0,1,2$. The _Peirce
rules_ are that

$\left\\{E_{i}(e),E_{j}(e),E_{k}(e)\right\\}\subseteq E_{i-j+k}(e)$

if $i-j+k\in\\{0,1,2\\},$ and
$\left\\{E_{i}(e),E_{j}(e),E_{k}(e)\right\\}=\\{0\\}$ otherwise. Moreover,
$\left\\{E_{2}(e),E_{0}(e),E\right\\}=\left\\{E_{0}(e),E_{2}(e),E\right\\}=\\{0\\}.$

The Peirce space $E_{2}(e)$ is a unital JB∗-algebra with unit $e$, product
$x\circ_{e}y:=\left\\{x,e,y\right\\}$ and involution
$x^{*_{e}}:=\left\\{e,x,e\right\\}$, respectively. The corresponding _Peirce
projections_ , $P_{i}(e):E\to E_{i}(e)$, $(i=0,1,2)$ are given by

$P_{2}(e)=Q(e)^{2},\ P_{1}(e)=2L(e,e)-2Q(e)^{2},\ \hbox{ and
}P_{0}(e)=Id-2L(e,e)+Q(e)^{2},$

where $Id$ is the identity map on $E$.

A JBW∗-triple is a JB∗-triple which is also a dual Banach space (with a unique
isometric predual [9]). Since the triple product of every JBW∗-triple is
separately weak∗ continuous (cf. [9]), it can be easily checked that Peirce
projections associated with a tripotent $e$ in a JBW∗-triple are weak∗
continuous. The second dual, $E^{**}$, of a JB∗-triple $E$ is a JBW∗-triple
[17].

Following standard notation, for each element $a$ in a JB∗-triple $E$ we
denote $a^{[1]}=a$ and $a^{[2n+1]}:=\left\\{a,a^{[2n-1]},a\right\\}$ $(\forall
n\in\mathbb{N})$. It follows from Jordan identity that JB∗-triples are power
associative, that is, $\left\\{a^{[k]},a^{[l]},a^{[m]}\right\\}=a^{[k+l+m]}$.
The JB∗-subtriple of $E$ generated by the element $a$ will be denoted by the
symbol $E_{a},$ and coincides with the closed linear span of the elements
$a^{[2n-1]}$ with $n\in\mathbb{N}$. The _local Gelfand theory_ for JB∗-triples
assures that $E_{a}$ is JB∗-triple isomorphic (and hence isometric) to
$C_{0}(L)$ for some locally compact Hausdorff space $L\subseteq(0,\|a\|],$
such that $L\cup\\{0\\}$ is compact and $\|a\|\in L$. It is further known that
there exists a triple isomorphism $\Psi$ from $E_{a}$ onto $C_{0}(L),$
satisfying $\Psi(a)(t)=t$ $(t\in L)$ (compare [28, Lemma 1.14]). This result
provides us with an spectral resolution for every element in a JB∗-triple $E$.

The local Gelfand theory for JB∗-triples is a powerful tool; among its
consequences, it follows that for an element $a$ in a JB∗-triple $E$ and each
natural $n$, there exists (a unique) element $a^{[{1}/({2n-1})]}$ in $E_{a}$
satisfying $(a^{[{1}/({2n-1})]})^{[2n-1]}=a.$ When $a$ is a norm one element,
the sequence $(a^{[{1}/({2n-1})]})$ converges in the weak∗ topology of
$E^{**}$ to a tripotent denoted by $r(a)$ and called the _range tripotent_ of
$a$. The tripotent $r(a)$ is the smallest tripotent $e$ in $E^{**}$ satisfying
that $a$ is positive in the JBW∗-algebra $E^{**}_{2}(e)$. It is also known
that the sequence $(a^{[2n-1]})$ converges in the weak∗ topology of $E^{**}$
to a tripotent (called the _support_ _tripotent_ of $a$) $s(a)$ in $E^{**}$,
which satisfies $s(a)\leq a\leq r(a)$ in $A^{**}_{2}(r(a))$ (compare [18,
Lemma 3.3]; the reader should be noted that in [19], $r(a)$ is called the
support tripotent of $a$).

Two elements $a,b$ in a JB∗-triple $E$ are said to be _orthogonal_ (written
$a\perp b$) if $L(a,b)=0$. It is known that $a\perp b$ if, and only if, one of
the following statements holds:

$b\perp a;$ | $\left\\{a,a,b\right\\}=0;$ | $a\perp r(b);$
---|---|---
$r(a)\perp r(b);$ |  $E^{**}_{2}(r(a))\perp E^{**}_{2}(r(b))$; |  $r(a)\in E^{**}_{0}(r(b))$;
$a\in E^{**}_{0}(r(b))$; |  $b\in E^{**}_{0}(r(a))$; |  $E_{a}\perp E_{b}$,

(see, for example, [13, Lemma 1]). The natural partial order in the set of
tripotents of a JB∗-triple $E$ is defined as follows: given two tripotents $e$
and $u$ in $E$ we say that $u\leq e$ if $e-u$ is a tripotent in $E$ and
$e-u\perp u$.

Range and support tripotent are examples of open and compact tripotents,
respectively. In attempt to generalize the studies of C.A. Akemann, L.G.
Brown, and G.K. Pedersen on open and compact projections in the bidual of a
C∗-algebra ([1, 2, 3, 12, 4]), C.M. Edwards and G.T. Rüttimann introduce in
[19] the notions of _open_ and _compact_ tripotents in the bidual of a
JB∗-triple. We recall that a tripotent $u$ in the bidual of a JB∗-triple $E$
is said to be _open_ when $E^{**}_{2}(u)\cap E$ is weak∗ dense in
$E^{**}_{2}(u)$. A tripotent $e$ in $E^{**}$ is said to be _compact-
$G_{\delta}$_ (relative to $E$) if there exists a norm one element $a$ in $E$
such that $e$ coincides with $s(a)$, the support tripotent of $a$. A tripotent
$e$ in $E^{**}$ is said to be _compact_ (relative to $E$) if there exists a
decreasing net $(e_{\lambda})$ of tripotents in $E^{**}$ which are
compact-$G_{\delta}$ with infimum $e$, or if $e$ is zero. Closed and bounded
tripotents in $E^{**}$ were introduced and studied by the second and third
authors of this note in [21] and [22]. A tripotent $e$ in $E^{**}$ such that
$E^{**}_{0}(e)\cap E$ is weak∗ dense in $E^{**}_{0}(e)$ is called _closed_
relative to $E$. When there exists a norm one element $a$ in $E$ such that
$a=e+P_{0}(e)(a)$, the tripotent $e$ is called _bounded_ (relative to $E$) and
we shall write $e\leq_{{}_{T}}a$ (c.f. [21]). The relation $\leq_{{}_{T}}$ is
consistent with the natural partial order on the set of tripotents, that is,
for any two tripotents $e$ and $u$ we have $e\leq u$ if and only if
$e\leq_{{}_{T}}u.$ An useful result established in [21, Theorem 2.6] (see also
[23, Theorem 3.2]) asserts that a tripotent $e$ in $E^{**}$ is compact if, and
only if, $e$ is closed and bounded.

