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Totally convex algebras. (English) Zbl 0763.46036

Having studied totally convex spaces in two earlier papers [Commun. Algebra. 12, 953-1019 (1984; Zbl 0552.46042) and ibid. 13, 1047-1113 (1985; Zbl 0574.46054)] the authors here turn to the study of totally convex algebras. This corresponds to the step from abelian groups to rings. Specifically, a totally convex algebra is a totally convex space together with an associative multiplication. For example, if \(C\) is a totally convex space, then the space of endomorphisms of \(C\) is a (unital) totally convex algebra. Much of the material is presented in the language of categories and functors. Indeed, the first stated result on totally convex algebras is that the category of these (together with their morphisms) arises as the category of Eilenberg-Moore algebras of the unit-ball functor on the category of Banach algebras and contractive homomorphisms. In subsequent sections of the paper, the authors introduce and discuss notions of ideal, tensor product, inverses (of several types), and spectrum; in other words, they investigate in this new context several of the standard concepts generally associated with Banach algebras. For instance, the element \(a\) is weakly invertible iff there is an element \(b\) and a scalar \(\rho\) with \(|\rho|\leq 1\) such that \(ab=ba=\rho(id)\). The spectrum of \(a\) is then defined as the set \(Sp(a)=\bigl\{ \lambda\mid\) \({\lambda \over {1+|\lambda|}}id- {\lambda\over {1+|\lambda|}}\) is not weakly invertible\(\bigr\}\). With these definitions, the authors prove, among other things, that \(Sp(a)\) is a non-empty compact set and that \(Sp(q(a))=\{q(\lambda)\mid\) \(\lambda\in Sp(a)\}\) (the spectral mapping theorem). The reader would do well to keep handy copies of the earlier papers mentioned above as there are numerous references to them throughout.

MSC:

46H05 General theory of topological algebras
46H10 Ideals and subalgebras
46M99 Methods of category theory in functional analysis
46H20 Structure, classification of topological algebras
46K05 General theory of topological algebras with involution
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