2. Derivations
The proof of the following theorem is almost as that of [20, Theorem 2.5], except we make the extension of the Jordan derivation on a universally reversible JC-algebra to its universal enveloping C*-algebra (see [1, Proposition 4.36], [8, 7.1.8] for its existance and propertites). Since this theorem plays an essential role in our results, we include the proof for completeness. But first note that if , where A and B are JC-algebras, and if is a Jordan derivation (i.e. , for all ), then defined by , for all , is a Jordan derivation.
Theorem 2.1. Suppose that is a Jordan derivation on a universall reversible JC-algebra A. Then D can be extended to a *-derivation of the C*-algebra .
Proof. Let
be a pure state of
, and let
be the cyclic reprentation of
induced by
, by the Gelfand-Neumark-Segal construction (GNS construction). By [
18, Theorem 9.22], [
10, Theorm 4.5.5],
is irreducible, so that,
, where
is the
-weak closure of
in
, and
is the double commutant of
[
12, Theorem 3.13.2], [
15, Proposition 1.21.9]. Having
and
as the second duals of
and
A, respectively, and deducing that
is a type
I factor representation, then by [
18, Lemma 2.2], [
15, Theorem 1.9.1],
extends to a normal representation
, which is irreducible, since
. Put
, then the restriction
is an irreducible representation. Note that
is universally reversible, since
A is universally reversible, [
1, Lemma 4.33], and so,
M is a reversible JW-factor of type
, for some
[
8, Corollary 5.3.7], which implies that the self-adjoint part
of the center
of the The universal enveloping von Neumann algebra
of of
M equals to the center
of
M by [
8, Theorem 7.3.5]. Therefore,
is a factor of type
I, by [
8, Theorem 7.4.2 (i)]. By [
2, Theorem 3.1], (see also [
9, Theorem 7.5.11]), either
with
a real flip on
, or
with
a quaternionian flipon
. This means that
for some
-Hilbert space on
, where
or
or
. Note that
defined by
for all
, is a Jordan derivation, which extends to a normal Jordan derivation
, by [
15, Lemma 4.1.4], then
for all
. Therefore,
for all
, and for some element
, by [
20, Lemma 2.6]. Let
,
,
, then, for all
, we have
That is,
. Since
A generates
as a C*-algebra, we have
for all
. It is clear that
defined by
, is a *-derivation. Since
is a faithful representation of
(see [
10, Proposition 4.5.5 and Theorem 4.5.6]), we can easily see that
defined by
is a *-derivation on
extending
D. □
Remark 2.2.
Given a *-derivation of the real C*-algebra . Then defined by , , is a *-derivation of . The linearity of is obvious. Let , . Then for some . Then , and so,
On the othe hand, we have,
from which we see that , and hence is a derivation on . Now, if , for some , then , and so, . That is, is a *-derivation, which clearly extends D to .
Theorem 2.3. Let A be a universall reversible JC-algebra , and let be a *-derivation of the C*-algebra . Then:
, is a Jordan derivation of A.
is *-derivation of the real C*-algebra .
Proof. Since A is universally reversible, we have , and [8, Proposition 7.3.3], and , where (see [7, p. 103]).
Note that defined by , for all , is a Jordan derivation, where [6, Remark 2.2], and , for all . Hence, for all , we have , since . That is, , implying that , is a Jordan derivation of A.
Let
, since
, and
, we have
implying that
. Therefore,
and so,
for all
. Hence,
is *-derivation. □
Corollary 2.4. Every Jordan derivation on a universal reversible JC-algebra A extends to a *-derivation on the real enveloping C*-algebra of A.
Proof. Let be a Jordan derivation. By Theorem 2.1, D extends to *-derivation of the C*-algebra , such that , and by Theorem 2.3, is *-derivation, which is obiously an extension of D to . □
The following Corllary is immediate by Theorems 2.1 and [14, Theorem 2].
Corollary 2.5. Every Jordan derivation on a universal reversible JC-algebra A is continuous.
Before establishing the relation between local derivations (resp., 2-local derivations, weak-local derivations , weak-2-local derivations) on a universally reversible JC-algebra A, and their extensions to its universal inveloping real and complex C*-algebras and , let us introduce a notion of a set-local (resp. set-2-local, set-weak-local, set-weak-2-local) derivation.