Given a JBW∗-triple $W,$ a norm one functional $\varphi$ in $W_{*}$ and a norm
one element $z$ in $W$ with $\varphi(z)=1$, Proposition 1.2 in [8] assures
that the mapping

$(x,y)\mapsto\varphi\left\\{x,y,z\right\\}$

is a positive sesquilinear form on $W.$ Furthermore, for every norm one
element $w$ in $W$ satisfying $\varphi(w)=1$, we have
$\varphi\left\\{x,y,z\right\\}=\varphi\left\\{x,y,w\right\\},$ for all $x,y\in
W$. Thus, the mapping
$x\mapsto\|x\|_{\varphi}:=\left(\varphi\left\\{x,x,z\right\\}\right)^{\frac{1}{2}},$
is a prehilbertian seminorm on $W$. The strong*-topology is the topology on
$W$ generated by the family
$\\{\|\cdot\|_{\varphi}:\varphi\in{W_{*}},\|\varphi\|=1\\}$ (c.f. [8]). For
later purposes, we recall that the triple product of a JBW∗-triple is jointly
strong∗-continuous on bounded sets (see [33, Theorem 9]). Since the
strong*-topology of a JBW∗-triple $W$ is compatible with the duality
$(W,W_{*})$ (cf. [33, Corollary 9]), it follows from the bipolar theorem that
for each convex $C\subseteq W$ we have

$\overline{C}^{\sigma(W,W_{*})}=\overline{C}^{S^{*}(W,W_{*})}.$

We shall also make use of the following fact due to L. Bunce (see [11]): Let
$F$ be a JBW∗-subtriple (i.e. a weak∗ closed JB∗-subtriple) of a JBW∗-triple
$W$, then the strong∗-topology of $F$ coincides with the restriction to $F$ of
the strong∗-topology of $W$, that is, $S^{*}(F,F_{*})=S^{*}(W,W_{*})|_{F}$.

Given a Banach space $X$, we habitually regard $X$ as being contained in
$X^{**}$ and we identify the weak∗ closure, in $X^{**}$, of a closed subspace
$Y$ of $X$ with $Y^{**}$. Let $F$ be a JB∗-subtriple of a JB∗-triple $E$ and
let $e$ be a tripotent in
$F^{**}\equiv\overline{F}^{\sigma(E^{**},E^{*})}\subseteq E^{**}$. Corollary
2.9 in [21] and [23, Proposition 3.3] prove that $u$ is compact in $F^{**}$
if, and only if, $u$ is compact in $E^{**}$.

The following lemma, whose statement is required later, can be directly
deduced from the joint strong∗-continuity of the triple product of every
JBW∗-triple on bounded sets and the definition of the corresponding Peirce
projection.

######  Lemma 1.5.

Let $W$ be a JBW∗-triple. Suppose that $(e_{\lambda})$ is a net (or a
sequence) of tripotents in $W$ converging, in the strong∗-topology of $W$ to a
tripotent $e$ in $W$. Let $(x_{\mu})$ be a net (or a sequence) in $W$,
converging to some $x\in W$ in the strong∗-topology. Then, for each
$i\in\\{0,1,2\\}$, the net (sequence) $P_{i}(e_{\lambda})(x_{\mu})$ tends to
$P_{i}(e)(x)$. $\hfill\Box$

##  2\. Local triple derivations on JB∗-triples are triple derivations

A triple derivation on a JB∗-triple $E$ is a linear mapping $\delta:E\to E$
satisfying that

$\delta\left\\{a,b,c\right\\}=\left\\{\delta(a),b,c\right\\}+\left\\{a,\delta(b),c\right\\}+\left\\{a,b,\delta(c)\right\\},$

for every $a,b,c\in E$. The Jordan identity implies that, for each $a,b$ in
$E$, the mapping $\delta(a,b)=L(a,b)-L(b,a)$ is a triple derivation on $E$.
Every triple derivation on a JB∗-triple is continuous (cf. [8, Corollary 2.2]
and [34, Corollary 10]). The separate weak∗ continuity of the triple product
of $E^{**}$ together with Goldstine’s theorem, imply that

(4) $\delta^{**}\hbox{ is a triple derivation on }E^{**}\hbox{ whenever
$\delta$ is a triple derivation on $E$.}$

This section contains the main result of the paper, which asserts that every
continuous local triple derivation $T$ on a JB∗-triple $E$ is a derivation.
Our strategy will consist in studying the behavior of $T^{**}$ on compact and
range tripotents in $E^{**}$. We start with a technical lemma borrowed from
[14].

######  Lemma 2.1.

[14, Lemma 4] Let $F$ be a JB∗-subtriple of a JB∗-triple $E$, where the latter
is regarded as a Jordan Banach triple $F$-module with respect to its natural
triple product. Let $T:F\to E$ be a local triple derivation. Then the products
of the form $\left\\{a,T(b),c\right\\}$ vanish for every $a,b,c$ in $F$ with
$a,c\perp b$. $\hfill\Box$

We shall survey now some of the properties of triple derivations. Let
$\delta:E\to E$ be a triple derivation on a JB∗-triple. Suppose that $e$ is a
tripotent in $E$. In such a case,

$\delta(e)=\delta\left\\{e,e,e\right\\}=2\left\\{\delta(e),e,e\right\\}+\left\\{e,\delta(e),e\right\\}=2P_{2}(e)(\delta(e))+P_{1}(e)(\delta(e))+Q(e)(\delta(e))$
$=2P_{2}(e)(\delta(e))+P_{1}(e)(\delta(e))+\Big{(}P_{2}(e)(\delta(e))\Big{)}^{*_{e}},$

which implies that

(5) $P_{0}(e)(\delta(e))=0\hbox{ and
}P_{2}(e)(\delta(e))=-\Big{(}P_{2}(e)(\delta(e))\Big{)}^{*_{e}}=-Q(e)(\delta(e)).$

If $T:E\to E$ is merely a local triple derivation on $E$, we can find a triple
derivation $\delta_{e}:E\to E$ such that $T(e)=\delta_{e}(e)$, which gives

(6) $P_{0}(e)(T(e))=0\hbox{ and
}P_{2}(e)(T(e))=-\Big{(}P_{2}(e)(T(e))\Big{)}^{*_{e}}=-Q(e)(T(e)).$

Though a JB∗-triple $E$ need not contain, in general, tripotent elements, its
bidual has a rich set of tripotents. The next proposition explains the
behavior of a continuous local triple derivation on a JB∗-triple $E$ on
compact tripotents in $E^{**}.$

######  Proposition 2.2.

Let $T:E\to E$ be a bounded local triple derivation on a JB∗-triple. Suppose
$e$ is a compact tripotent in $E^{**}.$ Then the following statements hold:

  1. $(a)$

$P_{0}(e)T^{**}(e)=0$;

  2. $(b)$

If $a$ is a norm one element in $E$ whose support tripotent is $e$ (that is,
$e$ is a compact-$G_{\delta}$ tripotent), then $Q(e)T(a)=Q(e)T^{**}(e);$

  3. $(c)$

$P_{2}(e)T^{**}(e)=-Q(e)(T^{**}(e)).$

###### Proof.

$(a)$ Let us assume that $e$ is a compact-$G_{\delta}$ tripotent. So, there
exists a norm one element $a$ in $E$ such that $s(a)=e$. Let $E_{a}$ denote
the JB∗-subtriple of $E$ generated by $a$. We have already mentioned that
there exists a subset $L\subseteq(0,1]$ with $1\in\\{0\\}\cup L$ compact and a
triple isomorphism $\Psi$ from $E_{a}$ onto $C_{0}(L)$ such that
$\Psi(a)(t)=t,$ $\forall t\in L$ (see page 1.1). Pick a sequence of norm one
elements $(b_{n})$ in $E_{a}$ such that $b_{n}=e+P_{0}(e)(b_{n}),$
$\left\\{b_{n},b_{n+1},b_{n}\right\\}=\left\\{b_{n},b_{n},b_{n+1}\right\\}=b_{n+1}$,
$(b_{n})\to e,$ in the strong∗-topology of $E^{**}$ (take, for example,

