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Abstract. We show that a real Banach algebra A such that a 2 = a 2 for a ∈ A is a subalgebra of the algebra C H (X) of continuous quaternion-valued functions on a compact set X.

In this paper the notion of universal algebra bundle is introduced. We show that several properties of local sections of a certain sheaf hold also for local sections of module bundles and universal algebra bundles. As a main result of this paper, we construct an equivalence between the category of module bundles (or universal algebra bundles) and a certain subcategory of the category of sheaves or a certain subcategory of the category of complete presheaves.