Abstract
Let A,B be standard operator algebras on complex Banach spaces X and Y of dimensions at least 3, respectively. In this paper we give the general form of a surjective (not assumed to be linear or unital) map ? : A ? B such that ?2 : M2(C)?A ? M2(C)?B defined by ?2((sij)2x2) = (?(sij))2x2 preserves nonzero idempotency of Jordan product of two operators in both directions. We also consider another specific kinds of products of operators, including usual product, Jordan semi-triple product and Jordan triple product. In either of these cases it turns out that ? is a scalar multiple of either an isomorphism or a conjugate isomorphism.
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