$b_{n}(t):=\left\\{\begin{array}[]{ll}0,&\hbox{if $t\in
L\cap[0,1-\frac{1}{n}]$;}\\\ n(n+1)(t-1+\frac{1}{n}),&\hbox{if $t\in
L\cap[1-\frac{1}{n},1-\frac{1}{n+1}]$;}\\\ 1,&\hbox{if $t\in
L\cap[1-\frac{1}{n+1},1]$.}\\\ \end{array}\right).$

Fix a natural $n.$ Since, the support tripotent of $b_{n},$ $s(b_{n})$, is a
compact tripotent in $E^{**}$, given $z,w\in E_{0}^{**}(s(b_{n}))$ we can find
(bounded) nets $(c_{\mu})$ and $(d_{\nu})$ in $E_{0}^{**}(s(b_{n}))\cap E$
converging to $z$ and $w$ in the strong∗-topology of $E^{**},$ respectively.
Clearly, $c_{\mu},d_{\nu}\perp b_{n+1}$ for every $\mu$ and $\nu$, and hence,
by Lemma 2.1,

$\left\\{c_{\mu},T(b_{n+1}),d_{\nu}\right\\}=0\ \ (\forall\mu,\nu).$

Taking strong∗-limits in $\mu$ and $\nu$ we have

$\left\\{z,T(b_{n+1}),w\right\\}=0,\hbox{ for every }n\in\mathbb{N},z,w\in
E_{0}^{**}(s(b_{n})),$

equivalently,

$\left\\{P_{0}^{**}(s(b_{n}))(x),T(b_{n+1}),P_{0}^{**}(s(b_{n}))(y)\right\\}=0,\hbox{
for every }n\in\mathbb{N},x,y\in E^{**}.$

Now, since the triple product of $E^{**}$ is jointly strong∗-continuous and
$T^{**}$ is strong∗-continuous, we can take strong∗-limit in the above
expression, and by Lemma 1.5, we have
$\left\\{P_{0}^{**}(e)(x),T^{**}(e),P_{0}^{**}(e)(y)\right\\}=0,$ for every
$x,y\in E^{**}$. It follows, for example, from Peirce arithmetic and the third
axiom in the definition of JB∗-triples, that $P_{0}^{**}(e)T^{**}(e)=0$.

Let us consider a compact tripotent $e\in E^{**}.$ By definition, there exists
a decreasing net $(e_{\lambda})$ of compact-$G_{\delta}$ tripotents in
$E^{**}$ converging to $e$ in the strong∗-topology of $E^{**}$. From the above
argument, $P_{0}(e_{\lambda})T^{**}(e_{\lambda})=0$ ($\forall\lambda$), and by
Lemma 1.5, $P_{0}(e)T^{**}(e)=0$, as we desired.

$(b)$ Again, let $a$ be a norm one element in $E$ such that $s(a)=e$. Let us
denote $a_{0}=P_{0}(e)(a)$. Adapting the previous arguments, we consider the
JB∗-subtriple, $E_{a},$ generated by $a$, and pick two sequences $(a_{n})$ and
$(b_{n})$ in the closed unit ball of $E_{a}$ defined by

$a_{n}(t):=\left\\{\begin{array}[]{ll}t,&\hbox{if $t\in
L\cap[0,1-\frac{1}{n}]$;}\\\ -n(n+1)(t-1+\frac{1}{n+1}),&\hbox{if $t\in
L\cap[1-\frac{1}{n},1-\frac{1}{n+1}]$;}\\\ 0,&\hbox{if $t\in
L\cap[1-\frac{1}{n+1},1]$}\\\ \end{array}\right.,$

and

$b_{n}(t):=\left\\{\begin{array}[]{ll}0,&\hbox{if $t\in
L\cap[0,1-\frac{1}{n+1}]$;}\\\ (n+1)(t-1+\frac{1}{n+1}),&\hbox{if $t\in
L\cap[1-\frac{1}{n+1},1]$.}\\\ \end{array}\right.$

Clearly, $a_{n}\perp b_{n}$ ($\forall n$), $(a_{n})\to a_{0}$ and $(b_{n})\to
e$ in the strong∗-topology of $E^{**}.$ Lemma 2.1 assures that

$\left\\{b_{n},T(a_{n}),b_{n}\right\\}=0\ \ (\forall n\in\mathbb{N}).$

Taking strong∗ limits in the above expression we have
$\left\\{e,T^{**}(a_{0}),e\right\\}=0$, and hence
$\left\\{e,T^{**}(a),e\right\\}=\left\\{e,T^{**}(e),e\right\\}.$

$(c)$ Assume, one more time, that $e$ is a compact-$G_{\delta}$ tripotent and
$a$ is a norm one element in $E$ such that $s(a)=e$. Since $T$ is a local
triple derivation, we can find a triple derivation $\delta_{a}:E\to E$ such
that $T(a)=\delta_{a}(a).$ We notice that $\delta_{a}^{**}:E^{**}\to E^{**}$
is a triple derivation on $E^{**}$ (see (4)). Since $\delta_{a}$ is triple
derivation, the identity in $(b)$ also holds whenever we replace $T$ with
$\delta_{a}.$ Therefore,

$P_{2}(e)T^{**}(e)=P_{2}(e)T(a)=P_{2}(e)\delta_{a}(a)=P_{2}(e)\delta_{a}^{**}(e)$
$\hbox{(by $(\ref{eq triple derivations on
tripotents})$)}=-Q(e)\Big{(}\delta_{a}^{**}(e)\Big{)}=-Q(e)\Big{(}\delta_{a}(a)\Big{)}=-Q(e)\Big{(}T(a)\Big{)}=-Q(e)\Big{(}T^{**}(e)\Big{)},$

which proves statement $(c)$ for compact-$G_{\delta}$ tripotents in $E^{**}$.

Let us consider a decreasing net $(e_{\lambda})$ of compact-$G_{\delta}$
tripotents in $E^{**}$ converging to $e$ in the strong∗-topology of $E^{**}$.
Since, for each $\lambda$,
$P_{2}(e_{\lambda})T^{**}(e_{\lambda})=-Q(e_{\lambda})(T^{**}(e_{\lambda}))$,
Lemma 1.5, assures the desired identity for $e$.∎

In the hypothesis of the above proposition, let $a$ be a norm one element in
$E$ and let $E_{a}$ denote the JB∗-subtriple of $E$ generated by $a$. By the
Gelfand theory for commutative JB∗-triples, there exists a subset
$L\subseteq(0,1]$ with $1\in\\{0\\}\cup L$ compact and a triple isomorphism
$\Psi$ from $E_{a}$ onto $C_{0}(L)$ such that $\Psi(a)(t)=t$ ($\forall t\in
L$). Clearly, the range tripotent of $a$ can be approximated, in the strong∗
topology of $E^{**}$, by a sequence $(e_{n})$ of compact-$G_{\delta}$
tripotents in $E^{**}$, that is, $(e_{n})\to r(a)$ in the strong∗-topology.
Since, by the above Proposition 2.2,

$P_{0}(e_{n})T^{**}(e_{n})=0,$

and

$P_{2}(e_{n})T^{**}(e_{n})=-Q(e_{n})(T^{**}(e_{n})),$

taking strong∗-limit in $n$ we deduce, by Lemma 1.5, that

(7) $P_{0}(r(a))T^{**}(r(a))=0,\hbox{ and,
}P_{2}(r(a))T^{**}(r(a))=-Q(r(a))(T^{**}(r(a))).$

Let $e$ be a compact or a range tripotent in $E^{**}$. It follows from
Proposition 2.2 and (7) that

