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totally convex algebra; Eilenberg-Moore algebra; Banach algebra; ideal; (weak) inverse; spectrum
By definition a totally convex algebra $A$ is a totally convex space $|A|$ equipped with an associative multiplication, i.e\. a morphism $\mu :|A|\otimes |A|\longrightarrow |A|$ of totally convex spaces. In this paper we introduce, for such algebras, the notions of ideal, tensor product, unitization, inverses, weak inverses, quasi-inverses, weak quasi-inverses and the spectrum of an element and investigate them in detail. This leads to a considerable generalization of the corresponding notions and results in the theory of Banach spaces.
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