Topologically nilpotent normed algebras
The paper under review is devoted to generalize some results on associative normed algebras to the nonassociative context. Further, some new results on associative and nonassociative normed algebras are also stated. The work is divided into ten sections. After an introductory Section 1, it is shown in the second section that nilpotence of a non-necessarily associative normed algebra \(A\) can be determined in terms of the equivalent algebra norms on \(A\). In Section 3, a notion of spectral radius of a bounded set of any normed algebra is introduced and some related results on associative normed algebras are extended to the general case. In the next section, it is considered the result asserting that, for each bounded and multiplicatively closed subset \(S\) of an associative normed algebra \(A\), there exists an equivalent algebra norm on \(A\) such that \(|||s||| \leq 1\) for any \(s \in S\), and the failure of this result in a nonassociative context is quantified. Section 5 is devoted to a preliminary study of topologically nilpotent non-necessarily associative algebras. The following two sections are centered on the development of the notions of quasi-nilpotent normed algebra and completeness. Section 8 is focused on the particular Lie normed case while in Section 9 the class of algebras almost arbitrarily prefixed is introduced to illustrate most of the notions introduced in the paper. Finally, the last section is a collection of remarks and complements.
scientific article; zbMATH DE number 6154883
scientific article