$T^{**}\left\\{e,e,e\right\\}=T^{**}(e)=P_{2}(e)T^{**}(e)+P_{1}(e)T^{**}(e),$
$2\left\\{e,e,T^{**}(e)\right\\}=2P_{2}(e)T^{**}(e)+P_{1}(e)T^{**}(e),$

and

$\left\\{e,T^{**}(e),e\right\\}=Q(e)T^{**}(e)=-P_{2}(e)T^{**}(e),$

which assures that

(8)
$T^{**}\left\\{e,e,e\right\\}=2\left\\{e,e,T^{**}(e)\right\\}+\left\\{e,T^{**}(e),e\right\\}.$

######  Corollary 2.3.

Let $T:E\to E$ be a continuous local triple derivation on a JB∗-triple.
Suppose that $e$ and $u$ are two orthogonal compact tripotents in $E^{**}$,
$r_{1}$ and $r_{2}$ are two orthogonal range tripotent in $E^{**}$ and $e\perp
r_{2}$. Then the following identities hold:

$T^{**}\left\\{e\pm u,e\pm u,e\pm u\right\\}=2\left\\{e\pm u,e\pm
u,T^{**}(e\pm u)\right\\}+\left\\{e\pm u,T^{**}(e\pm u),e\pm u\right\\},$
$T^{**}\left\\{r_{1}\pm r_{2},r_{1}\pm r_{2},r_{1}\pm
r_{2}\right\\}=2\left\\{r_{1}\pm r_{2},r_{1}\pm r_{2},T^{**}(r_{1}\pm
r_{2})\right\\}$ $+\left\\{r_{1}\pm r_{2},T^{**}(r_{1}\pm r_{2}),r_{1}\pm
r_{2}\right\\};$

and

$T^{**}\left\\{e\pm r_{2},e\pm r_{2},e\pm r_{2}\right\\}=2\left\\{e\pm
r_{2},e\pm r_{2},T^{**}(e\pm r_{2})\right\\}$ $+\left\\{e\pm r_{2},T^{**}(e\pm
r_{2}),e\pm r_{2}\right\\}.$

###### Proof.

Since $e$ and $u$ are two orthogonal compact tripotents in $E^{**},$ it
follows from Proposition 3.7 in [23] that $e\pm u$ is a compact tripotent in
$E^{**}.$ It is easy to see that the sum and the difference of two orthogonal
range tripotents in $E^{**}$ is again a range tripotent in $E^{**}$. Thus, the
first and the second identity have been proved in (8).

To see the last identity, we recall that since $e\perp r_{2}$ and $r_{2}$ is a
range projection, we can find a sequence of compact tripotents $(e_{n})$ in
$E^{**}$ such that $e_{n}\leq r_{2}$, and hence $e_{n}\perp e$ for every $n$,
and $(e_{n})\to r_{2}$ in the strong∗-topology of $E^{**}.$ The first identity
shows that

$T^{**}\left\\{e\pm e_{n},e\pm e_{n},e\pm e_{n}\right\\}=2\left\\{e\pm
e_{n},e\pm e_{n},T^{**}(e\pm e_{n})\right\\}$ $+\left\\{e\pm e_{n},T^{**}(e\pm
e_{n}),e\pm e_{n}\right\\},$

for every $n$. The desired equality follows by taking strong∗-limit in $n$. ∎

Let $T:E\to E$ be a bounded local triple derivation on a JB∗-triple. Another
application of the local Gelfand structure of JB∗-triples allows us to see
that each element $a$ in a JB∗-triple $E$ can be approximated in norm by a
finite real linear combination of mutually orthogonal range and compact
tripotents in $E_{a}^{**}\subseteq E^{**}$. Corollary 2.3 will imply that
$T^{**}$ behaves as a triple derivation on those elements which are finite
real linear combinations of mutually orthogonal range and compact tripotents
in $E^{**},$ and consequently, $T$ (and hence $T^{**}$) is a triple
derivation.

We can state now our main result, which solves a problem conjectured by M.
Mackey in [31, Conjecture (C3)] (and also posed in [14, Problem 1]).

######  Theorem 2.4.

Every bounded local triple derivation on a JB∗-triple is a triple derivation.

###### Proof.

Let $T:E\to E$ be a bounded local triple derivation on a JB∗-triple $E$. Let
$e_{1},\ldots,e_{m}$ be a family of mutually orthogonal range or compact
tripotents in $E^{**}$. Let us pick $i,j,k\in\\{1,\ldots,m\\}$ with $i,k\neq
j.$ By Proposition 2.2 and $(\ref{eq range tripotents in bidual})$ we have
$P_{0}(e_{j})T^{**}(e_{j})=0.$ By assumptions, $e_{i},e_{k}\in
E^{**}_{0}(e_{j})$ and hence

(9) $\left\\{e_{i},T^{**}(e_{j}),e_{k}\right\\}=0,$

by Peirce arithmetic.

Now, fix $i\neq j$ in $\\{1,\ldots,m\\}$. Since $e_{i}$ and $e_{j}$ are
compact or range tripotents in $E^{**}$, Corollary 2.3 implies that

(10)
$T^{**}\left\\{e_{i},e_{i},e_{i}\right\\}=2\left\\{e_{i},e_{i},T^{**}(e_{i})\right\\}+\left\\{e_{i},T^{**}(e_{i}),e_{i}\right\\};$

and

$T^{**}\left\\{e_{i}\pm e_{j},e_{i}\pm e_{j},e_{i}\pm
e_{j}\right\\}=2\left\\{e_{i}\pm e_{j},e_{i}\pm e_{j},T^{**}(e_{i}\pm
e_{j})\right\\}$ $+\left\\{e_{i}\pm e_{j},T^{**}(e_{i}\pm e_{j}),e_{i}\pm
e_{j}\right\\}.$

Combining the last two identities we get

$\pm
2\left\\{e_{i},e_{i},T^{**}(e_{j})\right\\}+2\left\\{e_{j},e_{j},T^{**}(e_{i})\right\\}\pm\left\\{e_{i},T^{**}(e_{j}),e_{i})\right\\}+\left\\{e_{j},T^{**}(e_{i}),e_{j}\right\\}$
$\pm
2\left\\{e_{i},T^{**}(e_{i}),e_{j}\right\\}+2\left\\{e_{i},T^{**}(e_{j}),e_{j}\right\\}=0,$

and consequently,

$+4\left\\{e_{j},e_{j},T^{**}(e_{i})\right\\}+2\left\\{e_{j},T^{**}(e_{i}),e_{j}\right\\}+4\left\\{e_{i},T^{**}(e_{j}),e_{j}\right\\}=0.$

Applying (9) we obtain

(11)
$\left\\{e_{j},e_{j},T^{**}(e_{i})\right\\}+\left\\{e_{i},T^{**}(e_{j}),e_{j}\right\\}=0.$

Consider now an element $\displaystyle b=\sum_{i=1}^{m}\lambda_{i}e_{i},$
where $e_{1},\ldots,e_{m}$ are as above. Having in mind that
$e_{1},\ldots,e_{m}$ are mutually orthogonal, we compute

(12)
$T^{**}(\left\\{b,b,b\right\\})=\sum_{i=1}^{m}\lambda_{i}^{3}T^{**}(\left\\{e_{i},e_{i},e_{i}\right\\});$
(13)
$2\left\\{T^{**}(b),b,b\right\\}=2\sum_{i,j=1}^{m}\lambda_{i}^{2}\lambda_{j}\left\\{e_{i},e_{i},T^{**}(e_{j})\right\\}$
$=2\sum_{i=1}^{m}\lambda_{i}^{3}\left\\{e_{i},e_{i},T^{**}(e_{i})\right\\}+2\sum_{i,j=1,i\neq
j}^{m}\lambda_{i}^{2}\lambda_{j}\left\\{e_{i},e_{i},T^{**}(e_{j})\right\\};$
(14) $\left\\{b,T^{**}(b),b\right\\}=\hbox{ (by ($\ref{eq 1 theorem main}$))
}=\sum_{i=1}^{m}\lambda_{i}^{3}\left\\{e_{i},T^{**}(e_{i}),e_{i}\right\\}+2\sum_{i,j=1,i\neq
j}^{m}\lambda_{i}^{2}\lambda_{j}\left\\{e_{i},T^{**}(e_{i}),e_{j}\right\\}.$

