### Maps Preserving 2-Idempotency of Certain Products of Operators

#### Abstract

Let $\mathcal{A}, \mathcal{B}$ be standard operator algebras on

complex Banach spaces $\mathcal X$ and $\mathcal Y$ 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 $\Phi:

\mathcal{A} \longrightarrow \mathcal{B}$ such that $\Phi_2

:M_2(\mathbb C)\otimes\mathcal {A}\rightarrow M_2(\mathbb C)

\otimes\mathcal {B}$ defined by $\Phi_2((s_{ij})_{2\times

2})=(\Phi(s_{ij}))_{2\times 2}$ 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 $\Phi$

is a scalar multiple of either an isomorphism or a conjugate

isomorphism.

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