This is an expository paper on the importance and applications of GB*-algebras in the theory of unbounded operators, which is closely related to quantum field theory and quantum mechanics. After recalling the definition and the main examples of GB*-algebras we exhibit their most important properties. Then, through concrete examples we are led to a question concerning the structure of the completion of a given C*-algebra 𝓐₀[||·||₀], under a locally convex *-algebra topology τ, making the multiplication of 𝓐₀ jointly continuous. We conclude that such a completion is a GB*-algebra over the τ-closure of the unit ball of 𝓐₀[||·||₀]. Further, we discuss some consequences of this result; we briefly comment the case when τ makes the multiplication of 𝓐₀ separately continuous and illustrate the results by examples.