Orthogonality Preservers Revisited

Abstract

We obtain a complete characterization of all orthogonality preserving operators from a JB *-algebra to a JB *-triple. If T : J → E is a bounded linear operator from a JB *-algebra (respectively, a C *-algebra) to a JB *-triple and h denotes the element T**(1), then T is orthogonality preserving, if and only if, T preserves zero-triple-products, if and only if, there exists a Jordan *-homomorphism [Formula: see text] such that S(x) and h operator commute and T(x) = h•r(h) S(x), for every x ∈ J, where r(h) is the range tripotent of h, [Formula: see text] is the Peirce-2 subspace associated to r(h) and •r(h) is the natural product making [Formula: see text] a JB *-algebra. This characterization culminates the description of all orthogonality preserving operators between C *-algebras and JB *-algebras and generalizes all the previously known results in this line of study.

Linear disjointness preservers of W*-algebras

“…In [30], it is shown that every bounded linear doubly orthogonality preserver θ between C*-algebra preserves the triple products {a, b, c} := ab * c + cb * a whenever θ * * (1) is a partial isometry. It is further investigated in [5,6] to extend this concept to JB*-algebras and JB*-triples. In [7], the following theorem is proved.…”

“…Finally, let us mention that some other kinds of disjointness in a W*-algebra can be defined by doubly orthogonality (see, e.g., [5,6,31]), and by its left (or right) ideals (see, e.g., [1,10,20,23]). We will also discuss them at the end of the paper.…”

Weakly compact orthogonality preservers on C*-algebras

“…We have already commented that continuous linear orthogonality preserving operators between C * -algebras (resp., JB * -algebras) have been recently characterised by Burgos, Martínez and the authors of this note in [6] and [7]. The characterization obtained in the just quoted papers was stated in Theorem 1.…”

“…Orthogonality preserving operators between general C * -algebras have been intensively studied by many researchers (compare [1,2,[6][7][8]21,35,36] and [9]). The studies on continuous orthogonality preserving operators between general C * -algebras culminate with the following description obtained in [6] and [7] (see §2 for the detailed definitions and concepts).…”

“…The studies on continuous orthogonality preserving operators between general C * -algebras culminate with the following description obtained in [6] and [7] (see §2 for the detailed definitions and concepts). A → E be a bounded linear mapping from a C * -algebra (resp., a JB * -algebra) to a JB * -triple.…”