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A characterization of $$B^*$$-algebras. (English) Zbl 0898.46050
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. F. F. Bonsall and J. Duncan, “Numerical ranges of operators on normed spaces and of elements of normed algebras”, Cambridge University Press (1971; Zbl 0207.44802)].
##### MSC:
 46K05 General theory of topological algebras with involution 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces
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