Centralizers for subsets of normed algebras (Q2703040)

Let \(A\) be a normed algebra with identity \(e\), \(\mathbf{N}_A\) the set of nilpotent elements of \(A\), \(Z_A\) the center of \(A\), \(G_A\) the set of all invertible elements of \(A\), \(H\subset G_A\) and \({\mathbf{F}}(H) = \{x\in A: \sup \{\|cxc^{-1}\|:s\in H\}<\infty\}.\) NEWLINENEWLINENEWLINEIt is shown that NEWLINENEWLINENEWLINE1) \({\mathbf{F}}(H)\) is the centralizer of the set \(H = \{ e-v:v\in \mathbf{N}_A\}\), NEWLINENEWLINENEWLINE2) \(\mathbf{F}(H)=A\) or the complement of \({\mathbf{F}}(H)\) is dense in \(A\) and NEWLINENEWLINENEWLINE3) \({\mathbf{F}}(H)\) is a closed subset of \(A\) if and only if \(\|x\|\) and \(M(x) =\sup \{\|cxc^{-1}\|: c\in H\}\) are equivalent norms on \({\mathbf{F}}(H)\). Moreover, \(\mathbf{F}(G_A) = Z_A\) if \(A\) is a normed \(Q\)-algebra (in case of Banach algebras this result follows from arguments due to Le Page) and the centralizer of the socle \(\mathbf{S}\) of a normed algebra \(B\) over complex numbers which has no non-zero nilpotent one-sided ideals is the same as the centralizer of the set \(\mathbf{P}\) of minimal idempotents of \(B\) (that is such idempotents \(i\) of \(B\) that minimal right ideal of \(B\) has the form \(iB\)).