Fourier and related algebras occur naturally in the harmonic analysis of locally compact groups (cf. also Harmonic analysis, abstract). They play an important role in the duality theories of these groups.
The Fourier–Stieltjes algebra and the Fourier algebra of a locally compact group were introduced by P. Eymard in 1964 in [a2] as respective replacements, in the case when is not Abelian, of the measure algebra of finite measures on and of the convolution algebra of integrable functions on , where is the character group of the Abelian group (cf. also Character of a group). Indeed, if is a locally compact Abelian group, the Fourier–Stieltjes transform of a finite measure on is the function on defined by
and the space of these functions is an algebra under pointwise multiplication, which is isomorphic to the measure algebra (cf. also Algebra of measures). Restricted to , viewed as a subspace of , the Fourier–Stieltjes transform is the Fourier transform on and its image is, by definition, the Fourier algebra . The generalized Bochner theorem states that a measurable function on is equal, almost everywhere, to the Fourier–Stieltjes transform of a non-negative finite measure on if and only if it is positive definite. Thus, can be defined as the linear span of the set of continuous positive-definite functions on . This definition is still valid when is not Abelian.
Let be a locally compact group. The elements of are exactly the matrix elements of the unitary representations of : if and only if there exist a unitary representation of in a Hilbert space and vectors such that
The elements of are the matrix elements . Because of the existence of the tensor product of unitary representations, is an algebra under pointwise multiplication. The norm defined as , where the infimum runs over all the representations , makes it into a Banach algebra. The Fourier algebra can be defined as the norm closure of the set of elements of with compact support. It consists exactly of the matrix elements of the regular representation on ; equivalently, its elements are the functions of the form , where and . It is a closed ideal in .
The most visible role of and with respect to duality is that is the dual of the -algebra of the group and is the pre-dual of the von Neumann algebra of its regular representation. The pairing is given by , where and . The comparison with a similar result for and , namely is the dual of the Banach space of continuous functions on vanishing at infinity and is the pre-dual of , leads to the theory of Kac algebras and a generalized Pontryagin theorem (see below). Two complementary results suggest to view as a dual object of the group ; namely, Eymard's theorem states that the topological space underlying can be recovered as the spectrum of the Fourier algebra and Walter's theorem states that a locally compact group is determined, up to topological isomorphism, by the normed algebra , or by ; the second result should be compared with theorems of J.G. Wendel and of B.E. Johnson, which establish the same property for the normed algebras and , respectively; see [a5] for a survey of these results.
The multipliers of the Fourier algebra reflect interesting properties of the group (cf. also Multiplier theory). First, the unit (i.e., the constant function ) belongs to if and only if the group is compact. Leptin's theorem (see [a3]) asserts that has a bounded approximate unit if and only if the group is amenable. A multiplier of the Fourier algebra is a function on such that the operator of multiplication by maps into itself. These multipliers form a Banach algebra under pointwise multiplication and the norm , denoted by . The transposed operator is a bounded linear mapping from into itself. One says that the multiplier is completely bounded if the mapping is completely bounded, meaning that is finite, where the supremum runs over all integers and is the identity operator from the -algebra of complex -matrices into itself. For example, the matrix elements of uniformly bounded representations of are such multipliers. The completely bounded multipliers form also a Banach algebra under pointwise multiplication and the norm , denoted by . There is an alternative description of completely bounded multipliers as Schur multipliers, initiated by M.G. Krein [a1] (cf. also Schur multiplicator) and related to the metric theory of Grothendieck's topological tensor products. Given a measure space , a measurable function on is called a Schur multiplier if pointwise, or Schur, multiplication of kernels by defines a bounded linear mapping from the space of bounded operators on into itself; its Schur norm is then . The Schur multipliers form a Banach algebra under pointwise multiplication. According to the Bożekjko–Fendler theorem, a continuous function on is a completely bounded multiplier of if and only if the function on defined by is a Schur multiplier; moreover, the Schur norm and the completely bounded norms are equal. The continuous right-invariant Schur multipliers on are called Herz–Schur multipliers; they form a subalgebra of , denoted by , which is isometrically isomorphic to . The following norm-decreasing inclusions hold:
When is amenable, these inclusions are equalities; on the other hand, according to Losert's theorem, if , then is amenable; the equality gives the same conclusion, at least when is discrete (M. Bożekjko and J. Wysoczanski). A locally compact group is called weakly amenable if there exists an approximate unit in which is bounded in the norm of . The Haagerup constant is defined as the infimum of these bounds over all -bounded approximate units. Free groups and, more generally simple Lie groups with finite centre and real rank one and their lattices, are weakly amenable and their Haagerup constants have been computed in [a4]. For example, for or and for (). Groups of real rank greater than one are not weakly amenable. See also [a4] for references to completely bounded multipliers.
An -version of the Fourier algebra has been developed for (see [a3] for a detailed account and references). Let be given by . The Herz–Figa–Talamanca algebra is the space of functions on of the form
with pointwise multiplication. It is the quotient of the projective tensor product with respect to the mapping defined by . Again, the amenability of is equivalent to the existence of a bounded approximate unit in . Just as above, one defines for a measure space the Schur multiplier algebra as the space of functions on such that the Schur multiplication sends the space of bounded operators on (or, equivalently, its pre-dual ) into itself, and the Herz–Schur multiplier algebra as the space of continuous functions on such that belongs to ; the product is pointwise multiplication. Since the mapping from onto intertwines and , a Herz–Schur multiplier is a multiplier of and the inclusion decreases the norm. It is an equality if is amenable. These algebras are also related to convolution operators. In particular, the dual of is the weak closure of in , where acts by left convolution. Banach algebra properties of the Fourier algebras and have been much studied; see [a3] for a bibliography up to 1984.
Fourier algebras are natural objects in the -algebraic theory of quantum groups and groupoids. In particular, Kac algebras (see [a5]) provide a symmetric framework for duality, which extends the classical Pontryagin duality theory for locally compact Abelian groups. Each Kac algebra has a dual Kac algebra and the dual of is isomorphic to . The Fourier algebra is the pre-dual of and the Fourier–Stieltjes algebra is the dual of the enveloping -algebra of . If is the Kac algebra of a locally compact group , then the dual Kac algebra is and the corresponding Fourier and Fourier–Stieltjes algebras are: , , and .
|[a1]||M. Krein, "Hermitian-positive kernels on homogeneous spaces I–II" Amer. Math. Soc. Transl. (2) , 34 (1963) pp. 69–164 Ukrain. Mat. Z. , 1–2 (1949/50) pp. 64–98; 10–59|
|[a2]||P. Eymard, "L'algèbre de Fourier d'un groupe localement compact" Bull. Soc. Math. France , 92 (1964) pp. 181–236|
|[a3]||J.-P. Pier, "Amenable locally compact groups" , Wiley–Interscience (1984)|
|[a4]||M. Cowling, U. Haagerup, "Completely bounded multipliers of the Fourier algebra of a simple Lie groups of real rank one" Invent. Math. , 96 : 3 (1989) pp. 507–549|
|[a5]||M. Enock, J.-M. Schwartz, "Kac algebras and duality of locally compact groups" , Springer (1992)|
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