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

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

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 et convoluteurs de " , Sem. Bourbaki 1969/70, Exp. 367 , Lecture Notes in Math. , 180 , Springer (1971) pp. 364–381
[a3]	A. Figà-Talamanca, "Translation invariant operators in " Duke Math. J. , 32 (1965) pp. 495–501
[a4]	C. Herz, "The theory of -spaces" Trans. Amer. Math. Soc. , 154 (1971) pp. 69–82
[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 et convoluteurs de " Thèse, Univ. Paris-Sud (1971)
[a8]	N. Lohoué, "Estimations des coefficients de représentations et opérateurs de convolution" Adv. Math. , 38 (1980) pp. 178–221
[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)