Finally, by $(\ref{eq 1b theorem main})$ and $(\ref{eq 2 theorem main})$ we
have

$T^{**}\left\\{b,b,b\right\\}=2\left\\{T^{**}(b),b,b\right\\}+\left\\{b,T^{**}(b),b\right\\}.$

Since every element $a\in E$ can be approximated in norm, by elements of the
form $\displaystyle b=\sum_{i=1}^{m}\lambda_{i}e_{i}$, we conclude that
$T\left\\{a,a,a\right\\}=2\left\\{T^{**}(a),a,a\right\\}+\left\\{a,T^{**}(a),a\right\\}$,
for every $a\in E.$ Finally, an standard polarisation argument, via formula

(15)
$\\{x,y,z\\}=8^{-1}\sum_{k=0}^{3}\sum_{j=1}^{2}(-1)^{j}i^{k}\Big{(}x+i^{k}\
y+(-1)^{j}z\Big{)}^{[3]}\ \ \ (x,y,z\in{E}),$

assures that $T$ is a triple derivation. ∎

In 1997, J.M. Isidro, W. Kaup and A. Rodrguez Palacios introduce a strictly
wider class of Jordan triples over the real field and called the elements of
this new category _real JB ∗-triples_. A _real JB ∗-triple_ is a norm closed
real subtriple of a JB∗-triple (cf. [25]). When restricted to real scalar
multiplication, every (complex) JB∗-triple is also a real JB∗-triple. The
class of real JB∗-triples also includes real C∗-algebras, JB-algebras, JC-
algebras, real JB∗-algebras, operator spaces between real, complex and
quaternionic Hilbert spaces and real Hilbert spaces. Every real JB∗-triple can
be complexified to become a JB∗-triple. Furthermore, every real JB∗-triple $A$
is a real form of a complex JB∗-triple, that is, there exist a (complex)
JB∗-triple $B\cong A\oplus iA$ and a period 2 conjugate linear isometry
$\tau:B\rightarrow B$ (which is also a JB∗-triple homomorphism) such that
$A=\\{b\in B\>:\>\tau(b)=b\\}$ (see [25]). Let us consider
$\widetilde{\tau}:B^{*}\rightarrow B^{*}$ defined by
$\widetilde{\tau}(\phi)(z)=\overline{\phi(\tau(z))}.$ The mapping
$\widetilde{\tau}$ is a conjugation on $B^{*}$, and the restriction mapping

$(B^{*})^{\widetilde{\tau}}\longrightarrow(B^{\tau})^{*}\ (\cong A^{*})$
$\phi\mapsto\phi|_{A}$

is an isometric bijection, where $(B^{*})^{\widetilde{\tau}}:=\\{\phi\in
B^{*}:\widetilde{\tau}(\phi)=\phi\\}.$ The second transpose
$\tau^{**}:B^{**}\to B^{**}$ is a period 2 conjugate linear isometry
satisfying $\left(B^{**}\right)^{\tau^{**}}=A^{**}$.

A real _JBW ∗-triple_ is a real JB∗-triple which is also a dual Banach space.
Every real JBW∗-triple has a unique (isometric) predual and separately weak∗
continuous triple product (see [32]). The bidual of every real JB∗-triple is a
JBW∗-triple (compare [25]).

The notions of range, support or compact-$G_{\delta}$, open, closed and
compact tripotents also make sense in the bidual of every real JB∗-triple.
When real JB∗-triples are regarded as real forms of complex JB∗-triples, the
generalised Urysohn’s lemmas proved by the second and third author of this
note in [23] remain valid for real JB∗-triples. Furthermore, an appropriate
local Gelfand theory for single-generated real JB∗-subtriples is also
available in the real setting (cf. [15, §3]). Thus, the arguments given above
to prove Theorem 2.4 can be applied to show that every bounded local triple
derivation $T$ on a real JB∗-triple $A$ satisfies that
$T\left\\{a,a,a\right\\}=2\left\\{T(a),a,a\right\\}+\left\\{a,T(a),a\right\\}$,
for every $a\in A.$ Unfortunately, the polarisation formula (15) employed at
the end of the proof of Theorem 2.4 is not valid for real JB∗-triples, so we
cannot obtain the conclusion of that theorem in the real setting.

######  Corollary 2.5.

Let $T$ be a continuous local triple derivation on a real JB∗-triple $A$. Then
$T\left\\{a,a,a\right\\}=2\left\\{T(a),a,a\right\\}+\left\\{a,T(a),a\right\\}$,
for every $a\in A,$ that is, $T$ is a triple derivation of the symmetrized
Jordan triple product
$3<\\!a,b,c\\!>:=\left\\{a,b,c\right\\}+\left\\{c,a,b\right\\}+\left\\{b,c,a\right\\}.$
$\hfill\Box$

In the light of the above corollary, it seems natural to consider the
following problem:

######  Problem 2.6.

Is every bounded local triple derivation on a real JB∗-triple a triple
derivation?

In a recent paper (see [34]), B. Russo and the third author of this note
initiated the study of triple module-valued triple derivations on (real and
complex) JB∗-triples. Let $E$ be a complex (resp. real) Jordan triple. We
recall that a _Jordan triple $E$-module_ (also called _triple $E$-module_) is
a vector space $X$ equipped with three mappings

$\\{.,.,.\\}_{1}:X\times E\times E\to X,\quad\\{.,.,.\\}_{2}:E\times X\times
E\to X$ $\hbox{ and }\\{.,.,.\\}_{3}:E\times E\times X\to X$

satisfying the following axioms:

  1. $(JTM1)$

$\\{x,a,b\\}_{1}$ is linear in $a$ and $x$ and conjugate linear in $b$ (resp.,
trilinear), $\\{a,b,x\\}_{3}$ is linear in $b$ and $x$ and conjugate linear in
$a$ (resp., trilinear) and $\\{a,x,b\\}_{2}$ is conjugate linear in $a,b,x$
(resp., trilinear)

  2. $(JTM2)$

$\\{x,b,a\\}_{1}=\\{a,b,x\\}_{3}$, and $\\{a,x,b\\}_{2}=\\{b,x,a\\}_{2}$ for
every $a,b\in E$ and $x\in X$.

  3. $(JTM3)$

Denoting by $\left\\{.,.,.\right\\}$ any of the products $\\{.,.,.\\}_{1}$,
$\\{.,.,.\\}_{2}$ and $\\{.,.,.\\}_{3}$, the Jordan identity (1) holds
whenever one of the elements is in $X$ and the rest are in $E$.

When the products $\\{.,.,.\\}_{1}$, $\\{.,.,.\\}_{2}$ and $\\{.,.,.\\}_{3}$
are (jointly) continuous we shall say that $X$ is a _Banach (Jordan) triple
$E$-module_. Henceforth, the triple products
$\left\\{\cdot,\cdot,\cdot\right\\}_{j}$, $j=1,2,3$, which occur in the
definition of Jordan triple module will be denoted simply by
$\left\\{\cdot,\cdot,\cdot\right\\}$ whenever the meaning is clear from the
context.

