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Semidirect products of locally convex algebras and the three-space-problem. (English) Zbl 0995.46032

An algebra (over \(\mathbb R\) or \(\mathbb C\)) containing an ideal \(C\) and a subalgebra \(B\) such that \(C\cap B= \{0\}\) and \(A= C+ B\) is called the semi-direct product of \(C\) and \(B\). If \({\mathcal T}\) is a locally convex topology on \(A\) and \((c,b)\mapsto b\) is a homeomorphism, then the preceding terminology is further amplified with the adjective topological. The authors present a method for constructing such algebras and they show that if both \(C\) and \(B\) are locally \(m\)-convex, then so is \(A\). However, they also construct an example in which \({\mathcal T}\) is a Banach-space topology, the ideal \(C\) in the topology \({\mathcal T}\cap C\), and the quotient space \(A/C\) in the quotient-topology induced by \({\mathcal T}\) are both Banach algebras, but \((A,{\mathcal T})\) is not a Banach algebra-multiplication in it is not \({\mathcal T}\)-continuous.

MSC:

46H10 Ideals and subalgebras
46H05 General theory of topological algebras
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