Brief communications

On the Radical for Some Class of Banach Algebras

H. S. Mustafaev

Yuzuncu Yil University

Abstract: Let $A$ be a complex Banach algebra. It is well known that the second dual $A^{**}$ of $A$ can be equipped with a multiplication that extends the original multiplication on $A$ and makes $A^{**}$ a Banach algebra. We show that $\operatorname{Rad}(A)={}^\bot(A^*\cdot A)$ and $\operatorname{Rad}(A^{**})=(A^*\cdot A)^\bot$ for some classes of Banach algebras $A$ with scattered structure space. Some applications of these results are given.

Keywords: Banach algebra, group algebra, radical, homomorphism, spectrum.

UDC: 517.98

Received: 28.06.2005

DOI: 10.4213/faa2870

 English version: Functional Analysis and Its Applications, 2007, 41:3, 241–244

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