Definition 1.
Let be a Banach algebra, and be a linear map. We call T a set-local derivation if for every , there exists a derivation, , depending on x, and a subset of containing x, satisfying for all . It is calleda set-2-local derivation if for every , there exists a derivation, , depending on x, and y, and a subset of containing x and y such that and for all . If for every , and every , there exists a derivation, , depending on x and φ, and a subset of containing x, satisfying for all , then it is called a set-weak-local derivation.If for every , and every , there exists a derivation, ), depending on and φ, and a subset of containing x and y, satisfying for all , then it is called a set-weak-2-local derivation
Theorem 2.6. Every local derivation on a universally reversible JC-algebra A extends to a set-local *-derivation on its universal inveloping real C*-algebra .
Proof. Let
be a fixed element. then
, for some
and
. Since
T be a local derivation on
A, there exists a Jordan derivation
, depending on
a, such that
. Let
be the extension of
to a *-derivationon
, by Corollary 2.4. It is then clear that the mapping
defined by
in
is a linear extension of
T. Let
, and let
, then
for some
, so we have,
That is, for
,
. Hence,
is a set-local *-derivation on
, since
is a subset of
containing
x. □
Theorem 2.7. Every local derivation T on a universally reversible JC-algebra A extends to a set-local derivation S on the self-adjoint part of its universal inveloping C*-algebra .
Proof. Note first that
, since
A is universally reversible, and
. Let
be a local derivation on
A, and let
x be a fixed element in
, say,
, for some
and
. Since
T is a local derivation on
A, then for
a, there exists a Jordan derivation
depending on
a such that
. By Theorem 2.1,
extends to a *-derivations
on
. It is then clear that
, where
,
,
,
defines a linear map
S on
which extends
T. Let
, then for any
, where
, for some
, we have
Hence, S is a set-local derivation on that extends T. □
Theorem 2.8. Every local derivation T on a universally reversible JC-algebra A extends to a set-local *-derivation on its universal inveloping C*-algebra .
Proof. Let
be a local derivation on a universally reversible JC algebra
A, and let
x be a fixed element in
, then
, for some
. Let
, where
, and
. Since
T is a local derivation on
A, then for
a, there exists a Jordan derivation
depending on
a such that
. By Theorem 3.1,
extends to a *-derivation
on
. Let
, and let
be the extention of
T to a set-local *- derivation on
arises in Theorem 2.6, then
for all
. Let
be the map defined by
,
,
,
. It is clear that
is a linear map extending
, and hence extends
T. Note that, for all
and all
, we have,
Since it is clear that
x belongs to the subset
of
, we see that
is a set-local *-derivation on
. □
The following Corllary is immediate by Theorems 2.1, Theorems 2.8, and [9, Theorem 5.3, p.318]
Corollary 2.9. Every local derivation T on a universally reversible JC-algebra A is contiuous.
The relation between 2-local derivations (resp., weak-2-local derivations) on a universally reversible JC-algebra, and their extensions on its universal inveloping real and complex C*-algebras and is established in the following two results.
Theorem 2.10. Let A be a universally reversible JC-algebra A, and let be a 2-local derivation. Then T extends to a set-local *-derivation on its universal inveloping real C*-algebra .
Proof. Let
be two fixed elements, then
and
, for some
and
. Since
T be a 2-local derivation on
A, there exists a Jordan derivation
, depending on
a and
b, such that
and
. Let
be the extension of
to a *-derivationon
, by Corollary 3.4. It is then clear that the mapping
defined by
,
and
, is a linear extension of
T. Let
and
, then
, and
. Note that for any
, we have,
and so,
for all
. Also, we can see that
, for all
. That is,
for all
. Hence,
is a set-local *-derivation on
. □
Theorem 2.11. Every 2-local derivation T on a universally reversible JC-algebra A extends to a set-2-local *-derivation on its universal inveloping C*-algebra .
