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Abstract
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Let $\G$ and $\GG$ be locally compact groups and $\h$ be a Hilbert
space. We study the generalized bi-Fourier-Stieltjes algebra
$B^2(\G\times\GG,\BH)$ by the space of all bi-operator
coefficients of unitary representations of $\G$ and $\GG$:
$\phi\in B^2(\G\times\GG,\BH)$ if there exists unitary
representations $\pi_i:\textbf{G}_i\to \textbf{B}(\h_i)$ and a
diagram of bounded operators
\[\h\stackrel{V_2}{\To}\hhh\stackrel{T}{\To}\hh\stackrel{V^*_1}{\To}\h\]
such that $\phi(s,t)=V^*_1\pi_1(s) T\pi_2(t) V_2 $. We extend the
pointwise product on $B^2(\G\times\GG,\BH)$ under which the
generalized bi-Fourier-Stieltjes algebra is a completely
contractive commutative Banach algebra. We obtain a (tensor type)
decomposition by which a well-behaved diagram of its subalgebras
will be introduced
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