Characterizations of Centralizers and Derivations on Some Algebras

Abstract

Abstract. A linear mapping φ on an algebra A is called a centralizable mapping at G ∈ A if φ(AB) = φ(A)B = Aφ(B) for each A and B in A with AB = G, and φ is called a derivable mapping at G ∈ A if φ(AB) = φ(A)B + Aφ(B) for each A and B in A with AB = G. A point G in A is called a full-centralizable point (resp. full-derivable point) if every centralizable (resp. derivable) mapping at G is a centralizer (resp. derivation). We prove that every point in a von Neumann algebra or a triangular algebra is a full-centralizable point. We also prove that a point in a von Neumann algebra is a full-derivable point if and only if its central carrier is the unit.

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

“…So far, many mathematicians have investigated Jordan centralizers on triangular rings (or algebras) with unity, see, e.g. [4,8,10,16,17,18], and the references therein. In the present paper, we shall initiate the study of Jordan centralizers, Jordan two-sided centralizers and some related mappings on triangular rings without assuming unity.…”

Characterizations of centralizable mappings on algebras of locally measurable operators

“…In [16], W. Xu, R. An and J. Hou prove that if H is a Hilbert space with dimH ≥ 2, then every element G in B(H) is a full-centralizable point of L(B(H)). In [8], J. He, J. Li and W. Qian prove that if M is a von Neumann algebra, then every element G in M is a full-centralizable point of L(M).…”