Saworotnow, Parfeny P. Another version of “Exotic characterization of a commutative \(H^*\)-algebra”. (English) Zbl 1092.46040 Int. J. Math. Math. Sci. 2005, No. 3, 483-486 (2005). The author characterizes commutative \(H^*\)-algebras without assuming commutativity and a Hilbert space structure. Theorem 1. Let \(A\) be a semisimple Banach algebra satisfying (i) for every closed right ideal \(R\) there is a closed left ideal \(L\) such that \(R\cap L=\{0\}\) and \(R+L=A\), (ii) if \(ab=ba\) then \(\| a+b\| ^2= \| a\| ^2+\| b\| ^2\). Then \(A\) is a commutative proper \(H^*\)-algebra. Reviewer: Wiesław Tadeusz Żelazko (Warszawa) MSC: 46K15 Hilbert algebras Keywords:\(H^*\)-algebras PDFBibTeX XMLCite \textit{P. P. Saworotnow}, Int. J. Math. Math. Sci. 2005, No. 3, 483--486 (2005; Zbl 1092.46040) Full Text: DOI EuDML