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# The Bochner-Schoenberg-Eberlein property and spectral synthesis for certain Banach algebra products

• Eberhard Kaniuth,
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## Abstract

Associated with two commutative Banach algebras $A$ and $B$ and a character $\theta$ of $B$ is a certain Banach algebra product $A\times_\theta B$, which is a splitting extension of $B$ by $A$. We investigate two topics for the algebra $A\times_\theta B$ in relation to the corresponding ones of $A$ and $B$. The first one is the Bochner-Schoenberg-Eberlein property and the algebra of Bochner-Schoenberg-Eberlein functions on the spectrum, whereas the second one concerns the wide range of spectral synthesis problems for $A\times_\theta B$.
 Keywords: commutative Banach algebra, splitting extension, Gelfand spectrum, set of synthesis, weak spectral set, multiplier algebra, BSE-algebra, BSE-function
 MSC Classifications: 46J10 - Banach algebras of continuous functions, function algebras [See also 46E25] 46J25 - Representations of commutative topological algebras 43A30 - Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A45 - Spectral synthesis on groups, semigroups, etc.