Journal of the
Korean Mathematical Society JKMS

A JB$ ^*$-triple is a Banach space $A$ on which the group ${\rm Aut}(B)$ of biholomorphic automorphisms acts transitively on the open unit ball $B$ of $A$. In this case, a triple product $\{\cdots\}$ from $A \times A \times A$ to $A$ can be defined in a canonical way. If $A$ is also the dual of some Banach space $A_*$, then $A$ is said to be a JBW$ ^*$-triple. A projection $R$ on $A$ is said to be structural if the identity $\{Ra, \; b, \; Rc\}=R \{a, \; Rb, \; c \}$ holds. On JBW$ ^*$-triples, structural projections being algebraic objects by definition have also some interesting metric properties, and it is possible to give a full characterization of structural projections in terms of the norm of the predual $A_*$ of $A$. It is shown, that the class of structural projections on $A$ coincides with the class of the adjoints of neutral GL-projections on $A_*$. Furthermore, the class of GL-projections on $A_*$ is naturally ordered and is completely ortho-additive with respect to L-orthogonality.

Keywords: JBW$^*$-triple, structural projection, GL-projection

MSC numbers: Primary 46L70; Secondary 17C65