By definition a totally convex algebra $A$ is a totally convex space $\left|A\right|$ equipped with an associative multiplication, i.eȧ morphism $\mu :\left|A\right|\otimes \left|A\right|⟶\left|A\right|$ 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.