P. Eymard equipped B(G),
the Fourier-Stieltjes algebra of a locally compact group G, with a norm
by considering it the dual Banach space of bounded linear functional on
another Banach space, namely the universal C∗-algebra, C∗(G). We show
that B(G) can be given the exact same norm if it is considered as a Banach
subalgebra of 𝒟(C∗(G)), the Banach algebra of completely bounded maps of
C∗(G) into itself equipped with the completely bounded norm. We show
here how the latter approach leads to a duality theory for finite (and, more
generally, discrete) groups which is not available if one restricts attention
to the “linear functional” [as opposed to the “completely bounded map”]
approach.
