Commuting (centralizing) traces and Lie (triple) isomorphisms on triangular algebras revisited.
This paper under review is motivated by two elegant works, which are \textit{D. Benkovič} and \textit{D. Eremita} [J. Algebra 280, No. 2, 797-824 (2004; Zbl 1076.16032)] and \textit{Z. Xiao, F. Wei} and \textit{A. Fošner} [Linear Multilinear Algebra 63, No. 7, 1309-1331 (2015; Zbl 1318.15012)]. In [loc. cit.] Benkovič and Eremita obtained a description of commuting traces of bilinear maps on certain triangular algebras with loyal bimodule. Applying the obtained results they gave a description of Lie isomorphisms of certain triangular algebras with loyal bimodule. As consequences, commuting traces of bilinear maps and Lie automorphisms on upper triangular matrix algebras over a 2-torsion free commutative domain and nest algebras are determined. More recently, Xiao et al. [loc. cit.] gave a description of centralizing traces of bilinear maps and Lie triple isomorphisms on certain triangular algebras with loyal bimodule. By virtue of these descriptions they characterized centralizing traces of bilinear maps and Lie triple automorphisms on upper triangular matrix algebras and nest algebras, respectively. In the current paper, the author proposes a more general notion -- weak loyal bimodules, which extends the notion of loyal bimodules. It is easy to verify that every upper triangular matrix algebra over a unital commutative ring contains weak loyal bimodules. Using some well-known conditions he describes commuting (centralizing) traces of bilinear maps and Lie (triple) isomorphisms on certain triangular algebras with weak loyal bimodule. Basing on these basic descriptions he determines the commuting (centralizing) traces of bilinear maps and Lie (triple) isomorphisms on upper triangular matrix algebras.
scientific article; zbMATH DE number 6521174
scientific article