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Extensions Of Derivations on JC-algebras

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29 November 2023

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30 November 2023

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Abstract
Extension of a derivation on a universally reversible JC-algebra A⊆B(H)_{sa} to the C*-algebra [A] generated by A in B(H) was studied by Upmier [20, Theorem 2.5]. In this article we study the extension of a Jordan derivation on a universally JC-algebra A to its universal enveloping real and complex C*-algebras R and U, respectively. Also, we establish the relationship between local derivations (resp., 2-local derivations, weak local derivations, weak-2-local derivations) of a universally JC-algebra A and the corresponding maps on its universal enveloping real and complex C*-algebras R and A, respectively.
Keywords: 
JC-algebras; JW-algebras; C*-algebras; von Neumann algebras; Banach bimodules; Derivations
Subject: 
Computer Science and Mathematics  -   Mathematics

MSC:  Primary 46L05; 46L10; 46L57; 47B47; Secondary 15A86; 47C15

1. Preliminaries

A uniformly closed Jordan subalgebra of the set of all continuous linear self adjoint operator B ( H ) s a on a complex Hilbert space H is called a JC-algebra, and its weak closure is called a JW-algebra. Let A be a JC-algebra, then there there is a unique (up to an isomorphism) a C*-algebra A , a Jordan isomrphism ψ : A A such that ψ ( A ) generates A as a C*-algebra, and A has a unique * antiautomrphism Φ of order 2 keeping the points of ψ ( A ) fixed. The set R = { x A : Φ ( x ) = x * } is a real C*-sublgebra of A which satisfies R i R = { 0 } , and A = R i R [1, Proposition 4.40, Lemma 4.41].
We refer the reader to [1, 4, 8, 13, 16, 17, 19] for the needed background of the theory of JC-algebras and JW-algebras, and to [10, 11, 12, 18] for the prerequisite background of C * -algebras and von Neumann algebras. Sufficient information about derivations can be found in [3, 5, 6]. Throughout this paper, we identify A with ψ ( A ) in A , and assume that A is a universally reversible JC-algebra.

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 φ : A B , where A and B are JC-algebras, and if δ : A A is a Jordan derivation (i.e. δ ( a b ) = δ ( a ) b + a δ ( b ) , for all a , b A ), then δ : φ ( A ) φ ( A ) defined by δ ( φ ( a ) ) = φ ( δ ( a ) ) , for all a A , is a Jordan derivation.
Theorem 2.1. 
Suppose that D : A A is a Jordan derivation on a universall reversible JC-algebra A. Then D can be extended to a *-derivation D : A A of the C*-algebra A .
Proof. 
Let ρ be a pure state of A , and let { π ρ , H ρ , ζ ρ } be the cyclic reprentation of A induced by ρ , by the Gelfand-Neumark-Segal construction (GNS construction). By [18, Theorem 9.22], [10, Theorm 4.5.5], π ρ : A B ( H ρ ) is irreducible, so that, π ρ ( A ) ¯ w = ( π ρ ( A ) = B ( H ρ ) , where π ρ ( A ) ¯ w is the σ -weak closure of π ρ ( A ) in B ( H ρ ) , and π ρ ( A ) is the double commutant of π ρ ( A ) [12, Theorem 3.13.2], [15, Proposition 1.21.9]. Having A * * and A * * as the second duals of A and A, respectively, and deducing that π ρ is a type I factor representation, then by [18, Lemma 2.2], [15, Theorem 1.9.1], π ρ extends to a normal representation π ρ * * : A * * B ( H ρ ) , which is irreducible, since π ρ * * ( A * * ) = π ρ ( A ) ¯ w = B ( H ρ ) . Put M = π ρ * * ( A * * ) = π ρ ( A ) = π ρ ( A ) ¯ w , then the restriction π ρ A : A B ( H ρ ) is an irreducible representation. Note that A * * is universally reversible, since A is universally reversible, [1, Lemma 4.33], and so, M is a reversible JW-factor of type I n , for some n 3 [8, Corollary 5.3.7], which implies that the self-adjoint part Z ( M ) s a of the center Z ( M ) of the The universal enveloping von Neumann algebra M of of M equals to the center Z ( M ) of M by [8, Theorem 7.3.5]. Therefore, M 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 M B ( H ρ ) s a α with α a real flip on B ( H ρ ) , or M B ( H ρ ) s a β with β a quaternionian flipon B ( H ρ ) . This means that M B ( K ) s a for some K -Hilbert space on H ρ , where K = ( R , or C , or H ) . Note that D 1 : π ρ ( A ) π ρ ( A ) defined by D 1 ( π ρ ( a ) ) = π ρ ( D ( a ) ) for all a A , is a Jordan derivation, which extends to a normal Jordan derivation D 2 : π ρ ( A ) ¯ w π ρ ( A ) ¯ w , by [15, Lemma 4.1.4], then π ρ ( D ( a ) ) = D 2 ( π ρ ( a ) ) for all a A . Therefore, D 2 ( d ) = [ w ρ , d ] for all d π ρ ( A ) ¯ w = π ρ * * ( A * * ) , and for some element w ρ B ( H ρ ) , by [20, Lemma 2.6]. Let H = ρ P ( A ) H ρ , π = ρ P ( A ) π ρ , w = ρ P ( A ) w ρ , then, for all a A , we have
π ( D ( a ) ) = ( ρ P ( A ) π ρ ) ( D ( a ) ) = ρ P ( A ) ( π ρ ( D ( a ) ) = ρ P ( A ) [ w ρ , π ρ ( a ) ] = ρ P ( A ) ( w ρ π ( a ) π ρ ( a ) w ρ ) = ( ρ P ( A ) w ρ ) ( ρ P ( A ) π ρ ( a ) ) ( ρ P ( A ) π ρ ( a ) ) ( ρ P ( A ) w ρ ) = w π ( a ) π ( a ) w = [ w , π ( a ) ] .
That is, [ w , π ( a ) ] = π ( D ( a ) ) π ( A ) π ( A ) . Since A generates A as a C*-algebra, we have [ w , π ( x ) ] π ( A ) for all x A . It is clear that D ¯ : π ( A ) π ( A ) defined by D ¯ ( π ( x ) ) = [ w , π ( x ) ] , x A , is a *-derivation. Since π : A B ( H ) is a faithful representation of A (see [10, Proposition 4.5.5 and Theorem 4.5.6]), we can easily see that D : A A defined by D ( x ) = ( π 1 D ¯ π ) ( x ) is a *-derivation on A extending D. □
Remark 2.2. 
Given a *-derivation δ : R R of the real C*-algebra R . Then δ : A A defined by δ ( x + i y ) = δ ( x ) + i δ ( y ) , x , y R , is a *-derivation of A . The linearity of δ is obvious. Let z j A , j = 1 , 2 . Then z j = x j + i y j for some x j , y j R . Then z 1 z 2 = ( x 1 + i y 1 ) ( x 2 + i y 2 ) = ( x 1 x 2 y 1 y 2 ) + i ( x 1 y 2 + y 1 x 2 ) , and so,
δ ( z 1 z 2 ) = δ ( ( x 1 x 2 y 1 y 2 ) + i ( x 1 y 2 + y 1 x 2 ) ) = δ ( x 1 x 2 y 1 y 2 ) + i δ ( x 1 y 2 + y 1 x 2 ) = δ ( x 1 x 2 ) δ ( y 1 y 2 ) + i δ ( x 1 y 2 ) + i δ γ ( y 1 x 2 ) = δ ( x 1 ) x 2 + x 1 δ ( x 2 ) δ ( y 1 ) y 2 y 1 δ ( y 2 ) + i δ ( x 1 ) y 2 + i x 1 δ ( y 2 ) + i δ ( y 1 ) x 2 + i y 1 δ ( x 2 )
On the othe hand, we have,
δ ( z 1 ) z 2 + z 1 δ ( z 2 ) = ( δ ( x 1 + i y 1 ) ) ( x 2 + i y 2 ) + ( x 1 + i y 1 ) ( δ ( x 2 + i y 2 ) ) = ( δ ( x 1 ) + i δ ( y 1 ) ) ( x 2 + i y 2 ) + ( x 1 + i y 1 ) ( δ ( x 2 ) + i δ ( y 2 ) ) = δ ( x 1 ) x 2 + i δ ( x 1 ) y 2 + i δ ( y 1 ) x 2 δ ( y 1 ) y 2 + x 1 δ ( x 2 ) + i x 1 δ ( y 2 ) + i y 1 δ ( x 2 ) y 1 δ ( y 2 ) ) ,
from which we see that δ ( z 1 z 2 ) = δ ( z 1 ) z 2 + z 1 δ ( z 2 ) , and hence δ is a derivation on A . Now, if z A , z = u + i w for some u , w R , then z * = u * i w * , and so, δ ( z * ) = δ ( u * ) i δ ( w * ) = δ ( u ) * i δ ( w ) * = ( δ ( u ) + i δ ( w ) ) * = ( δ ( z ) ) * . That is, δ is a *-derivation, which clearly extends D to A .
