On centralizers of reflexive algebras

On (m,n,l)-Jordan Centralizers of Some Algebras

“…It is easy to show that G 1 (GLG) ∨ G 2 (GLG) = G. So it follows from Case 1 that δ (1) is a centralizer on alg(GLG). (I − G)algL(I − G) is a von Neumann algebra and δ (2) is continuous, so by Corollary 2.7, δ (2) is a centralizer on alg((I − G)L(I − G)). Consequently, δ is a centralizer on algL.…”

“…Case 2: Suppose for every A ∈ algL. Therefore δ can be written as δ (1) ⊕ δ (2) , where δ (1) is a weak (m,n,l )-Jordan centralizer from alg(GLG) into itself and δ (2) is a weak (m,n,l )-Jordan centralizer from alg((I − G)L(I − G)) into itself. It is easy to show that G 1 (GLG) ∨ G 2 (GLG) = G. So it follows from Case 1 that δ (1) is a centralizer on alg(GLG).…”

“…Moreover, each Jordan centralizer is an (m, n)-Jordan centralizer and (1, 1)-Jordan centralizer satisfies the relation 2δ(x 2 ) = δ(x)x + xδ(x) for every x ∈ R. In [15], Vukman showed that (1, 1)-Jordan centralizer of a 2-torsion free semiprime ring R is a centralizer. In [2], Guo and Li studied (1, 1)-Jordan centralizer of some reflexive algebras. In [19], Vukman investigated (m, n)-Jordan centralizer and proved that for m ≥ 1 and n ≥ 1, every (m, n)-Jordan centralizer of a prime ring R with char(R) = 6mn(m + n) is a centralizer.…”