It is obvious that every real or complex Jordan triple $E$ is a real triple
$E$-module. The dual space, $E^{*}$, of a complex (resp., real) Jordan Banach
triple $E$ is a complex (resp., real) triple $E$-module with respect to the
products:

$\left\\{a,b,\varphi\right\\}(x)=\left\\{\varphi,b,a\right\\}(x):=\varphi\left\\{b,a,x\right\\},\hbox{
and,
}\left\\{a,\varphi,b\right\\}(x):=\overline{\varphi\left\\{a,x,b\right\\}},$

for all $\varphi\in E^{*}$, $a,b,x\in E$.

Let $E$ be a complex (respectively, real) JB∗-triple and let $X$ be a triple
$E$-module. We recall that a conjugate linear (respectively, linear) mapping
$\delta:E\to X$ is said to be a _triple or ternary derivation_ if

$\delta\\{a,b,c\\}=\left\\{\delta(a),b,c\right\\}+\left\\{a,\delta(b),c\right\\}+\left\\{a,b,\delta(c)\right\\}.$

A conjugate linear (respectively, linear) mapping $T:E\to X$ will be called a
_local triple derivation_ if for each $a$ in $E$ there exists a triple
derivation $\delta_{a}:E\to X$ such that $T(a)=\delta_{a}(a).$

######  Problem 2.7.

Is every continuous local triple derivation from a real or complex JB∗-triple
$E$ into a Banach triple $E$-module a triple derivation?

###  2.1. Automatic continuity

We establish now an automatic continuity result for local triple derivations,
giving a positive answer to Problem (1.3). We shall review the arguments given
by T. Barton and Y. Friedman to show that a triple derivation on a JB∗-triple
is a triple derivation. Let $X$ be a complex Banach space. We recall that a
linear mapping $T:X\to X$ is _dissipative_ if for each $x\in X$ and each
functional $\phi\in X^{*}$ with $\|x\|=\|\phi\|=\phi(x)=1$ we have
$\Re\hbox{e}\phi(T(x))\leq 0$. It is known that $T$ is continuous whenever it
is dissipative (compare [10, Proposition 3.1.15]). In [8, Theorem 2.1], Barton
and Friedman prove that every derivation on a JB∗-triple $E$ is dissipative
and hence continuous. Let us review some aspect in their proof. Let $x$ be an
element in $E$, let $\phi$ a functional in $E^{*}$ with
$\|x\|=\|\phi\|=\phi(x)=1$ and let $\delta:E\to E$ a triple derivation.
Applying Peirce arithmetic and support tripotents, the arguments in the proof
of Theorem 2.1 in [8] show that $\phi\delta(\left\\{x,x,x\right\\}-x)=0$ and
hence $\Re\hbox{e}\phi(\delta(x))=0$. It is also justified in the same result
that $\phi\left\\{a,x,x\right\\}=\phi(a)$ and
$\phi\left\\{x,a,x\right\\}=\overline{\phi(a)}$, for every $a$ in $E$. Let
$T:E\to E$ be a local triple derivation, not assumed a priori to be
continuous, on $E$. Pick a triple derivation $\delta_{x}$ satisfying
$\delta_{x}(x)=T(x)$. In this case, we have

$2\phi T(x)+\overline{\phi
T(x)}=2\phi\left\\{T(x),x,x\right\\}+\phi\left\\{x,T(x),x\right\\}$
$=2\phi\left\\{\delta_{x}(x),x,x\right\\}+\phi\left\\{x,\delta_{x}(x),x\right\\}=\phi\delta_{x}\left\\{x,x,x\right\\}.$

It follows from the above that

$2\Re\hbox{e}\phi T(x)=\phi\delta_{x}\left\\{x,x,x\right\\}-\phi
T(x)=\phi\delta_{x}\left\\{x,x,x\right\\}-\phi\delta_{x}(x)=\phi\delta_{x}(\left\\{x,x,x\right\\}-x)=0,$

because $\delta_{x}$ is a triple derivation on $E$. This shows that $T$ is
dissipative.

######  Theorem 2.8.

Every local triple derivation on a (real or complex) JB∗-triple is continuous.
Consequently, every local triple derivation on a JB∗-triple is a triple
derivation.$\hfill\Box$

##  3\. Local triple derivations and generalised derivations on C∗-algebras

We begin this section with the following corollary which solves Problem 2 in
[14].

######  Theorem 3.1.

Every local triple derivation on a C∗-algebra is a triple derivation.
$\hfill\Box$

In [14], it is established that every local triple derivation on a unital
C∗-algebra is a triple derivation. The strategy to prove this result relies on
the connections between (local) triple derivations and generalised derivations
on unital C∗-algebras in the sense introduced by J. Li and Z. Pan in [30]. We
recall that a linear mapping $D$ from a unital C∗-algebra $B$ to a (unital)
Banach $B$-bimodule $X$ is called a _generalised derivation_ whenever the
identity

$D(ab)=D(a)b+aD(b)-aD(1)b$

holds for every $a,b$ in $B$. A _generalised Jordan derivation_ from $B$ to
$X$ is a linear mapping $D:B\to X$ satisfying $D(a\circ b)=D(a)\circ b+a\circ
D(b)-U_{a,b}D(1)$, for every $a,b$ in $B$, where the Jordan product is given
by $a\circ b:=\frac{1}{2}(ab+ba)$ and $U_{a,b}(x):=\frac{1}{2}(axb+bxa)$ (cf.
[14]). Every generalised (Jordan) derivation $D:B\to X$ with $D(1)=0$ is a
(Jordan) derivation. As remarked in [14, Remark 8], generalised Jordan
derivations from $B$ to $X$ and a generalised derivations from $B$ to $X$
coincide. A linear mapping $T:B\to X$ is a _local generalised (Jordan)
derivation_ if for each $a\in B$, there exists a generalised (Jordan)
derivation $D_{a}:B\to X$ with $T(a)=D_{a}(a)$.

In the setting of unital C∗-algebras, J. Alaminos, M. Bresar, J. Extremera,
and A. Villena [5, Corollary 3.2] and J. Li and Z. Pan [30, Corollary 2.9]
established the following interesting result:

######  Theorem 3.2.

([5, Corollary 3.2], [30, Corollary 2.9]) Suppose that $B$ is a unital
C∗-algebra and $M$ is a unital Banach $B$-bimodule. Then for any bounded
linear map $T$ from $B$ to $M$, the following are equivalent:

  1. $(i)$

$T$ is a generalised (Jordan) derivation from $B$ to $M$;

  2. $(ii)$

$T$ is a local generalised (Jordan) derivation from $B$ to $M$;

  3. $(iii)$

$aT(b)c=0$, whenever $a,b,c\in B$ with $ab=bc=0$.$\hfill\Box$

Unfortunately, the above definitions of generalised (Jordan) derivations and
local generalised (Jordan) derivation make sense only when the domain is a
unital C∗-algebra. However, statement $(iii)$ in the above theorem makes sense
in the non-unital setting too. Our next goal is to generalise the above
theorem to the setting of not necessarily unital C∗-algebras. In a first step
we should consider a consequence derived from Theorem 4.5 in [6]. First we
recall a definition taken from [6, §4]: a generalised derivation from a Banach
algebra $A$ to a Banach $A$-bimodule $X$ is a linear operator $D:A\to X$ for
which there exists $\xi\in X^{**}$ satisfying

$D(ab)=D(a)b+aD(b)-a\xi b\hbox{ ($a,b\in A$).}$

######  Theorem 3.3.

[6, Theorem 4.5] Suppose that $A$ is a C∗-algebra and $X$ is an essential
Banach $A$-bimodule. Let $T:A\to X$ be a continuous linear operator satisfying

$a,b,c\in A,ab=bc=0\Rightarrow aT(b)c=0.$

Then $T$ is a generalized derivation.

