Jewell-Sinclair theorem and the automatic continuity of a derivation

Villena's theorem ensure the automatic continuity of a derivation on a non-associative H*-algebra with zero annihilator. In this paper, we formulate the Jewell-Sinclair theorem in a non-associative content, and we show that the Villena's theorem may be also followed from our formulation of such theorem. 1INTRODUCTION AND BASIC CONCEPTS: The range of derivations on Banach algebra are investigated, the fundamental work in this direction is due to Singer and Warmer in 1955 and it is in progress by another researchers, see [1],[2],[9] and [15]. Singer-Warmer theorem stated that: Let A be a commutative Banach algebra, and let δ be a continuous derivation on A . Then δ( A ) Jewell-Sinclair theorem and the automatic continuity of a derivation. 64 is contained in the radical of A . Singer and Warmer attained to improve the result mentioned above by delete the continuity of a derivation which is known as a Singer-Warmer conjecture: Let A be a commutative Banach algebra and let δ be a derivation on A . Then δ is a continuous derivation on A ? Johnson in 1969 proved that every derivation on a semi-simple commutative Banach algebra with identity is continuous [6]. This result is a partial answer of Singer-Warmer conjecture because the algebra has an identity. Kaplansky in 1958 see [8] generalized Singer-Warmer conjecture as follows: Is a derivation on semisimple Banach algebra be continuous? Johnson and Sinclair in 1968 see [7] confirmed Kaplansky conjecture, and the Johnson-Sinclair result it was generalized by Jewell and Sinclair in 1976 see [5] and Singer-Warmer conjecture confirmed by Thomas in 1988 see [13]. There are many studies of the automatic continuity of a derivation on non-associative Banach algebras. Particularly, the following problem is our interest. Is a derivation on a non-associative semisimple Banach algebra be continuous? For details about this problems see [10], [12], [14], [15]. One of the important theorem in this direction also is due to Villena see [14] and it is our concern in this paper which is state that (Villena Theorem): Every derivation on H*-algebra with zero annihilator is continuous. Our work can be divided in three directions: (1) State and study Jewell-Sinclair theorem: Every derivation on a Banach algebra A is continuous if A satisfies: iA has no non-zero finite-dimensional nilpotent ideals. iiFor each closed infinite-dimensional ideal I of A , there is a sequence { } A a N n n ⊆ ∈ such that the sequence { } N n n 2 1 I ) a .... a (a ∈ . . . of closed right ideals of A is constantly decreasing. (2) We put Jewell-Sinclair theorem on a non-associative content and we call it extended Jewel-Sinclair theorem as follows: Let B be a semiprime Banach algebra not necessary associative such that for each closed infinite-dimensional ideal I of B , there is a sequence { } ) (B M T N n n ⊆ ∈ (the multiplication algebra of B) such that the sequence { } N n n I T T T ∈ ) ...... . ( 2 1 of closed right ideals of B is constantly decreasing. Then, any derivation on B is continuous. (3) We show that Villena's proof of the automatic continuity of a derivation on H*-algebra with zero annihilator can be also followed by using extended Jewell-Sinclair theorem. To explain our goal mentioned above we need the following definitions, propositions, and theorems. Amir A. Mohammed & Yassin Abdulljabar & Nadia A. Abdulraziq 65 1-1 Definition [9], [11]: Let A be an algebra. A derivation δ on A is a linear mapping δ : A → A such that: ) ( ) ( ) ( y x y x xy δ δ δ + = for all x , y ∈ A. 1-2 Definition [12]: Let X and Y be normed spaces and let T be a linear mapping from X into Y, then sparating subspace of T (denoted by YT) defined as follows: { } { } { } { } N n y x T x X x Y y Y n n n T ∈ ∀ → → ⊆ ∃ ∈ = , , 0 , : . 1-3 Proposition [3], [12]: Let A be a normed algebra and δ be a derivation on A. Then Aδ is closed ideal of A. 1-4 Sinclair Theorem [12]: Let X and Y be Banach spaces and let Y X T ⎯→ : be a linear mapping. If the sequences { } ( ) { } ( ) Y BL R and X BL S n N n n ⊆ ⊆ ∈ satisfy TSn=RnT, then there is an integer N k ∈ such that ) Y ..R (R ) Y ..R (R T k T n ... = ... 1 1 k n ≥ ∀ . We recall from [4] that , an algebra A is semiprime if any ideal I of A such that I=0 implies I=0. Also an annihilator of A (denoted by Ann(A)) defined as follows: { } A a xa ax A x A Ann ∈ ∀ = = ∈ = , 0 : ) ( , and we say that A is of zero annihilator if Ann(A)={0}. Finally, the multiplication algebra of A denoted by M(A) is defined as a subalgebra of L(A) (the algebra of all linear mapping on A) generated by following operators: IdA : A A Lx : A A Rx : A A a a IdA (a)=a , a a Lx (a)=xa , a a Rx (a)=ax called identity, left, and right multiplication operators respectively. 