Maps preserving the idempotency of products of operators (Q996219)

Let \(X\) be a complex Banach space and denote \(B(X)\) the algebra of all bounded linear operators on \(X\). The authors consider surjective unital maps \(\phi:B(X) \to B(X)\) (without the assumption of linearity) which have the property that for any \(A,B\in B(X)\), the operator \(AB\) is a nonzero idempotent if and only if \(\phi(A)\phi(B)\) is a nonzero idempotent. It is proved that, in the case when \(X\) is infinite-dimensional, every such map is of the form \(\phi(A)=SAS^{-1}\) with some invertible bounded linear or conjugate-linear operator \(S\) on \(X\). This means that \(\phi\) is necessarily a linear or conjugate-linear algebra automorphism of \(B(X)\). As for the finite-dimensional case, when \(3\leq \dim X<\infty\), it is shown that the maps under consideration are necessarily semilinear ring automorphisms.