Characterization of Lie centralizable mappings on B(X)
Behrooz Fadaee, Hoger Ghahramani, Ayyoub Majidi
TL;DR
The paper characterizes additive Lie centralizable mappings on $\mathcal{B}(\mathcal{X})$ at a fixed $W$ with $\overline{W(\mathcal{X})} \neq \mathcal{X}$ and $W$ not a scalar multiple of the identity. It proves an if-and-only-if description: $(\mathbf{L}_{W})$ holds exactly when $\phi(A)=kA+h(A)$ with $k\in\mathbb{C}$ and $h:\mathcal{B}(\mathcal{X})\to \mathbb{C}I$ additive satisfying $h([A,B])=0$ for all $A,B$ with $AB=W$, and further shows that, in the invertible-central case, $\phi(A)=\lambda A+\mu(A)$ with $\mu([A,B])=0$ for $AB=\xi I$. The proofs rely on Pierce-type decompositions, block-operator analysis, and properties of prime algebras and the extended centroid, unifying and extending prior results for $W=0$ and nontrivial idempotents. These results elucidate the Lie structure of $\mathcal{B}(\mathcal{X})$ and advance the understanding of local Lie centralizers in operator algebras.
Abstract
Assume that B(X) is the algebra of all bounded linear operators on a complex Banach space X, and let W in B(X) is such that cl(W(X)) is not equal to X or W=zI, where z is a complex number and I is the identity operator. We show that if f: B(X) --> B(X) is an additive mapping Lie centralizable at W, then f(A)=kA+h(A) for all A in B(X), where k is a complex number and h:B(X)--> CI is an additive mapping such that h([A,B])=0 for all A,B in B(X) with AB=W.