A result concerning two-sided centralizers on algebras with involution (Q2869936)

Let \(R\) be an associative ring. An additive map \(T:R\rightarrow R\) is called a left centralizer when \(T(xy)=T(x)y\) holds for all pairs \(x,y\in R\). The definition of a right centralizer should be self-explanatory. We call a map \(T:R\rightarrow R\) a two-sided centralizer when \(T\) is both a left and a right centralizer. In the case that \(T:R\rightarrow R\) is a two-sided centralizer, where \(R\) is a semiprime ring with extended centroid \(C\), then it is known that \(T\) is of the form \(T(x)=\lambda x\) for all \(x\in R\), where \(\lambda \in C\) is some fixed element.NEWLINENEWLINELet now \(X\) be a real or complex Banach space and let \(\mathcal{L}(X)\) and \(\mathcal{F}(X)\) denote the algebra of all bounded linear operators on \(X\) and the ideal of all finite rank operators in \(\mathcal{L}(X)\), respectively. An algebra \(\mathcal{A}(X)\subset \) \(\mathcal{L}(X)\) is said to be standard when \(\mathcal{F}(X)\subset \mathcal{A}(X)\).NEWLINENEWLINE The authors prove the following result: Let \(X\) be a complex Hilbert space and \(\mathcal{A}(X)\) a standard operator algebra which is closed under the adjoint operation. Let \(A^{\ast }\) denote the adjoint operator of \(A\in \mathcal{L}(X)\). Suppose that \(T:\mathcal{A}(X)\rightarrow \mathcal{L}(X)\) is a linear map satisfying NEWLINE\[NEWLINE 3T(AA^{\ast }A)=T(A)A^{\ast }A+AT(A^{\ast })A+AA^{\ast }T(A) NEWLINE\]NEWLINE for all \(A\in \mathcal{A}(X)\). Then \(T\) is of the form \(T(A)=\lambda A\) for all \(A\in \) \(\mathcal{A}(X)\), where \(\lambda \) is a fixed complex number.NEWLINENEWLINENote that any standard operator algebra is prime. Given an integer \(n\geq 2\), a ring is said to be \(n\)-torsion free if, for \(x\in R\), \(nx=0\) implies \(x=0\).NEWLINENEWLINE To place their result in a broader perspective, the authors conclude the paper with the following conjecture: Let \(R\) be a semiprime ring, equipped with an involution, having suitable torsion restrictions, and let \(T:R\rightarrow R\) be an additive map satisfying the relation NEWLINE\[NEWLINE3T(xx^{\ast }x)=T(x)x^{\ast }x+xT(x^{\ast })x+xx^{\ast }T(x)NEWLINE\]NEWLINE for all \(x\in R\). In this case, \(T\) is a two-sided centralizer.