On locally \(A\)-convex modules (Q2067222)

The paper deals with the inheritance of the property of being some type of an \(\mathcal A\)-module for a locally convex algebra \(({\mathcal A}, (\mid\!\cdot\!\mid_i)_{i\in I})\). More precisely, the authors are interested in the following classes of \(\mathcal A\)-modules:\par Let \((E, (\mid\!\cdot\!\mid_\lambda)_{\lambda\in\Lambda})\) be a left \(\mathcal A\)-module for a locally convex algebra \(({\mathcal A}, (\mid\!\cdot\!\mid_i)_{i\in I})\). Then \(E\) is\par a) a left \(\mathcal A\)-module with a jointly continuous action if, for each \(\lambda\in \Lambda\), there exist \(i(\lambda)\in I\) and \(\lambda'\in\Lambda\) such that \(\mid ax\!\mid_\lambda\leqslant\mid \!a\!\mid_{i(\lambda)}\mid\!x\mid_{\lambda'}\) for every \(a\in\mathcal A\) and \(x\in E\);\par b) a locally \(A\)-convex \(\mathcal A\)-module if, for each \(a\in\mathcal A\) and each \(\lambda\in\Lambda\), there exists \(\alpha(a, \lambda)>0\) such that \(\mid ax\!\mid_\lambda\leqslant\alpha(a, \lambda)\mid\! x\mid_\lambda\) for every \(x\in E\);\par c) a locally uniformly \(A\)-convex left \(\mathcal A\)-module if, for each \(a\in\mathcal A\), there exists \(\alpha(a)>0\) such that \(\mid ax\!\mid_\lambda\leqslant\alpha(a)\mid\! x\mid_\lambda\) for every \(\lambda\in\Lambda\) and \(x\in E\);\par d) a locally \(m\)-convex left \(\mathcal A\)-module if, for each \(\lambda\in\Lambda\), there exists \(i(\lambda)\in I\) such that \(\mid ax\!\mid_\lambda\leqslant\mid\!a\!\mid_{i(\lambda)}\mid\! x\mid_\lambda\) for every \(a\in\mathcal A\) and \(x\in E\).\par Many examples of algebras, belonging to the one of the classes and not in the another, are provided.\par The authors show that the property of being locally \(A\)-convex left \(\mathcal A\)-module is inherited by the operations of\par (i) taking the quotient by a left \(\mathcal A\)-submodule;\par (ii) considering instead of a non-unital algebra \(\mathcal A\) its unitization;\par (iii) taking a completion of a locally \(A\)-convex left \(\mathcal A\)-module;\par (iv) taking the (direct) product left \(\mathcal A\)-module of locally \(A\)-convex left \(\mathcal A\)-modules;\par (v) taking the projective limit of a projective system of locally \(A\)-convex left \(\mathcal A\)-modules;\par and that the property of being locally \(m\)-convex left \(\mathcal A\)-module is inherited by the operation of\par (vi) taking the strict inductive limit of a sequence of locally \(m\)-convex left \(\mathcal A\)-modules.\par The authors also claim that the properties (i)-(v) are also true for left \(\mathcal A\)-modules with jointly continuous action, locally uniformly \(A\)-convex left \(\mathcal A\)-modules and locally \(m\)-convex left \(\mathcal A\)-modules.\par At the end of the paper, the authors offer some Arens-Michael-like decomposition for some projective sytems of \(m\)-normed or \(A\)-normed left \(\mathcal A\)-modules.