Orthogonality preservers in C∗-algebras, JB∗-algebras and JB∗-triples

We study orthogonality preserving operators between C∗-algebras, JB∗-algebras and JB∗- triples. Let T : A → E be an orthogonality preserving bounded linear operator from a C∗- algebra to a JB∗-triple satisfying that T ∗∗(1) = d is a von Neumann regular element. Then T (A) ⊆ E∗∗ 2 (r(d)), every element in T (A) and d operator commute in the JB∗- algebra E∗∗ 2 (r(d)), and there exists a triple homomorphism S : A → E∗∗ 2 (r(d)), such that T = L(d, r(d))S, where r(d) denotes the range tripotent of d in E∗∗. An analogous result for A being a JB∗-algebra is also obtained. When T : A → B is an operator between two C∗-algebras, we show that, denoting h = T ∗∗(1) then, T orthogonality preserving if and only if there exists a triple homomorphism S : A → B∗∗ satisfying h∗ S(z) = S(z∗)∗h, hS(z∗)∗ = S(z)h∗, and T (z) = L h, r(h) S(z) = 1 2 hr(h)∗ S(z)+ S(z)r(h)∗h . This allows us to prove that a bounded linear operator between two C∗-algebras is orthogonality preserving if and only if it preserves zero-triple-products.