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

We study orthogonality preserving operators between C∗C∗-algebras, JB∗JB∗-algebras and JB∗JB∗-triples. Let T:A→E be an orthogonality preserving bounded linear operator from a C∗C∗-algebra to a JB∗JB∗-triple satisfying that T∗∗(1)=dT∗∗(1)=d is a von Neumann regular element. Then T(A)⊆E2∗∗(r(d)), every element in T(A)T(A) and d   operator commute in the JB∗JB∗-algebra E2∗∗(r(d)), and there exists a triple homomorphism S:A→E2∗∗(r(d)), such that T=L(d,r(d))ST=L(d,r(d))S, where r(d)r(d) denotes the range tripotent of d   in E∗∗E∗∗. An analogous result for A   being a JB∗JB∗-algebra is also obtained. When T:A→B is an operator between two C∗C∗-algebras, we show that, denoting h=T∗∗(1)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∗)∗hh∗S(z)=S(z∗)∗h, hS(z∗)∗=S(z)h∗hS(z∗)∗=S(z)h∗, andT(z)=L(h,r(h))(S(z))=12(hr(h)∗S(z)+S(z)r(h)∗h). This allows us to prove that a bounded linear operator between two C∗C∗-algebras is orthogonality preserving if and only if it preserves zero-triple-products.