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 $A\times_\theta B$.
commutative Banach algebra, splitting extension, Gelfand spectrum, set of synthesis, weak spectral set, multiplier algebra, BSE-algebra, BSE-function
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.