2EXTENDED JEWELL-SINCLAIR THEOREM . We extending the Jewell-Sinclair theorem in the following way. 2-1 Extended Jewell-Sinclair Theorem: Let B be a semiprime Banach algebra not necessary associative such that for each closed infinitedimensional ideals I ⊆ B, there is a sequence { } ) (B M T N n n ⊆ ∈ such that the sequence { } N n n I T T T ∈ ) ...... . ( 2 1 of closed right ideals of B is constantly decreasing. Then, any derivation on B is continuous. Proof : Let B be a semiprime Banach algebra not necessary associative and let δ be a derivation on B. By proposition (1-3), Bδ is closed ideal of B. If Bδ is infinite-dimensional, then by assumption there is a sequence { } ) (B M T N n n ⊆ ∈ such that the sequence { } N n n B T T T ∈ δ ) ...... . ( 2 1 is constantly decreasing. By applying Sinclair theorem (1-4) and by setting: X = Y = B, T = δ, Rn = Tn= Sn We get, there exist a natural number N K∈ such that: K n B T T B T T K n ≥ ∀ = ) .... . ( ) ..... . ( 1 1 δ δ . This condition implies that {Tn}n0N not constantly decreasing, and this is a contradiction. So, Bδ must be finite-dimensional. Note that, if Bδ is finite-dimensional, then δ δ B is continuous. Now we claim that } 0 { 2 = δ B . Let a, b ∈ Bδ , Jewell-Sinclair theorem and the automatic continuity of a derivation. 66 then there is { } B a N n n ⊆ ∈ , such that 0 = ∞ → n n a Lim and a a Lim n n = ∞ → ) ( δ . Now, ) ( ) ( ) ( b a Lim b a Lim b a Lim n n n n n n δ δ δ ∞ → ∞ → ∞ → + = . Since 0 = ∞ → n n a Lim and b ∈ Bδ , and that Bδ is an ideal of B, we have 0 = ∞ → b a Lim n n , anb ∈ Bδ and take into account that Bδ finite-dimensional and δ δ B is continuous we get 0 ) ( = ∞ → b a Lim n n δ . Now, a b = [ ] 0 ) ( ) ( ) ( = − = ∞ → ∞ → b a b a Lim b a Lim n n n n n δ δ δ , we find that } 0 { 2 = δ B and since B is semiprime by assumption, so we have } 0 { = δ B and by using closed graph theorem we obtain that δ is continuous. This completes the proof ● 3AN APPLICATION EXAMPLE . Before we give our example we need some propositions and theorems. Recall from [4] that an involution of an algebra A is a mapping x → x* of A into A such that the mapping * satisfies the following conditions: For all x , y ∈ X , α ∈ C (complex field) (i) (x + y)* = x* + y* (ii) (αx)* = α* x* (iii) (x*)* = x (iv) (xy)* = y* x* Also, recall that from [11] if A is a semiprime algebra and if C(A) is the extended centroid of A, then A is said to be centrally closed if and only if C(A) is equal to the base field. 3-1 Proposition [14]: Let δ be a derivation on a topologically simple Banach algebra A not necessary associative, and suppose that there exists non zero T in M(A) with finite dimensional range. Then δ is continuous. 3-2 Proposition [14]: Let A be a centrally closed prime algebra such that dim(T(A)) > 1 for all non zero T in the multiplication algebra M(A) of A . Then there is a sequence {bn}n∈N in A and {Tn}n∈N in M(A) such that Tn ... T1 bn ≠ 0 and Tn+1 ... T1 bn = 0 for all n in N. 3-3 Theorem [14]: Every H*-algebra with zero annihilation is the closure of the orthogonal sum of its minimal closed ideal, and these are topologically simple H*-algebra. 3-4 Theorem [14]: Every topologically simple H*-algebra is centrally closed prime algebra. 3-5 Lemma [11]: Let A be an algebra, and assume the existence of a non-degenerate symmetric associative bilinear form < . , . > on A. Then we have: Amir A. Mohammed & Yassin Abdulljabar & Nadia A. Abdulraziq 67 (i) There exist a unique linear algebra involution # on the multiplication algebra M(A) of A satisfying d d R L = # and A d L R d d ∈ = all for # (ii) for x , y in A and T in M(A). The equality holds ) ( , , # > < = > < y T x y x T . Now, we present our application example. 3-6 Example (Villena's Theorem): Every derivation on H*-algebra A with zero annihilator is continuous. Proof: Let δ be a derivation on H*-algebra A with zero annihilator. We assume first that A is topologically simple, and by applying theorem (3-4) we have, A is a centrally closed prime algebra. Now, M(A) satisfying one of the following case: (i) There exist an element in M(A) has finite dimensional range. (ii) Every element in M(A) has infinite dimensional range. First case: if (i) is true, then δ is continuous by using proposition (3-1). Second case: If every element in M(A) has infinite-dimensional range, then by using proposition (3-2) there exist a sequence {Cn}n∈N ⊆ A and {Tn}n∈N ⊆ M(A) such that: Tn+1 Tn ... T1 Cn = 0 ...(1) Tn Tn-1 ... T1 Cn ≠ 0 ...(2) For all n in N. By using the fact that every topologically simple H*algebra contain a non degenerate symmetric associative bilinear continuous form < . , . >, also by lemma (3-5), there exist algebra involution # on M(A) such that : A b L R R L b b b b ∈ ∀ = = , , # # ...(3) Now, suppose that n ∈ N is a positive integer number satisfying : ) ( ... ) ( ... # 1 # 1 # # 1 A T T A T T n n + = ...(4) This implies that a sequence ) ( } { # A M T N n n ⊆ ∈ is not constantly decreasing . Now, from equation (1) we can have Tn+1 Tn ... T1 Cn = 0. The mapping < . , . > is non degenerate, continuous [by lemma (3-5)] and from equation (4) we can get: 0 > < = > < = > < = > < = > < = > < = > < = > < = > < =

