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**A characterization of Q-algebras.**
*(English)*
Zbl 0876.46035

Rassias, John M. (ed.), Functional analysis, approximation theory and numerical analysis. Dedicated to Stefan Banach on his 100th birthday, Alexander Markowiç Ostrowski on his 99th birthday, Stanislaw Marcin Ulam on his 83rd birthday. Singapore: World Scientific. 277-280 (1994).

We will prove in this work the following:

(I) A t.a. (: topological algebra) \((E,\tau)\) is “\(Q\)” iff there exists \(V\in W_0\) such that \(r_E\leq g_V\).

(II) A l.c.(: locally convex) algebra \((E,\Gamma)\) is “\(Q\)” iff there exist \(\mu>0\) and \(p\in\Gamma\) such that \(r_E\leq p\).

(III) A l.m.c. (: locally multiplicatively convex) t.a. \((E,\Gamma)\) is “\(Q\)” iff there exists \(p\in\Gamma\) such that \(r_E\leq p\).

Applications.

(IV) A “\(Q\)” locally convex \(*\)-algebra \((E,\Gamma)\) with the \(B^*\)-property is normed (: there exists \(p\in \Gamma\) such that \(q\leq p\) for all \(q\in\Gamma\).

(V) (A. Mallios) A l.c. \(*\)-algebra \((E,\Gamma)\) with the \(B^*\)-property whose completion is a “\(Q\)”-algebra is normed.

For the entire collection see [Zbl 0867.00015].

(I) A t.a. (: topological algebra) \((E,\tau)\) is “\(Q\)” iff there exists \(V\in W_0\) such that \(r_E\leq g_V\).

(II) A l.c.(: locally convex) algebra \((E,\Gamma)\) is “\(Q\)” iff there exist \(\mu>0\) and \(p\in\Gamma\) such that \(r_E\leq p\).

(III) A l.m.c. (: locally multiplicatively convex) t.a. \((E,\Gamma)\) is “\(Q\)” iff there exists \(p\in\Gamma\) such that \(r_E\leq p\).

Applications.

(IV) A “\(Q\)” locally convex \(*\)-algebra \((E,\Gamma)\) with the \(B^*\)-property is normed (: there exists \(p\in \Gamma\) such that \(q\leq p\) for all \(q\in\Gamma\).

(V) (A. Mallios) A l.c. \(*\)-algebra \((E,\Gamma)\) with the \(B^*\)-property whose completion is a “\(Q\)”-algebra is normed.

For the entire collection see [Zbl 0867.00015].

### MSC:

46H05 | General theory of topological algebras |

### Keywords:

\(Q\)-algebras; locally convex algebra; \(Q\) locally convex \(*\)-algebra; topological algebra; locally multiplicatively convex; \(B^*\)-property
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\textit{Y. Tsertos}, in: Functional analysis, approximation theory and numerical analysis. Dedicated to Stefan Banach on his 100th birthday, Alexander Markowiç Ostrowski on his 99th birthday, Stanislaw Marcin Ulam on his 83rd birthday. Singapore: World Scientific. 277--280 (1994; Zbl 0876.46035)