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Orthogonality preservers in C ∗ -algebras, JB ∗ -algebras and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow…

2008; Elsevier BV; Volume: 348; Issue: 1 Linguagem: Inglês

10.1016/j.jmaa.2008.07.020

1096-0813

María Burgos, Francisco J. Fernández-Polo, Jorge J. Garcés, Juan Martı́nez Moreno, Antonio M. Peralta,

Holomorphic and Operator Theory

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)⊆E2∗∗(r(d)), every element in T(A) and d operator commute in the JB∗-algebra E2∗∗(r(d)), and there exists a triple homomorphism S:A→E2∗∗(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∗, 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∗-algebras is orthogonality preserving if and only if it preserves zero-triple-products.