A geometric characterization of structural projections on a JBW∗-triple

A structural projection R on a Jordan∗-triple A is a linear projection such that, for all elements a, b and c in A,R{a Rb c}={Ra b Rc}.The L-orthogonal complement G◊ of a subset G of a complex Banach space E is the set of elements x in E such that, for all elements y in G,||x±y||=||x||+||y||.A contractive projection P on E is said to be neutral if the condition that||Px||=||x||implies that the elements Px and x coincide, and is said to be a GL-projection if the L-orthogonal complement (PE)◊ of the range PE of P is contained in the kernel ker(P) of P. It is shown that, for a JBW∗-triple A, with predual A∗, a linear projection R on A is structural if and only if it is the adjoint of a neutral GL-projection on A∗, thereby giving a purely geometric characterization of structural projections.