Proof. Let
be a weak 2-local derivation on a universally reversible JC algebra
A, and let
,
a fixed element in
, then
, for some
, since
. Let
and
, where
, and
. Since
T is a 2-local derivation on
A, then for
, there exists a Jordan derivation
depending on
such that
and
. By Theorem 2.1,
extends to a *-derivation
on
. Let
be the extention of
T to a set-2-local *- derivation on
arises in Theorem 2.10, where
for all
. Then
defined by
,
,
,
is an extention of
, and hence of
T. As in the proof of Theorem 2.10, we can easily see that for all
,
, and all
, we have,
that is, for all
, we have,
. Hence,
is a set-2-local *-derivation on
. □
Theorem 2.12. Every weak-local derivation T on a universally reversible JC-algebra A extends to a set-weak-local *-derivation on its universal inveloping real C*-algebra .
Proof. Let
be a weak-local derivation on a universally reversible JC algebra
A, and let
be a fixed element, then
, for some
, and
. Let
, then
. Since
T is a weak-local derivation on
A, then for
a and
, there exists a derivation
, depending on
a and
, such that
. Let
be the extention of
D to a *-derivation on
, by Corollary 2.4, and define
by
for each
in
. It is then clear that
, and for all
. A similar argument as in the the proof of Theorem 2.6. shows that
So that is a set-weak-local *-derivation on . □
Theorem 2.13. Every weak local derivation T on a universally reversible JC-algebra A extends to a set-weak-local *-derivation on its universal inveloping C*-algebra .
Proof. Let
be a weak local derivation on a universally reversible JC algebra
A, and let
x be a fixed element in
, and
, then Put
, then
, and
, for some
. Let
where
, and
. Since
T is a weak-local derivation on
A, then for
a and
, there exists a Jordan derivation
depending on
a and
such that
. By Theorem 2.1,
extends to a *-derivation
on
. Let
be the extention of
T to a set-weak-local *- derivation on
arises in Theorem 2.7, where
for all
. Then
defined by
,
,
,
is an extention of
. As in the proof of Theorem 2.8, we can easily see that for all
, and all
, we have,
that is, for all
,
. Hence,
is a set-weak-local *-derivation on
. □
The following Corllary is immediate by Theorems 2.1 and Theorem 2.8, and [6, Theorem 2.1].
Corollary 2.14. Every weak-local derivation T on a universally reversible JC-algebra A is contiuous.
Theorem 2.15. Every weak-2-local derivation T on a universally reversible JC-algebra A extends to a set-weak-2-local *-derivation on its universal inveloping real C*-algebra .
Proof. Let
be a weak-local derivation on a universally reversible JC algebra
A, and let
be a fixed elements, then
and
for some
, and
. Let
, then
. Since
T is a weak–2-local derivation on
A, then for
and
, there exists a Jordan derivation
, depending on
and
, such that
and
. Let
be the extention of
to a *-derivation on
, by Corollary 2.4, and define
by
for each
in
, where
. It is then clear that
. A similar argument in the proof of Theorem 2.10 and Theorem 2.13, we see that for all
or
,
That is,
, for all
. So that
is a set-weak-local *-derivation on
. □
The proof of the next Theorem is similar to the proof of Theorem 2.15, and using a similar argument in the proof of Theorem 2.13.
Theorem 2.16. Every weak-2-local derivation T on a universally reversible JC-algebra A extends to a set-weak-2-local *-derivation on its universal inveloping C*-algebra .
Proof. Let be a weak 2-local derivation on a universally reversible JC algebra A, and let , a fixed element in , then , for some . Let , then . Let and , where , and . Since T is a weak-2-local derivation on A, then for and , there exists a Jordan derivation depending on and such that and . By Theorem 2.1, extends to a *-derivation on . Let be the extention of T to a set-weak 2-local *- derivation on arises in Theorem 2.14, where for all . Then defined by , , , is an extention of T. As in the proof of Theorem 2.11, we can easily see that for all , we have, . Hence, is a set-2-weak local *-derivation on . □
Theorem 2.17. Let A be a universally reversible JC algebra, and be a local *-derivation (resp. 2-local *-derivation, weak-local *-derivation, weak-2-local *-derivation).
If , then is a local derivation (resp. 2-local derivation, weak-local derivation, weak-2-local derivation).
If , then is a local *-derivation (resp. 2-local *-derivation, weak-local *-derivation, weak-2-local *-derivation).
Proof. It immediate by Theorem 2.3. □