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In this paper we introduce a family of weakly contractive maps on the space \(B(S)\) of bounded real valued functions and use it to show that a fundamental step in the proof of the well-known Stone-Weierstrass approximation theorem can be achieved via Rakotch’s fixed point theorem for weakly contractive maps. With the same technique, we obtain Zemanek’s theorem on the existence of square roots in certain Banach subalgebras of \(B(S)\), and, finally, in the context of abstract Banach algebras, we exhibit some relationship between weakly contractive maps on the closed unit ball and the geometry of the spheres.

Rakotch type contractive maps, square roots and uniform convexity