Basic theory of Fourier and Fourier-Stieltjes algebras

In this chapter the Fourier and Fourier-Stieltjes algebras, ApGq and BpGq, associated to a locally compact group G, are introduced and studied almost to the extent of Eymard’s fundamental paper [73]. In particular, BpGq is identified as the Banach space dual of the group C ̊-algebra C ̊pGq and a fair number of basic functorial properties are presented. Similarly, for the Fourier algebra ApGq, the elements of which are shown to be precisely the convolution products of Lfunctions on G. Given a commutative Banach algebra A, immediate problems arising are to determine the spectrum (or Gelfand space) of A and to check whether the range of the Gelfand transform is a regular function algebra. As we show in Section 2.3, the spectrum σpApGqq turns out to be homeomorphic to G and the Gelfand homomorphism is then nothing but the identity mapping. Moreover, ApGq is regular. We next identify, following Eymard’s seminal paper [73], the Banach space dual of ApGq as the von Neumann subalgebra V NpGq of BpL2pGqq generated by the left regular representation of G. The fact that ApGq is the predual of a von Neumann algebra will prove to be of great importance. For instance, it allows us to equip ApGq with a natural operator space structure and employing the theory of operator spaces has led to significant progress, as will be shown in Chapters 4 and 6. In Section 2.5 the very important notion of support of an operator in V NpGq is introduced and several properties of these supports, which are extremely useful later on, are shown. An immediate consequence of one of the results about the support is that singletons in G are sets of synthesis for ApGq. Let H be a closed subgroup of the locally compact group G. A challenging problem is whether functions in ApHq and BpHq extend to functions in ApGq and BpGq, respectively. For the Fourier algebras there is a very satisfactory solution to the effect that every function in ApHq extends to a function in ApGq of the same norm (Section 2.6). For Fourier-Stieltjes algebras, however, the problem is considerably more difficult and its investigation will cover a major portion of Chapter 7. If A is a nonunital Banach algebra, then often the existence of a bounded approximate identity in A proves useful. In Section 2.7 we present Leptin’s theorem [191] saying that ApGq has a bounded approximate identity precisely when the group G is amenable. The proof uses several different characterizations of amenability of a locally compact group. The notion of Fourier algebra has been generalized by Arsac [5]. He associated to any unitary representation π of G a closed subspace AπpGq of BpGq and studied these spaces extensively. When π is the left regular representation of G, then