Banach Algebra of Bounded Complex-Valued Functionals

Banach Algebra of Bounded Complex-Valued Functionals In this article, we describe some basic properties of the Banach algebra which is constructed from all bounded complex-valued functionals.

Let V be a complex algebra. A complex algebra is called a complex subalgebra of V if it satisfies the conditions (Def. 1). (Def. 1)(i) The carrier of it ⊆ the carrier of V , (ii) the addition of it = (the addition of V ) (the carrier of it), (iii) the multiplication of it = (the multiplication of V ) (the carrier of it), (iv) the external multiplication of it = (the external multiplication of V ) (C × the carrier of it), (v) 1 it = 1 V , and (vi) 0 it = 0 V . We now state the proposition (1) Let X be a non empty set, V be a complex algebra, V 1 be a non empty subset of V , d 1 , d 2 be elements of X, A be a binary operation on X, M be a function from X × X into X, and M 1 be a function from C × X into X. Suppose that V 1 = X and d 1 = 0 V and d 2 = 1 V and A = (the addition of V ) (V 1 ) and M = (the multiplication of V ) (V 1 ) and Let V be a complex algebra. One can check that there exists a complex subalgebra of V which is strict.
Let V be a complex algebra and let V 1 be a subset of V . We say that V 1 is C-additively-linearly-closed if and only if: (Def. 2) V 1 is add closed and has inverse and for every complex number a and for every element v of V such that v ∈ V 1 holds a · v ∈ V 1 . Let V be a complex algebra and let V 1 be a subset of V . Let us assume that V 1 is C-additively-linearly-closed and non empty. The functor Mult(V 1 , V ) yielding a function from C × V 1 into V 1 is defined as follows: Let X be a non empty set. The functor C-BoundedFunctions X yielding a non empty subset of CAlgebra(X) is defined by: Let X be a non empty set. Note that CAlgebra(X) is scalar unital. Let X be a non empty set. One can verify that C-BoundedFunctions X is C-additively-linearly-closed and multiplicatively-closed. Let V be a complex algebra. Observe that there exists a non empty subset of V which is C-additively-linearly-closed and multiplicatively-closed.
Let V be a non empty CLS structure. We say that V is scalar-multiplcationcancelable if and only if: (Def. 5) For every complex number a and for every element v of V such that a · v = 0 V holds a = 0 or v = 0 V . One can prove the following two propositions: (2) Let V be a complex algebra and V 1 be a C-additively-linearly-closed multiplicatively-closed non empty subset of V .
One can prove the following proposition (4) For every non empty set X holds the C-algebra of bounded functions of X is a complex subalgebra of CAlgebra(X).
Let X be a non empty set. Note that the C-algebra of bounded functions of X is vector distributive and scalar unital.
Next we state several propositions:  Let X be a non empty set and let F be a set. Let us assume that F ∈ C-BoundedFunctions X. The functor modetrans(F, X) yields a function from X into C and is defined by: (Def. 7) modetrans(F, X) = F and modetrans(F, X) X is bounded.
Let X be a non empty set and let f be a function from X into C. The functor PreNorms(f ) yields a non empty subset of R and is defined by: We now state two propositions: (10) For every non empty set X and for every function f from X into C such that f X is bounded holds PreNorms(f ) is upper bounded. Let X be a non empty set. The functor C-BoundedFunctionsNorm X yields a function from C-BoundedFunctions X into R and is defined by: (Def. 9) For every set x such that x ∈ C-BoundedFunctions X holds (C-BoundedFunctionsNorm X)(x) = sup PreNorms(modetrans(x, X)). One can prove the following two propositions: (13) 1 For every non empty set X and for every function f from X into C such that f X is bounded holds modetrans(f, X) = f. (14) For every non empty set X and for every function f from X into C such that f X is bounded holds (C-BoundedFunctionsNorm X)(f ) = sup PreNorms(f ). Let X be a non empty set. The C-normed algebra of bounded functions of X yielding a normed complex algebra structure is defined by: (Def. 10) The C-normed algebra of bounded functions of X = C-BoundedFunctions X, mult(C-BoundedFunctions X, CAlgebra(X)), Add(C-BoundedFunctions X, CAlgebra(X)), Mult(C-BoundedFunctions X, CAlgebra(X)), One(C-BoundedFunctions X, CAlgebra(X)), Zero(C-BoundedFunctions X, CAlgebra(X)), C-BoundedFunctionsNorm X . Let X be a non empty set. One can verify that the C-normed algebra of bounded functions of X is non empty.
Let X be a non empty set. One can check that the C-normed algebra of bounded functions of X is unital.
We now state a number of propositions: (15) Let W be a normed complex algebra structure and V be a complex algebra. Suppose the carrier of W , the multiplication of W , the addition of W , the external multiplication of W , the one of W , the zero of W = V. Then W is a complex algebra. (16) For every non empty set X holds the C-normed algebra of bounded functions of X is a complex algebra. (17) Let X be a non empty set and F be a point of the C-normed algebra of bounded functions of X. Then (Mult(C-BoundedFunctions X, CAlgebra(X)))(1 C , F ) = F.  (26) Let X be a non empty set, a be a complex number, and F , G be points of the C-normed algebra of bounded functions of X. Then (i) if F = 0, then F = 0 the C-normed algebra of bounded functions of X , (ii) if F = 0 the C-normed algebra of bounded functions of X , then F = 0, (iii) a · F = |a| · F , and (iv) Let X be a non empty set. Note that the C-normed algebra of bounded functions of X is right complementable, Abelian, add-associative, right zeroed, vector distributive, scalar distributive, scalar associative, scalar unital, discernible, reflexive, and complex normed space-like.
We now state two propositions: (27) Let X be a non empty set, f , g, h be functions from X into C, and F , G, H be points of the C-normed algebra of bounded functions of X. Suppose f = F and g = G and h = H. Then H = F − G if and only if for every element x of X holds h(x) = f (x) − g(x).
(28) Let X be a non empty set and s 1 be a sequence of the C-normed algebra of bounded functions of X. If s 1 is CCauchy, then s 1 is convergent.
Let X be a non empty set. Observe that the C-normed algebra of bounded functions of X is complete.
Next we state the proposition (29) For every non empty set X holds the C-normed algebra of bounded functions of X is a complex Banach algebra.