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Many properties of two-sided algebras remain valid for one-sided algebras. Namely, any one sided Banach algebra is commutative modulo its Jacobson radical. In the case of normed algebras, the one-sidedness is not inherited by a sub-algebra, nor by the completion of a normed algebra. | Turk J Math 26 (2002) , 305 – 316. ¨ ITAK ˙ c TUB One Sided Banach Algebras A. El Kinani, A. Najmi, M. Oudadess Abstract Many properties of two-sided algebras remain valid for one-sided algebras. Namely, any one sided Banach algebra is commutative modulo its Jacobson radical. Key Words: Right-sidedness, two-sidedness, commutativity, Banach algebra. Introduction In [1], the authors have proceeded to a study of algebras said to be two-sided by E. Hille and R. S. Philips ([3]). We consider here the left (or right) sidedness, where the notions of two-sidedness and one-sidedness are distinct (Example I-3). Many algebraic properties of [1] are still true. In the case of normed algebras, the one-sidedness is not inherited by a sub-algebra, nor by the completion of a normed algebra. About the structure of these algebras, every rightsided finite dimensional algebra A (and, more generally every, Artinian Banach algebra) is written as A = Rad(A) ⊕ Cn , where Rad(A) is the (Jacobson) radical. It is two-sided if, and only if, Rad(A) is two-sided. We examine the case of a right-sided Banach algebra A such that Rad(A) is finite dimensional and A/Rad(A) is a B(∞) direct sum of total matrix algebras. We prove also that a right-sided Banach algebra is commutative modulo the Jacobson radical like in the two-sided case ([1]). Some conditions for the converse to be true are equally given. For example, if Rad(A) is right-sided and A/Rad(A) is a 1991 Mathematics Subject Classification: 46H05, 46H20 305 EL KINANI, NAJMI, OUDADESS C ∗ -algebra or an l1 -algebra, then A is right-sided. Recall that RadA is the intersection of all regular right (or all regular left) ideals of A. 1. Algebraic properties All algebras considered here are complex. In the sequel, we put A2 = {xy : x, y ∈ A}. A zero-algebra is an algebra A such that A2 = {0}. For every fixed x ∈ A, we write Annd (x) for the right annihilator of x and Bx for an algebraic complementary of Annd (x) in A. Definition 1.1 . A