Given a C∗-algebra, $A$, the _multiplier algebra_ of $A$, $M(A)$, is the set
of all elements $x\in A^{**}$ such that, for each element $a\in A$, $xa$ and
$ax$ both lie in $A$. We notice that $M(A)$ is a C∗-algebra and contains the
unit element of $A^{**}$. Clearly, $A=M(A)$ whenever $A$ is unital.

For later purposes, we recall some basic results on Arens regularity (cf.
[7]). Let $X$, $Y$ and $Z$ be Banach spaces and let $m:X\times Y\to Z$ be a
bounded bilinear mapping. Defining
$m^{*}(z^{\prime},x)(y):=z^{\prime}(m(x,y))$ $(x\in X,y\in Y,z^{\prime}\in
Z^{*})$, we obtain a bounded bilinear mapping $m^{*}:Z^{*}\times X\to Y^{*}.$
Iterating the process, we define a mapping $m^{***}:X^{**}\times Y^{**}\to
Z^{**}.$ The mapping $x^{\prime\prime}\mapsto
m^{***}(x^{\prime\prime},y^{\prime\prime})$ is weak∗ to weak∗ continuous
whenever we fix $y^{\prime\prime}\in Y^{**}$, and the mapping
$y^{\prime\prime}\mapsto m^{***}(x,y^{\prime\prime})$ is weak∗ to weak∗
continuous for every $x\in X$. One can consider the transposed mapping
$m^{t}:Y\times X\to Z,$ $m^{t}(y,x)=m(x,y)$ and the extended mapping
$m^{t***t}:X^{**}\times Y^{**}\to Z^{**}.$ In this case, the mapping
$x^{\prime\prime}\mapsto m^{t***t}(x^{\prime\prime},y)$ is weak∗ to weak∗
continuous whenever we fix $y\in Y$, and the mapping $y^{\prime\prime}\mapsto
m^{t***t}(x^{\prime\prime},y^{\prime\prime})$ is weak∗ to weak∗ continuous for
every $x^{\prime\prime}\in X^{**}$.

In general, the mappings $m^{t***t}$ and $m^{***}$ do not coincide (cf. [7]).
When $m^{t***t}=m^{***},$ we say that $m$ is Arens regular. It is well known
that the product of every C∗-algebra is Arens regular and the unique Arens
extension of the product of $A$ to $A^{**}\times A^{**}$ coincides with the
product of its enveloping von Neumann algebra (cf. [16, Corollary 3.2.37]).

Let $X$ be a Banach $A$-bimodule over a C∗-algebra $A.$ Let us denote by
$\pi_{1}:A\times X\to X$ and $\pi_{2}:X\times A\to X$ the corresponding module
operations given by $\pi_{1}(a,x)=ax$ and $\pi_{2}(x,a)=xa$, respectively. By
an abuse of notation, given $a\in A^{**}$ and $z\in X^{**},$ we shall
frequently write $az=\pi_{1}^{***}(a,z)$ and $za=\pi_{2}^{***}(z,a)$. It is
known that $X^{**}$ is a Banach $A^{**}$-bimodule for the just defined
operations ([16, Theorem 2.6.15$(iii)$]). It is also known that whenever
$(a_{\lambda})$ and $(x_{\mu})$ are nets in $A$ and $X$, respectively, such
that $a_{\lambda}\to a\in A^{**}$ in the weak∗ topology of $A^{**}$ and
$x_{\mu}\to x\in X^{**}$ in the weak∗ topology of $X^{**}$, then

(16) $ax=\pi_{1}^{***}(a,x)=\lim_{\lambda}\lim_{\mu}a_{\lambda}x_{\mu}\hbox{
and }xa=\pi_{2}^{***}(x,a)=\lim_{\mu}\lim_{\lambda}x_{\mu}a_{\lambda}$

in the weak∗ topology of $X^{**}$ (cf. [16, 2.6.26]).

Our next proposition completes the whole picture.

######  Proposition 3.4.

Let $X$ be an essential Banach $A$-bimodule over a C∗-algebra $A$ and let
$T:A\to X$ be a bounded linear operator. The following are equivalent:

  1. $(a)$

$T^{**}|_{M(A)}:M(A)\to X^{**}$ is a generalised (Jordan) derivation;

  2. $(b)$

$T^{**}:A^{**}\to X^{**}$ is a generalised (Jordan) derivation;

  3. $(c)$

$T^{**}=d+M_{T^{**}(1)}$, where $d:A^{**}\to X^{**}$ is a derivation and
$M_{T^{**}(1)}(a)=T^{**}(1)\circ a=\frac{1}{2}(aT^{**}(1)+T^{**}(1)a)$;

  4. $(d)$

$T^{**}:A^{**}\to X^{**}$ is a local generalised (Jordan) derivation;

  5. $(e)$

$T^{**}|_{M(A)}:M(A)\to X^{**}$ is a local generalised (Jordan) derivation;

  6. $(f)$

$T$ is a generalised derivation;

  7. $(g)$

$aT(b)c=0$, whenever $ab=bc=0$ in $A$.

###### Proof.

The implications
$(b)\Leftrightarrow(c)\Rightarrow(d)\Rightarrow(e)\Rightarrow(g)$ are clear.
The equivalence $(f)\Leftrightarrow(g)$ was established in Theorem 3.3 ([6,
Theorem 4.5]). To see $(a)\Rightarrow(b)$ suppose that $T^{**}|_{M(A)}$ is a
generalised derivation, i.e.,

$T(ab)=T(a)b+aT(b)-aT^{**}(1)b,$

for every $a,b$ in $M(A)$. Since, by Goldstine’s theorem, the closed unit ball
of $A$ is weak∗ dense in the closed unit ball of $A^{**}$, we deduce from (16)
that the above equality also holds when $a$ and $b$ are in $A^{**}$ and $T$ is
replaced with $T^{**}.$ Thus, $T^{**}$ is a generalised derivation.

We shall finally prove the implication $(g)\Rightarrow(a)$. Let $a,b,c$ be
elements in $M(A)$ with $ab=bc=0$. We may assume that $a,b$ and $c$ lie in the
closed unit ball of $M(A)$. We observe that $a^{[2n-1]}b=0$ for every natural
$n$. Therefore, $\alpha b=0$, for every $\alpha$ in the JB∗-subtriple,
$M(A)_{a},$ of $M(A)$ generated by $a$. We can similarly show that

(17) $\alpha\beta=\beta\gamma=0$

for every $\alpha\in M(A)_{a}$, $\beta\in M(A)_{b}$ and $\gamma\in M(A)_{c}$.

