An equation on operator algebras and semisimple ▫$H^\ast$▫-algebras
Vukman, Joso
In this paper we prove the following result: Let ▫$X$▫ be a Banach space over the real or complex field ▫$F$▫ and let $▫L(X)$▫ be the algebra of all bounded linear operators on $▫X$▫. Suppose there ...exists an additive mapping ▫$T:A(X) \to L(X)$▫, where ▫$A(X) \subset L(X)$▫ is a standard operator algebra. Suppose that ▫$T(A^3) = AT(A)A$▫ holds for all ▫$A \in A(X)$▫. In this case ▫$T$▫ is of the form ▫$T(A)=\lambda A$▫ for any ▫$A \in A(X)$▫ and some ▫$\lambda \in F$▫. This result is applied to semisimple ▫$H^\ast$▫-algebras.
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