Conditions for a function to be a centralizer on an \(H^*\)-algebra (Q1911347)

Let \(A\) be a \(H^*\)-algebra. A function \(S:A\to A\) is a centralizer, if \(S(xy)= (Sx)y\), \(\forall x,y\in A\). Denote by \(R(A)\) the \(C^*\)-algebra of centralizers and by \(C(A)\) its closed subalgebra generated by the set of all left multipliers \(\{L_a: a\in A\), \(L_a x=ax\), \(\forall x\in A\}\). Main Theorem. Let \(F:A\to A\) be a function such that \(CFC\in R(A)\) whenever \(C\in C(A)\). Then \(F\in R(A)\).