Let H be an ultraspherical hypergroup associated to a locally compact group G
and let A(H) be the Fourier algebra of H. For a left Banach A(H)-submodule X
of VN(H), define QX to be the norm closure of the linear span of the set {u
f : u ?A(H), f ? X} in BA(H)(A(H),X
A locally compact group G is discrete if and only if the Fourier algebra A(G) has a non-zero (weakly) compact multiplier. We partially extend this result to the setting of ultraspherical hypergroups. Let H be an ultraspherical hypergroup and let A(H) denote the corresponding Fourier algebra. We will give several characterizations of discreteness of H in the terms of the algebraic properties of A(H). We also study Arens regularity of closed ideals of A(H).
Let [Formula: see text] be a locally compact quantum group. Then the space [Formula: see text] of trace class operators on [Formula: see text] is a Banach algebra with the convolution induced by the right fundamental unitary of [Formula: see text]. We show that properties of [Formula: see text] such as amenability, triviality and compactness are equivalent to the existence of left or right invariant means on the convolution Banach algebra [Formula: see text]. We also investigate the relation between the existence of certain (weakly) compact right and left multipliers of [Formula: see text] and some properties of [Formula: see text].
Let H be an ultraspherical hypergroup and let
$A(H)$
be the Fourier algebra associated with
$H.$
In this paper, we study the dual and the double dual of
$A(H).$
We prove among other things that the subspace of all uniformly continuous functionals on
$A(H)$
forms a
$C^*$
-algebra. We also prove that the double dual
$A(H)^{\ast \ast }$
is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of
$A(H)^{\ast \ast }.$
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