Theorem 2.3. 
Let A be a universall reversible JC-algebra , and let D : A A be a *-derivation of the C*-algebra A . Then:
( i ) D A , is a Jordan derivation of A.
( i i ) D R is *-derivation of the real C*-algebra R .
Proof. 
Since A is universally reversible, we have A = R s a , and A = R s a = A s a Φ [8, Proposition 7.3.3], and R = R s . a B , where B = { b R * ( A ) : b * = b } (see [7, p. 103]).
( i ) Note that D A s a : A s a A s a defined by D ( x y ) = x D ( y ) + D ( x ) y , for all x , y A s a , is a Jordan derivation, where x y = x y + y x 2 [6, Remark 2.2], and D ( x ) = ( Φ D Φ ) ( x ) , for all x A . Hence, for all a A = A s a Φ , we have D ( a ) = ( Φ D Φ ) ( a ) = Φ ( D ( a ) ) , since D ( a ) * = D ( a * ) = D ( a ) . That is, D ( a ) A s a Φ = R s a = A , implying that D A , is a Jordan derivation of A.
( i i ) Let b B , since Φ ( b ) = b * = b , and D ( b ) = ( Φ D Φ ) ( b ) , we have
Φ ( D ( b ) ) = ( Φ D ) ( b ) = ( Φ D ) ( Φ ( b ) ) = ( Φ D Φ ) ( b ) = D ( b ) = D ( b ) = D ( b * ) = ( D ( b ) ) * ,
implying that D ( b ) B R . Therefore,
Φ ( D ( x ) ) = Φ ( D ( a + b ) ) = Φ ( D ( a ) ) + Φ ( D ( a ) ) = D ( a ) + ( D ( b ) ) * = ( D ( a + b ) ) * = ( D ( x ) ) * ,
and so, D ( x ) R for all x R . Hence, D R 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 R of A.
Proof. 
Let D : A A be a Jordan derivation. By Theorem 2.1, D extends to *-derivation D : A A of the C*-algebra A , such that D | A = D , and by Theorem 2.3, D | R is *-derivation, which is obiously an extension of D to R . □
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 R and A , let us introduce a notion of a set-local (resp. set-2-local, set-weak-local, set-weak-2-local) derivation.
Definition 1. 
Let A be a Banach algebra, and T : A A be a linear map. We call T a set-local derivation if for every x A , there exists a derivation, D x : A A , depending on x, and a subset C of A containing x, satisfying T ( y ) = D x ( y ) for all y C . It is calleda set-2-local derivation if for every x , y A , there exists a derivation, D x , y : A A , depending on x, and y, and a subset C of A containing x and y such that T ( z ) = D x , y ( z ) and for all z C . If for every x A , and every φ A * , there exists a derivation, D x , φ : A A , depending on x and φ, and a subset C of A containing x, satisfying φ T ( y ) = φ D x , φ ( y ) for all y C , then it is called a set-weak-local derivation.If for every x , y A , and every φ A * , there exists a derivation, D x , y , φ : A A ), depending on x , y and φ, and a subset C of A containing x and y, satisfying φ T ( z ) = φ D x , y , φ ( z ) for all z C , then it is called a set-weak-2-local derivation
Theorem 2.6. 
