Some Results on Strongly Fully Stable Banach Γ –Algebra Modules Related To ΓA -deal

The main objective of this research is to study and to introduce a concept of strong fully stable Banach Γ-algebra modules related to an ideal.. Some properties and characterizations of full stability are studied.


Introduction
The theory of Banach algebras is an abstract mathematical theory. Banach algebras started in the early twentieth century, when abstract concepts and structures were introduced, transforming both of the mathematical language and practice. A non-empty set is an algebra with ( , , ) over a field is a vector space and is a ring with , and a ( b) = (a b) = ( a) b for all , for each [1]. In [2], a ring is an algebra where is a ring and and , unary is and nullary element is , when a commutative group is and a semi-group is , module with the module multiplication which is taken to be the multiplication in . A Banach space is a -module with the module multiplication which is taken to be the scalar multiplication. So every closed left ideal of a Banach algebra is in a natural way is a left -module. If U is a submodule of an -module V, then the quotient space V/U with the quotient norm is in a natural way an -module. If is a Banach algebra, and V is a Banach space, then V becomes an -module if we define av = 0 for all a A, v V. We call V a trivial -module [3]. A function : → (not necessarily is commutative) is said to be (homomorphism) multiplier if ( ) [4]. Following [5], let be sub-module of a module ( ) , is said to be stable, for each homomorphism : ⇾ an module is called a fully stable module , if each sub-module of is stable". In [6] , a B-A module is said to be full stability B-algebramodule if each sub-module of B-algebra-module and for every multiplier from to such that ( ). From [7], a left B -algebra-module is generated if there exists such that for all , can be written as for some . A 1-generated is called cyclic module . Following [8,9,10] Let be B-algebra module , is called fully stable Banach module if for every submodule of and for each multiplier θ such that θ( ) ", suppose that is B-algebra module, if for each sub-module of and for every multiplier θ from , such that θ( ) is said to be fully stable Banach algebramodule relative to an ideal of It is easy to see every fully stable Banach module is fully stable Banach module relative to an ideal. Assume that is B-algebra module , if for every sub-module of and for every multiplier θ such that θ( ) is said to be strongly fully stable B-algebra module relative to an ideal of , It is an easy mater to see that every strongly fully stable B-algebra module relative to an ideal is fully stable Banach algebra module. Following [11], suppose that -is a groupoid and is a space of vectors over . Hence , over is said to be a -algebra, if there is a mapping from × × to (we denoted the image by α for , in and α in ) such that :(1)) (α + β) = α + β , (2) (c )α = c( α ) = α(c ), (3)( + ) α = α + α , α ( + ) = α + α , (4) 0α = α0 = 0, α, β and for all , , , c . A -algebra is said to be associative if (5) ( α )β = α( β ), and unital if for every α, β , there is an element 1 α in V such that 1 α α = = α1 α for every nonzero elements of . The concept of strongly fully stable B-A-modules related to an ideal have been introduced and is proving anther characterization of strong fully stable B-A-modules related to A-ideal A Banach A-module is strong fully stable B-A-modules related to A-ideal if and only if for each subsets of , implies ( ) ( ) .

2-Strongly Fully Stability Banach -Algebra Modules Related to an -ideal.
In this section the concept of strongly fully stable Banach -Algebra Modules Related to an ideal is introduced and other characterizations of this concept have been studied.

Definition:
Suppose that is B-algebra-module , is said to be strongly fully stable Bmodule related to ideal of , if for each sub-module of B-algebra-module and for all -multiplier from to such that ( ) It is easy to see that every strongly fully stable Banach -modules related to an ideal is fully stable Banach -modules. Therefore is strongly fully stable Banach -modules related to ideal, if and if for each 1-generated sub-module in and for every multiplier such that ( ) Let be a Banach -modules and be a nonzero ideal of algebra . If is fully stable B--modules and = then is strong fully stable B-A-modules related to an ideal K, since for each 1-generated sub-module N of and A-homomorphism f from to , ∩ =. ∩ X f( ). Suppose that is a B-module, let and be two subsets of , then 1) * +,and 2) ( ) * +. for each ideal of . When the sub-module of strong fully stable B-A-modules related to A-ideal have been partial answer in the next result.

Proposition : Suppose that
is a strong fully stable B-A-modules related to a nonzero ideal of . Then every pure submodule is strong fully stable B-Amodules related to A-ideal. Proof : Assume that is pure sub-module in . For all sub-module L in and f : L → N a multiplier, set i o f = g : L → X , i is the inclusion mapping from to , then from assumption f(L) = g(L) , and f(L) N. Hence f (L) L ∩ ∩ . Because of is pure sub-module in ,we have ∩ = , for each ideal of , therefore f(L) L ∩ . Thus is strong fully stable B-Amodules related to A-ideal .

6 Definition : A Banach
module is called Baer criterion relative to an ideal in A satisfied, if every sub-module of Baer criterion relative to an ideal in A satisfied, this mean that, for each 1-generated sub-module in and multiplier, there is in s.t ( ) for all The next proposition and corollary give new characterization of strong fully stable B-Amodules related to A-ideal.

Proposition
:If is a B-algebra -module, then the Baer criterion relative to an ideal in A is satisfied for 1-generated sub-module in if and only if ( ( )) for each Proof :-Suppose that the Baer criterion relative to an ideal in A holds for 1-generated submodule of . Let ( ( )) and define by ,and , then is well define. It is easy to see is an multiplier. There exists an element from the assumption that ( ) we have in particular, ( ) , therefore ( ( )) , and ( ( )) . Conversely, assume that ( ( )) , for each , then for each multiplier , and μs and ( ) for some hence Baer criterion relative to an ideal in A holds.

Corollary: is strong fully stable B-A-modules related to A-ideal K if and only if
( ( )) In [8], the authors assume that be a unital B-algebra. Algebra-module is said to be quasi α-injective if, φ from to is algebra-module homomorphism (multiplier) such that || φ || ≤ 1, there is algebra-module homomorphism (multiplier) θ from to , θ o i = φ and || θ || ≤ α , i is an iso-metry ,algebra-module isomorphism is an iso-metry algebra-multiplier , from sub-module in to , and is said to be quasi injective if is quasi α -injective for some α The concept of strongly quasi α-injective related to an ideal of is introduced. 2.9 Definition: Assume that be a unital B-algebra. module is said to be strongly quasi α-injective related to an ideal of if, φ from to is multiplier such that || φ || ≤ 1, there exists multiplier θ from to , such that (θ o i)(n) = φ(n) and || θ || ≤ α i is an iso-metry from sub-module to . is said to be strongly quasi injective related to ideal if it is strongly quasi αinjective related to ideal for some α. In the following proposition we give the relationship between strongly quasi α-injective Bmodule related to ideal and strongly fully stable B-module related to an ideal K of A. 2.10 Proposition: Assume that be B-module and be a non-zero ideal of . If is strong fully stable B-A-modules related to A-ideal then is strongly quasi injective Bmodule related to ideal. Proof: Suppose that is sub-module in and f : ⇾ be any algebra-module homomrphism. Because is a fully stable B--module related to ideal , therefore f( ) ∩ , hence there exists λt such that f(n) = λtn . Define g : ⇾ by λtx = g ( x ), it is easy to see that g is a well defined multipler, f(x) = g(x) = λtx , and for all y in , (f o i) (y)g(y) = f(y)g(y) , i is iso-metry, and for some α, || g || ≤ α Therefore is strongly quasi injective B-module related to ideal.