ORTHOGONALLY ADDITIVE POLYNOMIALS ON DEDEKIND σ-COMPLETE VECTOR LATTICES

The only topology on vector lattices under consideration is the relatively uniform topology. Let A be vector lattice and let B be a topological space. A map P : A ?> B is called a homogeneous polynomial of degree n (or a n-homogeneous polynomial) if P (x) = $ (x,x), where ^ is a n-multilinear map from An = A x ... x A (n-times) into B. Throughout the paper, 'operator' (linear, multilinear or polynomial) will mean 'continuous operator'. A homogeneous polynomial (of degree n) P : A ?> B is said to be orthogonally additive if P(x + y) ? P(x) -f P(y) whenever x, y G A are orthogonally (i.e. \x\ A \y\ = 0). We denote by P0 (nA, B) the set of n-homogeneous orthogonally additive polynomials from A to B. Interest in orthogonally additive polynomials on Banach lattices originates in the work of K. Sundaresan [15], in which it has been characterised as the space of n-homogeneous orthogonally additive polynomials on Lp and ?p. More precisely, K. Sundaresan proved that every n-homogeneous orthogonally additive polynomial P : Lp -? R is determined by some g G L~ via the formula P (f) ? f fng d/i, for all / G Lp. Very recently, D. Perez-Garcia and I. Villanueva in [13], D. Carando, S. Lassalle and I. Zalduendo in [9] proved the following analogous result for C (X) spaces: Let Y be a Banach space, let P : C (X) ?> Y be an orthogonally additive