×

Another version of “Exotic characterization of a commutative \(H^*\)-algebra”. (English) Zbl 1092.46040

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.

MSC:

46K15 Hilbert algebras
PDFBibTeX XMLCite
Full Text: DOI EuDML