Every local derivation T : A A on a universally reversible JC-algebra A extends to a set-local *-derivation T on its universal inveloping real C*-algebra R .
Proof. 
Let x R be a fixed element. then x = a + b , for some a A = R s a and b B . Since T be a local derivation on A, there exists a Jordan derivation D a : A A , depending on a, such that T ( a ) = D a ( a ) . Let D a be the extension of D a to a *-derivationon R , by Corollary 2.4. It is then clear that the mapping T : R R defined by T ( y ) = T ( c ) + D a ( d ) , y = c + d in R s a B = R is a linear extension of T. Let E a = { a + d : d B } , and let y E a , then y = a + d for some d B , so we have,
T ( a + d ) = T ( a ) + D a ( d ) = D a ( a ) + D a ( d ) = D a ( a ) + D a ( d ) = D a ( a + d ) .
That is, for y E a , T ( y ) = D a ( y ) . Hence, T is a set-local *-derivation on R , since E a is a subset of R s a B 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 A s a of its universal inveloping C*-algebra C * ( A ) .
Proof. 
Note first that A s a = A i B , since A is universally reversible, and A = R s a . Let T : A A be a local derivation on A, and let x be a fixed element in A s a , say, x = a i b , for some a A and b B . Since T is a local derivation on A, then for a, there exists a Jordan derivation D a : A A depending on a such that T ( a ) = D a ( a ) . By Theorem 2.1, D a extends to a *-derivations D a on A . It is then clear that S : y T ( c ) + i D a ( d ) , where y A s a , y = c + i d , c A , d B defines a linear map S on A s a which extends T. Let F a = { a + i d : d B } , then for any y F a , where y = a + i d , for some d B , we have
S ( y ) = S ( a + i d ) = T ( a ) + i D a ( d ) = D a ( a ) + i D a ( d ) = D a ( a ) + i D a ( d ) = D a ( a + i d ) = D a ( y ) .
Hence, S is a set-local derivation on A s a that extends T. □
Theorem 2.8. 
Every local derivation T on a universally reversible JC-algebra A extends to a set-local *-derivation T on its universal inveloping C*-algebra A .
Proof. 
Let T : A A be a local derivation on a universally reversible JC algebra A, and let x be a fixed element in A , then x = y + i z , for some y , z R . Let y = a + b , where a A = R s a , and b B . Since T is a local derivation on A, then for a, there exists a Jordan derivation D a : A A depending on a such that T ( a ) = D a ( a ) . By Theorem 3.1, D a extends to a *-derivation D a on A . Let E a = { a + d : d B } , and let T be the extention of T to a set-local *- derivation on R arises in Theorem 2.6, then T ( y ) = D a ( y ) for all y E a = { a + d : d B } . Let T : A A be the map defined by T ( w ) = T ( u ) + i D a ( v ) , w A , w = u + i v , u , v R . It is clear that T is a linear map extending T , and hence extends T. Note that, for all u E a and all v R , we have,
T ( u + i v ) = T ( u ) + i D a ( v ) = D a ( u ) + i D a ( v ) = D a ( u + i v ) .
Since it is clear that x belongs to the subset E a + i R of A , we see that T is a set-local *-derivation on A . □
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 R and A is established in the following two results.
Theorem 2.10. 
Let A be a universally reversible JC-algebra A, and let T : A A be a 2-local derivation. Then T extends to a set-local *-derivation T on its universal inveloping real C*-algebra R .
Proof. 
