Orthogonally Additive Polynomials and Orthosymmetric Maps in Banach Algebras with Properties A and B

This paper considers Banach algebras with properties A or B, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in particular, it includes C ∗-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from Aⁿ into C.