is contained in the radical of A .Singer and Warmer attained to improve the result mentioned above by delete the continuity of a derivation which is known as a Singer-Warmer conjecture: Let A be a commutative Banach algebra and let δ be a derivation on A .Then δ is a continuous derivation on A ? Johnson in 1969 proved that every derivation on a semi-simple commutative Banach algebra with identity is continuous [6].This result is a partial answer of Singer-Warmer conjecture because the algebra has an identity.Kaplansky in 1958 see [8] generalized Singer-Warmer conjecture as follows: Is a derivation on semisimple Banach algebra be continuous?Johnson and Sinclair in 1968 see [7] confirmed Kaplansky conjecture, and the Johnson-Sinclair result it was generalized by Jewell and Sinclair in 1976 see [5] and Singer-Warmer conjecture confirmed by Thomas in 1988 see [13].
There are many studies of the automatic continuity of a derivation on non-associative Banach algebras.Particularly, the following problem is our interest.Is a derivation on a non-associative semisimple Banach algebra be continuous?
For details about this problems see [10], [12], [14], [15].One of the important theorem in this direction also is due to Villena see [14] and it is our concern in this paper which is state that (Villena Theorem): Every derivation on H*-algebra with zero annihilator is continuous.
Our work can be divided in three directions: (1) State and study Jewell-Sinclair theorem: Every derivation on a Banach algebra A is continuous if A satisfies: i-A has no non-zero finite-dimensional nilpotent ideals.
ii-For each closed infinite-dimensional ideal I of A , there is a sequence { } (3) We show that Villena's proof of the automatic continuity of a derivation on H*-algebra with zero annihilator can be also followed by using extended Jewell-Sinclair theorem.To explain our goal mentioned above we need the following definitions, propositions, and theorems.[9], [11]: Let A be an algebra.A derivation δ on A is a linear mapping δ : A → A such that:

1-1 Definition
for all x , y ∈ A. [12]: Let X and Y be normed spaces and let T be a linear mapping from X into Y, then sparating subspace of T (denoted by Y T ) defined as follows:

1-2 Definition
. 1-3 Proposition [3], [12]: Let A be a normed algebra and δ be a derivation on A. Then A δ is closed ideal of A. 1-4 Sinclair Theorem [12]: Let X and Y be Banach spaces and let We recall from [4] that , an algebra A is semiprime if any ideal I of A such that I 2 =0 implies I=0.Also an annihilator of A (denoted by Ann(A)) defined as follows: , and we say that A is of zero annihilator if Ann(A)={0}.Finally, the multiplication algebra of A denoted by M(A) is defined as a subalgebra of L(A) (the algebra of all linear mapping on A) generated by following operators: called identity, left, and right multiplication operators respectively.