Let x , y R be two fixed elements, then x = a + c and y = b + d , for some a , b A = R s a and c , d B . Since T be a 2-local derivation on A, there exists a Jordan derivation D a , b : A A , depending on a and b, such that T ( a ) = D a , b ( a ) and T ( b ) = D a , b ( b ) . Let D a , b be the extension of D a , b to a *-derivationon R , by Corollary 3.4. It is then clear that the mapping T : R R defined by T ( w ) = T ( u ) + D a , b ( v ) , w = u + v , u R * ( A ) s a and v B , is a linear extension of T. Let E a = { a + v : v B } and E b = { b + v : v B } , then x E a , and y E b . Note that for any v B , we have,
T ( a + v ) = T ( a ) + D a , b ( v ) = D a , b ( a ) + D a , b ( v ) = D a , b ( a ) + D a , b ( v ) = D a , b ( a + v ) ,
and so, T ( w ) = D a , b ( w ) for all w E a . Also, we can see that T ( w ) = D a , b ( w ) , for all w E b . That is, T ( w ) = D a , b ( w ) for all w E a E b . Hence, T is a set-local *-derivation on R . □
Theorem 2.11. 
Every 2-local derivation T on a universally reversible JC-algebra A extends to a set-2-local *-derivation T on its universal inveloping C*-algebra A .
Proof. 
Let T : A A be a weak 2-local derivation on a universally reversible JC algebra A, and let x j , j = 1 , 2 a fixed element in A , then x j = y j + i z j , for some y j , z j R , since A = R i R . Let y j = a j + c j and z j = b j + d j , where a j , b j A , and c j , d j B . Since T is a 2-local derivation on A, then for a 1 , a 2 , there exists a Jordan derivation D a 1 , a 2 : A A depending on a 1 , a 2 such that T ( a 1 ) = D a 1 , a 2 ( a 1 ) and T ( a 2 ) = D a 1 , a 2 , ( a 2 ) . By Theorem 2.1, D a 1 , a 2 extends to a *-derivation D a 1 , a 2 on A . Let T be the extention of T to a set-2-local *- derivation on R arises in Theorem 2.10, where T ( w ) = D a 1 , a 2 ( w ) for all w E a 1 E a 2 . Then T : A A defined by T ( w ) = T ( u ) + i D a 1 , a 2 ( v ) , w A , w = u + i v , u , v R is an extention of T , and hence of T. As in the proof of Theorem 2.10, we can easily see that for all u E a j , j = 1 , 2 , and all v R , we have,
T ( u + i v ) = T ( u ) + i D a 1 , a 2 ( v ) = D a 1 , a 2 ( u ) + i D a 1 a 2 ( v ) = D a 1 , a 2 ( u + i v ) ,
that is, for all w ( ( E a 2 E a 2 ) + R ) , we have, T ( w ) = D a 1 , a 2 ( w ) . Hence, T is a set-2-local *-derivation on A . □
Theorem 2.12. 
Every weak-local derivation T on a universally reversible JC-algebra A extends to a set-weak-local *-derivation T on its universal inveloping real C*-algebra R .
Proof. 
Let T : A A be a weak-local derivation on a universally reversible JC algebra A, and let x R be a fixed element, then x = a + b , for some a A , and b B . Let θ R * , then φ = θ | A A * . Since T is a weak-local derivation on A, then for a and φ , there exists a derivation D a , φ : A A , depending on a and φ , such that φ T ( a ) = φ D a , φ ( a ) . Let D a , φ be the extention of D to a *-derivation on R , by Corollary 2.4, and define T : R R by T ( y ) = T ( c ) + D a , φ ( d ) for each y = c + d in R = R s a B = A B . It is then clear that T A = T , and for all y E a = { a + d : d B } . A similar argument as in the the proof of Theorem 2.6. shows that
θ ( T ( y ) ) = θ ( D a , φ ( y ) ) , for all y E a .
So that T is a set-weak-local *-derivation on R . □
Theorem 2.13. 
Every weak local derivation T on a universally reversible JC-algebra A extends to a set-weak-local *-derivation T on its universal inveloping C*-algebra A .
Proof. 
