Resumen de Order Bounded Separating Linear Maps on F-Algebras

Karim Boulabiar

• A F-algebra is an Archimedean lattice ordered algebra with a weak order unit. Let be A and B be F -algebras and let T be a separating linear map from A into B, that is, T is a linear map such that T(f)T(g) = 0 in B whenever fg = 0 in A. It is proven by an order theoretical and purely algebraic method that there exist a 'weight' element w in B and a positive algebra homomorphism S from A into the maximal ring of quotients Q(B) of B such that T(f) = wS(f) holds for all f in A. Both real and complex cases are considered. This result generalizes the following theorem proved by W. Arendt in his paper [Spectral properties of Lamperti operators, Indiana Univ. J. Math., 32 (1983), 199-215]. Let C(X) and C(Y) be the F-algebras of all scalar-valued continuous functions on compact Hausdorff topological spaces X and Y, respectively. Then for every separating linear map T from C(X) into C(Y) there exist a 'weight' function w in C(Y) and a function h from Y into X (continuous on the cozero set of w) such that T(f)(y) = w(y)f(h(y)) holds for all f in C(X) and y in Y.