Commuting traces and commutativity preserving maps on triangular algebras

Abstract

Let A be a triangular algebra. The problem of describing the form of a bilinear map B :for all x ∈ A is considered. As an application, commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.  2004 Elsevier Inc. All rights reserved.

Characterizations of Lie Higher and Lie Triple Derivations on Triangular Algebras

“…Basic examples of triangular algebras are upper triangular matrix algebras and nest algebras. Many authors have made important contributions to the related topics, see [2,4,5,6,13]. Cheung in [5] initiated the study of linear mappings of abstract triangular algebras and obtained a number of elegant results.…”

Centralizing traces and Lie triple isomorphisms on triangular algebras

“…When we investigate the above-mentioned mappings, the principal task is to describe their forms. This is demonstrated by various works, see [8,10,11,12,13,15,20,21,28,29,33,36,41,44,45,49,50]. We encourage the reader to read the well-written survey paper [13], in which the author presented the development of the theory of semi-centralizing mappings and their applications in details.…”

Nonlinear maps preserving Lie products on triangular algebras

“…(4), we get that (e 2 0 − e 0 )n = 0 for all n ∈ N. This and Eq. (2) imply that e 0 is an idempotent of C and ( …”

“…for all X ∈ τ(M), and so by the injectivity of ϕ, we have [E 1 , X] = [E 2 , X] for all X ∈ τ(M). This implies that E 1 = E 2 + λI for some λ ∈ C. Hence {0, 1} = σ(E 1 ) = σ(E 2 + λI) = {λ, λ + 1}.…”

“…The R-algebra Cheung [6,7] described commuting maps and Lie derivations of these algebras. Benkovič and Eremita [2] studied commuting traces of bilinear maps and Lie isomorphisms of triangular algebras. Benkovič [3] investigated biderivations of triangular algebras.…”