Since $M(A)$ is a C∗-subalgebra of $A^{**}$, by Goldstine’s theorem, we can
find nets $(x_{\lambda})$, $(y_{\mu})$ and $(z_{\nu})$ in the closed unit ball
of $A$, converging in the weak∗ topology of $A^{**}$ to $a^{[\frac{1}{3}]}$,
$b^{[\frac{1}{3}]}$ and $c^{[\frac{1}{3}]}$, respectively. The nets
$\left(a^{[\frac{1}{3}]}x_{\lambda}^{*}a^{[\frac{1}{3}]}\right)$,
$\left(b^{[\frac{1}{3}]}y_{\mu}^{*}b^{[\frac{1}{3}]}\right)$, and
$\left(c^{[\frac{1}{3}]}z_{\nu}^{*}c^{[\frac{1}{3}]}\right)$ lie in $A$, and
by $(\ref{eq 3 prop multipliers})$, we have

$\left(a^{[\frac{1}{3}]}x_{\lambda}^{*}a^{[\frac{1}{3}]}\right)\left(b^{[\frac{1}{3}]}y_{\mu}^{*}b^{[\frac{1}{3}]}\right)=0=\left(b^{[\frac{1}{3}]}y_{\mu}^{*}b^{[\frac{1}{3}]}\right)\left(c^{[\frac{1}{3}]}z_{\nu}^{*}c^{[\frac{1}{3}]}\right),$

for every $\lambda,\mu$ and $\nu$. Our hypothesis assures that

$\left(a^{[\frac{1}{3}]}x_{\lambda}^{*}a^{[\frac{1}{3}]}\right)T\left(b^{[\frac{1}{3}]}y_{\mu}^{*}b^{[\frac{1}{3}]}\right)\left(c^{[\frac{1}{3}]}z_{\nu}^{*}c^{[\frac{1}{3}]}\right)=0,$

for every $\lambda,\mu$ and $\nu$. Taking weak∗ limit in $\nu$, it follows
from the properties of $\pi_{2}^{***}$ that

$\left(a^{[\frac{1}{3}]}x_{\lambda}^{*}a^{[\frac{1}{3}]}\right)T\left(b^{[\frac{1}{3}]}y_{\mu}^{*}b^{[\frac{1}{3}]}\right)c=0,$

for every $\lambda,$ and $\mu$. Finally, taking weak∗ limits first in $\mu$
and later in $\lambda$ we have $aT^{**}(b)c=0.$ We have therefore shown that
$aT^{**}(b)c=0$ whenever $ab=bc=0$ in $M(A)$. Since $X$ is an essential
$A$-bimodule, and $A^{**}$ is unital, it can be easily checked that $X^{**}$
is a unital $M(A)$-bimodule. Thus, the statement $(a)$ follows from Theorem
3.2 above. ∎

It is worth to notice that the proof of the above theorem makes use of the
local structure of a C∗-algebra when it is regarded as a JB∗-triple. Jordan
techniques are also employed in the next remark.

######  Remark 3.5.

In the hypothesis above, every statement in Proposition 3.4 is equivalent to:

  1. $(h)$

For each right closed ideal $R$ of $A$ and each left closed ideal $L$ of $A$,
we have

$T(R\cap L)\subseteq XL+RX.$

Indeed, suppose that $T^{**}:A^{**}\to X^{**}$ is a generalised derivation.
Let $a$ be an element in the intersection $R\cap L.$ The element
$a^{[3]}=aa^{*}a$ lies in $R$ and in $L$. By induction on $n$, we proved that
$a^{[2n-1]}$ belongs to $L\cap R$ for every natural $n$. Therefore, the
JB∗-subtriple, $A_{a},$ of $A$ generated by $a$ is contained in $R\cap L$. Let
us take $b\in A_{a}$ satisfying $b^{[3]}=a$. Thus,

$T(a)=T(b^{[3]})=T(bb^{*}b)=T(b)b^{*}b+bT(b^{*}b)-bT^{**}(1)b^{*}b\in XL+RX,$

which proves $(b)\Rightarrow(h)$.

Assume now that $T(R\cap L)\subseteq XL+RX$ whenever $R$ is a right closed
ideal of $A$ and $L$ is a left closed ideal of $A$. Pick $a,b$ and $c$ in $A$
with $ab=0=bc$. Let $R=\\{x\in A:ax=0\\}$ and $L=\\{x\in A:xc=0\\}$. In this
case, $R$ is a right closed ideal, $L$ is a left closed ideal and $b\in L\cap
R$, then, by assumptions, $T(b)\in XL+RX$. Therefore, $aT(b)c\in
aXLc+aRXc=\\{0\\}$, which gives the desired equivalence.

When, in the hypothesis of the above Remark 3.5, $X$ coincides with $A$, we
have $XL+RX=L+R$. Under the additional assumption of $A$ being unital, V.
Shul’man established in [35, Theorem 1] that a bounded linear mapping $T:A\to
A$ satisfies statement $(h)$ in Remark 3.5 if, and only if, $T=D+L_{a}$, where
$D:A\to A$ is a derivation and $L_{a}$ is the left multiplication operator by
the element $a=T(1)$ (if and only if $T$ is a generalised derivation). So, our
Proposition 3.4 and the equivalence with Remark 3.5$(h)$ provides us a
generalisation of the result proved by Shul’man.

We culminate this section exploring the connections with (local) triple
derivations on a C∗-algebra. Let $\delta:A\to A$ be a triple derivation on a
C∗-algebra. We have already commented that $\delta^{**}:A^{**}\to A^{**}$ is a
triple derivation (see (4)). By Lemma 1 in [24]
$\delta^{**}(1)^{*}=-\delta^{**}(1)$. It can be easily seen that

$\delta^{**}(a\circ
b)=\delta^{**}(\left\\{a,1,b\right\\})=\left\\{\delta^{**}(a),1,b\right\\}+\left\\{a,\delta^{**}(1),b\right\\}+\left\\{a,1,\delta^{**}(b)\right\\}$
$={\delta^{**}(a)}\circ b+a\circ{\delta^{**}(b)}-U_{a,b}\delta^{**}(1),$

which shows that $\delta^{**}$ is a generalised Jordan derivation and hence a
generalised derivation (compare [14, Comments before Proposition 5 and Remark
8]). Moreover,

$\delta^{**}(a^{*})=\delta^{**}\left\\{1,a,1\right\\}=2\left\\{\delta^{**}(1),a,1\right\\}+\left\\{1,\delta^{**}(a),1\right\\}=2\delta^{**}(1)\circ
a^{*}+\delta^{**}(a)^{*}.$

In particular, every triple derivation $\delta$ with $\delta^{**}(1)=0$ is a
symmetric derivation or a ∗-derivation (i.e. $\delta$ is a derivation with
$\delta(a^{*})=\delta(a)^{*}$, for every $a\in A$). In particular, for every
triple derivation $\delta$, $\delta^{**}-\delta(\frac{1}{2}\delta^{**}(1),1)$
is a ∗-derivation on $A$.

######  Corollary 3.6.

Let $T:A\to A$ be a linear operator on a C∗-algebra. The following are
equivalent:

  1. $(a)$

$T$ is a local triple derivation;

  2. $(b)$

$T$ is a triple derivation;

  3. $(c)$

$T^{**}$ is a bounded generalised derivation with $T^{**}(1)=-T^{**}(1)^{*}$
and $T^{**}-L_{T^{**}(1)}$ is a symmetric operator;

  4. $(d)$

$T^{**}-L_{T^{**}(1)}$ is a bounded ∗-derivation and
$T^{**}(1)=-T^{**}(1)^{*}$;

  5. $(e)$

$T^{**}$ is a bounded generalised derivation with $T^{**}(1)=-T^{**}(1)^{*}$
and $T^{**}-\delta(\frac{1}{2}T^{**}(1),1)$ is a symmetric operator;

  6. $(f)$

$T^{**}-\delta(\frac{1}{2}\delta^{**}(1),1)$ is a bounded ∗-derivation and
$T^{**}(1)=-T^{**}(1)^{*}$;

  7. $(g)$

$T^{**}(1)=-T^{**}(1)^{*}$, $T^{**}-L_{T^{**}(1)}$ is a symmetric operator,
$T$ is bounded, and $aT(b)c=0$, whenever $ab=bc=0$ in $A$.

  8. $(h)$

$T^{**}(1)=-T^{**}(1)^{*}$, $T^{**}-\delta(\frac{1}{2}\delta^{**}(1),1)$ is a
symmetric operator, $T$ is bounded, and $aT(b)c=0$, whenever $ab=bc=0$ in $A$.

$\hfill\Box$

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

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