In this paper we study the tensor product of two f -algebras. We show that the Riesz Subspace generated by a subalgebra in an f -algebra is an algebra in order to prove that the Riesz tensor product of two f -algebras has a structure of an f -algebra.

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It is shown that the order bidual X ∼∼ of an Archimedean semiprime f -algebra X has a unit element for the Arens multiplication if and only if every positive linear functional on X extends to a positive linear functional on the f -algebra Orth (X ) of all orthomorphisms on X .

We investigate representations : A −→ L b (X ), where A is a unital function algebra and L b (X ) is the space of all order bounded operators on a vector lattice X . Given an element x ∈ X , the orbit space [x] generated by at x is the subspaceIn this paper we make a detailed study of the orbite space [x] , x ∈ X . It turn out that they are vector lattices with a weak order unit. Moreover, It is proved that for any representation : A −→ L b (X ) can be extended to a representation of the order bidual A ∼∼ of A.

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