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| Paper IPM / M / 18135 |
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| Abstract: | |
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A well-known theorem of Bochner, generalized by Eberlein, characterizes the image
of Fourier-Stieltjes transform of the measure algebra of a locally compact abelian
group. With the approach that arose from this theorem, the concept of BSE-Algebras
was created by Takahasi and Hatori to study the Gelfand image of commutative
Banach algebras. Let A and B be two semisimple commutative Banach algebras. We
denote by A~B a Banach algebra obtained by the completion of A B w.r.t. a
submultiplicative cross-norm which dominates the injective norm. In this article, we
provide necessary and sufficient conditions for A~B to be a BSE-algebra. We give
an extension of Bochner-Eberlein theorem. We also provide another proof for the
well-known fact that the injective tensor product of A and B is isomorphic to a C-
algebra if and only if A and B are so. For a discrete space X, it is proved that the
space C0�ðX; A�?, the set of all continuous functions f : X ! A that vanish at infinity,
is a BSE-algebra if and only if A is a BSE-algebra.
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