A characterization of \(B^*\)-algebras (Q1365266)

The following theorem is proved: if \(A\) is a \(B^*\)-algebra with a bounded approximate identity of norm less or equal to 1, then an element of \(A\) is hermitian if and only if it is selfadjoint. It follows that for Banach algebras with a bounded approximate identity of norm less than 1, the condition \(A=H(A)+iH(A)\) is a characterization of \(B^*\) algebras. This result is well-known [see e.g. \textit{F. F. Bonsall} and \textit{J. Duncan}, ``Numerical ranges of operators on normed spaces and of elements of normed algebras'', Cambridge University Press (1971; Zbl 0207.44802)].