The characterization of generalized Jordan centralizers on algebras

Abstract

In this paper, it is shown that if A is a CSL subalgebra of a von Neumann algebr and φ is a continuous mapping on A such that (m + n + k + l)φ(A 2 ) − (mφ(A)A + nAφ(A) + kφ(I)A 2 + lA 2 φ(I)) ∈ FI for any A ∈ A, where F is the real field or the complex field, then φ is a centralizer. It is also shown that if φ is an additive mapping on A such that (m + n + k + l)φ(A 2 ) = mφ(A)A + nAφ(A) + kφ(I)A 2 + lA 2 φ(I) for any A ∈ A, then φ is a centralizer.

Characterization of Jordan centralizers and Jordan two-sided centralizers on triangular rings without assuming unity

“…Vukman [22] showed that an additive mapping T : R → R, where R is a 2-torsion free semiprime ring, satisfying 2T (x 2 ) = T (x)x + xT (x) for all x ∈ R, is a centralizer. To read more about centralizers, we refer the readers to some recent papers [2,3,6,7,13,14,15,19,20,21,23], where further references can be found. So far, many mathematicians have investigated Jordan centralizers on triangular rings (or algebras) with unity, see, e.g.…”

The Characterization of Generalized Jordan Centralizers on Triangular Algebras

“…Li et al ( [10]) proved that a Jordan centralizer on a CSL subalgebra of a von Neumann algebra is a centralizer. Chen et al ( [11]) proved that an additive mapping on a CSL subalgebra of a von Neumann algebra such that ( + + + ) ( 2 ) − ( ( ) + ( ) + ( ) 2 + 2 ( )) ∈ F for any ∈ A is a centralizer. Lately several authors investigated ( , )− Jordan centralizers or their related mappings on rings and algebras.…”