Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Amalgamated duplication of the Banach algebra $\bf{\frak A}$ along a
  ${\frak A}$-bimodule ${\mathcal A}$   Let ${\mathcal A}$ and ${\frak A}$ be Banach algebras such that ${\mathcal
A}$ is a Banach ${\frak A}$-bimodule with compatible actions. We define the
product ${\cal A}\rtimes{\frak A}$, which is a strongly splitting Banach
algebra extension of ${\frak A}$ by $\cal A$. After characterization of the
multiplier algebra, topological centre, (maximal) ideals and spectrum of ${\cal
A}\rtimes{\frak A}$, we restrict our investigation to the study of
semisimplicity, regularity, Arens regularity of ${\cal A}\rtimes{\frak A}$ in
relation to that of the algebras $\cal A$, $\frak A$ and the action of $\frak
A$ on $\cal A$. We also compute the first cohomology group $H^1{(}{\cal
A}\rtimes{\frak A},({\cal A}\rtimes{\frak A})^{(n)}{)}$ for all $n\in {\Bbb
N}\cup\{0\}$ as well as the first-order cyclic cohomology group
$H_\lambda^1{(}{\cal A}\rtimes{\frak A},({\cal A}\rtimes{\frak A})^{(1)}{)}$,
where $({\cal A}\rtimes{\frak A})^{(n)}$ is the n-th dual space of ${\cal
A}\rtimes{\frak A}$ when $n\in{\Bbb N}$ and ${\cal A}\rtimes{\frak A}$ itself
when $n=0$. These results are not only of interest in their own right, but also
they pave the way for obtaining some new results for Lau products and module
extensions of Banach algebras as well as triangular Banach algebra. Finally,
special attention is devoted to the cyclic and $n$-weak amenability of ${\cal
A}\rtimes{\frak A}$.

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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ex
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The categories of this paper are: eess