Maps preserving the idempotency of Jordan products

TL;DR

This paper characterizes mappings on operator algebras that preserve the idempotency of scaled Jordan products, without assuming linearity, for complex Banach space operators.

Contribution

It determines the form of non-linear, range-large mappings preserving idempotency of scaled Jordan products in operator algebras.

Findings

Identifies the structure of such mappings on B(X)

Shows these mappings preserve the idempotency of scaled Jordan products

Provides conditions under which the mappings are characterized

Abstract

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X of dimension at least three. For an arbitrary nonzero complex number t we determine the form of mappings f: B(X)-->B(X) with sufficiently large range such that t(AB+BA) is idempotent if and only if t(f(A)f(B)+f(B)f(A)) is idempotent, for all A, B in B(X). Note that f is not assumed to be linear or additive.