2-EXTENDED JEWELL-SINCLAIR THEOREM .
We extending the Jewell-Sinclair theorem in the following way.

2-1 Extended Jewell-Sinclair Theorem:
Let B be a semiprime Banach algebra not necessary associative such that for each closed infinitedimensional ideals I ⊆ B, there is a sequence { } of closed right ideals of B is constantly decreasing.Then, any derivation on B is continuous.

Proof :
Let B be a semiprime Banach algebra not necessary associative and let δ be a derivation on is constantly decreasing.By applying Sinclair theorem (1-4) and by setting: We get, there exist a natural number . This condition implies that {T n } n0N not constantly decreasing, and this is a contradiction.So, B δ must be finite-dimensional.Note that, if B δ is finite-dimensional, then , we find that

3-AN APPLICATION EXAMPLE .
Before we give our example we need some propositions and theorems.
Recall from [4] that an involution of an algebra A is a mapping x → x* of A into A such that the mapping * satisfies the following conditions: For all x , y ∈ X , α ∈ C (complex field) Also, recall that from [11] if A is a semiprime algebra and if C(A) is the extended centroid of A, then A is said to be centrally closed if and only if C(A) is equal to the base field.

3-1 Proposition [14]:
Let δ be a derivation on a topologically simple Banach algebra A not necessary associative, and suppose that there exists non zero T in M(A) with finite dimensional range.Then δ is continuous.

3-2 Proposition [14]:
Let A be a centrally closed prime algebra such that dim(T(A)) > 1 for all non zero T in the multiplication algebra M(A) of A .Then there is a sequence

3-3 Theorem [14]:
Every H*-algebra with zero annihilation is the closure of the orthogonal sum of its minimal closed ideal, and these are topologically simple H*-algebra.

3-5 Lemma [11]:
Let A be an algebra, and assume the existence of a non-degenerate symmetric associative bilinear form < ., .> on A. Then we have: Let δ be a derivation on H*-algebra A with zero annihilator.We assume first that A is topologically simple, and by applying theorem (3)(4) we have, A is a centrally closed prime algebra.Now, M(A) satisfying one of the following case: (i) There exist an element in M(A) has finite dimensional range.
(ii) Every element in M(A) has infinite dimensional range.
First case: if (i) is true, then δ is continuous by using proposition (3-1).
Second case: If every element in M(A) has infinite-dimensional range, then by using proposition (3-2) there exist a sequence {C n } n∈N ⊆ A and {T n } n∈N ⊆ M(A) such that: ) For all n in N. By using the fact that every topologically simple H*algebra contain a non degenerate symmetric associative bilinear continuous form < ., .>, also by lemma (3)(4)(5), there exist algebra involution # on M(A) such that : Now, suppose that n ∈ N is a positive integer number satisfying : ideals of A is constantly decreasing.(2)We put Jewell-Sinclair theorem on a non-associative content and we call it extended Jewel-Sinclair theorem as follows: Let B be a semiprime Banach algebra not necessary associative such that for each closed infinite-dimensional ideal I of B , there is a sequence ideals of B is constantly decreasing.Then, any derivation on B is continuous.

,
b ∈ B δ , and that B δ is an ideal of B, a n b ∈ B δ and take into account that B δ finite-dimensional and δ δ B is continuous we get closed graph theorem we obtain that δ is continuous.This completes the proof • (i) There exist a unique linear algebra involution # on the multiplication algebra M(A) of A satisfying for x , y in A and T in M(A).present our application example.3-6Example (Villena's Theorem):Every derivation on H*-algebra A with zero annihilator is continuous.Proof: decreasing .Now, from equation (1) we can have T n+1 T n … T 1 C n = 0.The mapping < ., .> is non degenerate, continuous [by lemma(3)(4)(5)] and from equation (4) we can get: 0