Advertible complétude et structure de \(Q\)-algèbre (Advertible completeness and \(Q\)-algebra structure)
Let \(E\) be a real or complex locally pseudo-convex algebra having a Hausdorff topology defined by a family of \(p_{i}\)-pseudo-seminorms, \(0 <p_{i} \leq 1\) (so that \(\|tx\|= |t|^{p_{i}}\|x\|\)). The product may be assumed separately continuous. It seems that the author is almost always assuming that \(E\) has a unit element which he denotes by \(e\). Much of the article comprises a plethora of (sometimes poorly stated) definitions and there is a mix of nets and sequences. The spectral radius of \(x \in E\) is denoted by \(\rho(x)\) and the author calls \(\beta(x) = \inf \{ \alpha > 0: {\lim_{n \to \infty}(\alpha^{-1}x)^{n}} = 0 \}\), with the convention that \(\inf{\emptyset} = +\infty\), the radius of regularity of \(x\). A Q-algebra [cf. \textit{I. Kaplansky}, Amer. J. Math. 69, 153-183 (1947; Zbl 0034.16604)] is here one for which the group \(G\) of invertible elements is open, and for a Mackey \(Q\)-algebra openness should somehow involve the Mackey topology. To have \(\rho \leq \beta\) it is seen sufficient that either \(E\) is multiplicatively pseudo-complete or a Mackey \(Q\)--algebra. A net (\(x_\lambda\)) in \(E\) is called advertibly convergent [see \textit{S. Warner}, Duke Math J. 23, 1-11 (1956; Zbl 0070.11801)] if there is an element \(y \in E\) such that both (\(yx_\lambda\)) and (\(x_{\lambda}y\)) converge to \(e\); this is something like (\(x_\lambda\)) being an approximate quasi-inverse [cf. \textit{I. Kaplansky}, Amer. J. Math. 69, 153-183 (1947; Zbl 0034.16604)]. For \(E\) advertibly complete, it is seen that \(\rho \leq \beta\) (on all of \(E\)). The algebra is called sequential if for any sequence (\(x_n\)) converging to \(0\) there exists an \(n_{0}\) such that \((x_{n_{0}})^{k}\) converges to \(0\) as \(k \to \infty\); sequential, strongly sequential and infra-sequential algebras were defined by \textit{T. Husain} and \textit{S. Ng} [J. Aust. Math. Soc., Ser. A 21, 498-503 (1976; Zbl 0328.46048)] and \textit{T. Husain} [Can. Math. Bull. 22, 413-418 (1979; Zbl 0427.46028)]. The author shows that sequential, infra-sequential and strongly sequential algebras satisfying \(\rho \leq \beta\) are sequentially advertibly complete, \(Q\)-algebras and Mackey \(Q\)-algebras, respectively. For a pseudo seminormed algebra \(E\) with \(\rho \leq \beta\) the distinction between advertibly complete, \(Q\)-algebra and Mackey \(Q\)-algebra falls away.
scientific article; zbMATH DE number 1985435
scientific article