The purpose of this note is to describe when a general complex algebraic $^*$-algebra pre-$C^*$-normed, and investigate their structure the $^*$-algebras are Baer $^*$-rings in addition algebraicity. As main result we prove following theorem for $^*$-algebras: every kind can be decomposed as direct sum $M\oplus B$, where $M$ finite dimensional $B$ commutative $^*$-algebra. summand $^*$-isomorph...

For two normed algebras $A$ and $B$ with the character space   $bigtriangleup(B)neq emptyset$  and a left $B-$module $X,$  a certain class of bounded linear maps from $A$ into $X$ is introduced. We set $CMH_B(A, X)$  as the set of all non-zero $B-$character module homomorphisms from $A$ into $X$. In the case where $bigtriangleup(B)=lbrace varphirbrace$ then $CMH_B(A, X)bigcup lbrace 0rbrace$ is...

The theory of fuzzy sets, which was initiated by Zadeh in his seminal paper [33] in 1965, was applied to generalize some of the basic concepts of algebra. The fuzzy algebraic structures play a vital role in mathematics with wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, logic, set theory, real an...

In this article, we formalize topological properties of real normed spaces. In the first few parts, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. In the middle of the article, we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, ima...

in this paper we initiate the study of real group algebras and investigate some of its aspects.let l1 (g) be a group algebra of a locally compact group g,τ :g →g be a group homeomorphismsuch that τ 2 =τοτ = 1, the identity map, and lp (g,τ ) = { f ∈ lp (g) : fοτ = f } ( p ≥ 1) . in thispaper, among other results, we clarify the structure of lp (g,τ ) and characterize amenability ofl1 (g,τ ) and...

The concept of doubling, introduced around 1840 by Hamilton and Graves, associates with any quadratic algebra A over a field k of characterstic not 2 its double V(A) = A×A, with multiplication (w, x)(y, z) = (wy− z̄x, xȳ + zw). It yields an endofunctor on the category of all quadratic k-algebras which is faithful but not full. We study in which respect the division property of a quadratic k-alge...

In this paper, we give a characterization of strongly Jordan zero-product preserving maps on normed algebras as a generalization of  Jordan zero-product preserving maps. In this direction, we give some illustrative examples to show that the notions of strongly zero-product preserving maps and strongly Jordan zero-product preserving maps are completely different. Also, we prove that the direct p...

In this paper, we define what we call (non)degenerate valued and graded division algebras [Definition 3.1] and use them to give examples of division p-algebras that are not tensor product of cyclic algebras [Corollary 3.17] and examples of indecomposable division algebras of prime exponent [Theorem 5.2, Corollary 5.3 and Remark 5.5]. We give also, many results concerning subfields of these divi...

A short review on infinite-dimensional Grassmann-Banach algebras (IDGBA) is presented. Starting with the simplest IDGBA over K = R with l 1-norm (suggested by A. Rogers), we define a more general IDGBA over complete normed field K with l 1-norm and set of generators of arbitrary power. Any l 1-type IDGBA may be obtained by action of Grassmann-Banach functor of projective type on certain l 1-spa...