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The goal of this article is to present some method of optimal extension of positive order continuous and $ \sigma $ -order continuous operators on quasi-Banach function spaces with values in Dedekind complete quasi-Banach lattices. The optimal extension of such an operator is the smallest extension of the Bartle–Dunford–Schwartz type integral. It is also shown that if a positive operator sends order convergent sequences to quasinorm convergent sequences, then its optimal extension is the Bartle–Dunford–Schwartz type integral.

We construct a functional calculus on unital uniformly complete f-algebras for continuous center-valued functions of polynomial growth and study the connection with the Minkowski duality.