Figà-Talamanca algebra
Let be a locally compact group,
and
. Consider the set
of all pairs
, with
a sequence in
and
a sequence in
such that
. Here,
is defined by
, where
is some left-invariant Haar measure on
. Let
denote the set of all
for which there is a pair
such that
, where
. The set
is a linear subspace of the
-vector space of all continuous complex-valued functions on
vanishing at infinity. For
one sets
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1) For the pointwise product on ,
is a Banach algebra. This algebra is called the Figà-Talamanca algebra of
. If
is Abelian,
is isometrically isomorphic to
, where
is the dual group of
. For
not necessarily Abelian,
is precisely the Fourier algebra of
.
2) If is amenable, then
. The algebra
is a useful tool for studying the
-convolution operators of
(see [a2], [a7], [a8]). For a function
on
and
one sets
. A continuous linear operator
on
is said to be a
-convolution operator of
if
for every
and every
. Let
be the set of all
-convolution operators of
. It is a closed subalgebra of the Banach algebra
of all continuous linear operators on
. For a complex bounded measure
on
(i.e.
) and a continuous complex-valued function
with compact support on
(
), the rule
defines a
-convolution operator
. Of course, for
,
denotes the set of all
with
-almost everywhere.
Even for one has
. Let
be the closure in
of
with respect to the ultraweak operator topology on
.
3) The dual of the Banach space
is canonically isometrically isomorphic to
. Also,
with the topology
is homeomorphic to
with the ultraweak operator topology on
. As a consequence, for
amenable or for
arbitrary but with
,
.
This duality between and
also permits one to develop (see [a1]) a kind of "non-commutative harmonic analysis on G" , where (for
Abelian)
replaces
and
replaces
. (Cf. also Harmonic analysis, abstract.)
Let . Then the support of
, denoted by
, is the set of all
for which for all open subsets
,
, of
with
and
there are
with
,
and
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If , then
. For
Abelian, let
be the canonical mapping from
onto
. Then
, where
, is an isometric isomorphism of the Banach algebra
onto
. Let
and
. Then
"belongs to the spectrum of u" (written as
) if
lies in the closure of the linear span of
in
, for the weak topology
. Let
; then
. For
not necessarily amenable and
,
if and only
is empty. This assertion is a non-commutative version of the Wiener theorem! Similarly, there is also a version of the Carleman–Kaplansky theorem: for
,
if and only there exist
such that
, where
denotes the Dirac measure in
(cf. also Dirac distribution). In fact, even for
or for
(but
) the situation is not classical!
The Banach space has been first introduced by A. Figà-Talamanca in [a3] for
Abelian or
non-Abelian but compact. For these classes of groups he obtained assertion 3) above. The statement for a general locally compact group is due to C.S. Herz [a5]. Assertion 1) is also due to Herz [a4].
The Banach algebra also satisfies the following properties:
a) Let be a closed subgroup of
. Then
. More precisely, for every
and for every
there is a
with
and
(see [a5]).
b) The Banach algebra has bounded approximate units (i.e. there is a
such that for every
and for every
there is a
with
and
) if and only if the locally compact group
is amenable (see [a5] and [a6] for
). This algebra is often called the Figà-Talamanca–Herz algebra.
See also Fourier algebra.
References
[a1] | A. Derighetti, "Quelques observations concernant les ensembles de Ditkin d'un groupe localement compact" Monatsh. Math. , 101 (1986) pp. 95–113 |
[a2] | P. Eymard, "Algèbres ![]() ![]() |
[a3] | A. Figà-Talamanca, "Translation invariant operators in ![]() |
[a4] | C. Herz, "The theory of ![]() |
[a5] | C. Herz, "Harmonic synthesis for subgroups" Ann. Inst. Fourier (Grenoble) , 23 : 3 (1973) pp. 91–123 |
[a6] | H. Leptin, "Sur l'algèbre de Fourier d'un groupe localement compact" C.R. Acad. Sci. Paris Sér. A , 266 (1968) pp. 1180–1182 |
[a7] | N. Lohoué, "Algèbres ![]() ![]() |
[a8] | N. Lohoué, "Estimations ![]() |
[a9] | J.R. McMullen, "Extensions of positive-definite functions" Memoirs Amer. Math. Soc. , 117 (1972) |
[a10] | J.-P. Pier, "Amenable locally compact groups" , Wiley (1984) |
Figà-Talamanca algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fig%C3%A0-Talamanca_algebra&oldid=50183