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.