Let T : A A be a weak local derivation on a universally reversible JC algebra A, and let x be a fixed element in A , and σ A * , then Put φ = σ | A , then φ A * , and x = y + i z , for some y , z R * ( A ) . Let y = a + b where a A , and b B . Since T is a weak-local derivation on A, then for a and φ , there exists a Jordan derivation D a , φ : A A depending on a and φ such that φ T ( a ) = φ D a , φ ( a ) . By Theorem 2.1, D a , φ extends to a *-derivation D a , φ on A . Let T be the extention of T to a set-weak-local *- derivation on R arises in Theorem 2.7, where T ( y ) = D a , φ ( y ) for all y E a = { a + d : d B } . Then T : A A defined by T ( w ) = T ( u ) + i D a , φ ( v ) , w A , w = u + i v , u , v R is an extention of T . As in the proof of Theorem 2.8, we can easily see that for all u E a , and all v R , we have,
σ T ( u + i v ) = σ ( T ( u ) + i D a , φ ( v ) ) = σ ( D a , φ ( u ) + i D a , φ ( v ) ) = σ ( D a , φ ( u + i v ) ) ,
that is, for all w ( E a + i R * ( A ) ) , ψ T ( w ) = ψ D a , φ ( w ) . Hence, T is a set-weak-local *-derivation on A . □
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 T on its universal inveloping real C*-algebra R .
Proof. 
Let T : A A be a weak-local derivation on a universally reversible JC algebra A, and let x , y R be a fixed elements, then x = a + c and y = b + d for some a , b R s a = A , and c , d B . Let ρ R * , then ρ = ρ | A A * . Since T is a weak–2-local derivation on A, then for a , b and ρ , there exists a Jordan derivation D a , b , ρ : A A , depending on a , b and ρ , such that ρ T ( a ) = ρ D a , b , ρ ( a ) and ρ T ( b ) = ρ D a , b , ρ ( b ) . Let D a , b , ρ be the extention of D a , b , ρ to a *-derivation on R , by Corollary 2.4, and define T : R R by T ( w ) = T ( u ) + D a , b , ρ ( v ) for each w = u + v in R , where u A , v B . It is then clear that T A = T . A similar argument in the proof of Theorem 2.10 and Theorem 2.13, we see that for all z E a = { a + v : v B } or z E b = { b + v : v B } ,
ρ ( T ( z ) ) = ρ ( D a , b , ρ ( y ) ) .
That is, ρ ( T ( z ) ) = ρ ( D a , b , φ ( y ) ) , for all z E a E b . So that T is a set-weak-local *-derivation on R . □
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 T on its universal inveloping C*-algebra A .
Proof. 
Let T : A A be a weak 2-local derivation on a universally reversible JC algebra A, and let x j , j = 1 , 2 a fixed element in A , then x j = y j + i z j , for some y j , z j R . Let σ A * , then ρ = σ | A A * . Let y j = a j + c j and z j = b j + d j , where a j , b j A , and c j , d j B . Since T is a weak-2-local derivation on A, then for a 1 , a 2 and ρ , there exists a Jordan derivation D a 1 , a 2 , ρ : A A depending on a 1 , a 2 and ρ such that ρ T ( a 1 ) = ρ D a 1 , a 2 , ρ ( a 1 ) and ρ T ( a 2 ) = ρ D a 1 , a 2 , ρ ( a 2 ) . By Theorem 2.1, D a 1 , a 2 , ρ extends to a *-derivation D a 1 , a 2 , ρ on A . Let T be the extention of T to a set-weak 2-local *- derivation on R arises in Theorem 2.14, where T ( w ) = D a , b ( w ) for all w E a E b . Then T : A A defined by T ( w ) = T ( u ) + i D a 1 , a 2 , ρ ( v ) , w A , w = u + i v , u , v R is an extention of T. As in the proof of Theorem 2.11, we can easily see that for all z ( ( E a E ) + R , we have, σ T ( z ) = σ D a 1 a 2 , ρ ( z ) . Hence, T is a set-2-weak local *-derivation on A . □
Theorem 2.17. 
Let A be a universally reversible JC algebra, and T : A A be a local *-derivation (resp. 2-local *-derivation, weak-local *-derivation, weak-2-local *-derivation).
( i ) If T ( A ) A , then T A : A A is a local derivation (resp. 2-local derivation, weak-local derivation, weak-2-local derivation).
( i i ) If T ( R ) ) R , then T R : R R is a local *-derivation (resp. 2-local *-derivation, weak-local *-derivation, weak-2-local *-derivation).
Proof. 
It immediate by Theorem 2.3. □

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