Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

Abstract

Abstract. Let G be a generalized matrix algebra over a commutative ring R and Z(G) be the center of G. Suppose that q : G × G −→ G is an R-bilinear mapping and Tq : G −→ G is the trace of q. We describe the form of Tq satisfying the condition [Tq(G), G] ∈ Z(G) for all G ∈ G. The question of when Tq has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of G to be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article.

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

“…Benkovič and Eremita in [4] directly applied the theory of commuting traces to the study of Lie isomorphisms on a triangular algebra has the standard form (♠). These results are further extended to the case of generalized matrix algebras, see [43], [62], [63]. Lie triple isomorphisms between rings and between (non-)self-adjoint operator algebras have received a fair amount of attention and have also been intensively studied, see [2], [13], [14], [15], [16], [20], [21], [32], [46], [48], [51], [52], [53], [56], [55], [57], [58], [59], [65], [66].…”

“…That is, m −1 (0) is a Jordan ideal of Z(G). However, by [43], Lemma 4.1 it follows that m −1 (0) = 0.…”

“…It should be remarked that Corollary 4.4 and Corollary 4.5 removes the assumption that R is a domain in [43], Corollary 3.22 and Corollary 3.23. 2) be the upper triangular matrix algebra over R. Suppose that q : T n (R) × T n (R) → T n (R) is an R-bilinear mapping.…”

“…Simultaneously, some researchers engage in characterizing k-commuting mappings of generalized matrix algebras and those of unital algebras with notrivial idempotents, see [22], [40], [41], [52]. Motivated by Benkovič and Eremita's work (see [4]), the present authors (see [43], [62], [63]) undertook the study of centralizing (or commuting) traces of bilinear mappings on triangular algebras and generalized matrix algebras. It is shown that every centralizing trace of an arbitrary bilinear mapping on a class of generalized matrix algebra has the so-called proper from.…”