Inverse-producing extensions of normed algebras
Richard F. Arens · Transactions of the American Mathematical Society · 1958
1. Introduction. The first result presented here is that an element c of a commutative normed algebra A has an inverse in some norm-preserving extension B (superalgebra) of A if and only if can—>0 implies an—*0 lor every sequence in A, i.e., c is not a topological zero-divisor. The method may be described as consisting in constructing the algebra A (x) of polynomials with coefficients in A, and then dividing out the ideal generated by the (first degree!) polynomial 1—cx. The resulting algebra B is always of infinite degree over A, except when c already has an inverse in A. In the classical applications of this method (usually, to fields) the coefficient of the highest power may be taken as 1, and that sort of polynomial extension goes over in manageable fashion to normed algebras [see bibliography: Hoffman and Arens]. However, the present construction is nontrivial only because c has no inverse (and effective only because it is not a topological zero-divisor). Most of our space is taken up with quantitative refinements, with systems of elements, and finally with a result on semi-simplicity, interesting perhaps mainly in view of the complexity of our proof. The question (unsolved at present) is whether, when A is semi-simple, it can be arranged that B is semi-simple (if it can be arranged, then it can be done by our method (see 3.71)). We are able to answer affirmatively only in the case where c generates A over the scalars (or some other similarly-behaving subalgebra). We remark that all problems treated here are trivial when A has the sup norm, i.e., ||a||=||a||M [see Loomis]: the algebra of all bounded (complex valued) functions on the Shilov boundary provides an extension solving all reasonable problems. C. E. Rickart has exhibited a technique for simultaneously advancing to regularity a certain class of elements. This class is more or less the obvious one for his technique—reminiscent of the making a field out of an integral domain. Our individual- (or finitely many) element-method takes care of some elements not in Rickart's class. Some unsolved problems already familiar to people in this field are men