By Ronald Larsen
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Extra info for Banach Algebras: An Introduction
Let A be a Banach algebra with identity e. If x ~ A is topologically nilpotent, then x is a two-sided topological zero divisor. Proof. Let (akl be any sequence of distinct nonzero complex numbers such that limklakl = 0 and consider the sequence (x/~J contained in A. Since x is topologically nilpotent, it follews at once that limll (~ )nlli/n = Ii~ x = 0 n ak n ak l/n Ck = I. 2 , 3 , •.. 2. we deduce that each x/a k is quasi-regular. 3, .... • We note next that xYk = akYk (ake - x)y k = akYk - e = akYk - YkCake - x) = Yk x (k = 1,2,3, ...
The theorem of this section assert! that we can weaken one of the requirements in the definition of a Banach algebra A and still essentially obtain the same sort of mathematical object. In particular. if we assume only that multiplication in A is separately continuous, that is, the mapping (x)y) ~ xy, x,y E A, is continuous in y for each x and continuou! 2. In this sense we need not require the validity of the nora inequality IIxyll < IIxlillyll, X,Y E A, in order to obtain a B8nach algebra. This weaker bypothesis was the one originally used in [GRkl].
Furtherwith identity, then it is easily group. 4, we shall not investigate in any detail the topological properties of A_lor A-I. Nevertheless a certain amount is known, and the interested reader is referred to [Ga, pp. 90 and 91; Ri, pp. 13-15, 19-23, 280-283, 293-296; 5, pp. 7-90, 99-103]. It is also worth noting that, if A is a commutative Banach algebra with identity, yEA-I, and lIxll < l/lly-lU, then 1. Fundamentals of Banach Algebras 32 and (y - x) -1 =y -1 + y -1 CD E (y k=l -1 k x) • This can be established via a direct computation.