Centralisers on rings and algebras

Abstract

In this paper we investigate identities related to centralisers in rings and algebras. We prove, for example, the following result. Let A be a semisimple H* -algebra and let T: A → A be an additive mapping satisfying the relation T(xm+n+1) = xmT(x)xn for all x ∈ A and some integers m ≥ 1, n ≥ 1. In this case T is a left and a right centraliser.

On a Functional Equation Related to Two-Sided Centralizers

“…Lately several authors investigated centralizers on rings and algebras. Some of the results can be found in [3,11,12,13,14,15,16].…”

On (m,n)-Jordan centralizers in rings and algebras

“…4.in [6] that D 1 (x) = D 3 (x) = 0 for all x ∈ R, whence it follows that D 2 (x) = 0 because of (12). In other words, we have…”

“…Let us recall that a semisimple H * −algebra is a complex semisimple Banach * −algebra whose norm is a Hilbert space norm such that (x, yz * ) = (xz, y) = (z, x * y) is fulfilled for all x, y, z ∈ A (see [1]). For results concerning centralizers in rings and algebras we refer to [10][11][12][13] where further references can be found. Let X be a real or complex Banach space and let L(X) and F (X) denote the algebra of all bounded linear operators on X and the ideal of all finite rank operators in L(X), respectively.…”

The characterization of generalized Jordan centralizers on algebras

“…Benkovic and Eremita ( [1]) proved that if R is a prime ring with Ch(R) = 0 or Ch(R) ≥ n, where n is a fixed positive integer and n ≥ 2, and φ is an additive mapping on R such that φ(A n ) = φ(A)A n−1 for any A ∈ R, then φ is a centralizer. Vukman and Kosi-Ulbl ( [17]) proved that if X is a Banach space over the field F, and A is a standard subalgebra of B(X) and φ : A → B(X) is an additive mapping such that φ(A m+n+1 ) = A m φ(A)A n for any A ∈ A, where m, n ∈ Z + and then φ is a centralizer. Qi etc.…”