## 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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