The Fourier-Stieltjes and Fourier algebras for locally compact groupoids

摘要

The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locallycompact group G, first studied by P. Eymard, have played an important role inharmonic analysis and in the study of operator algebras generated by G.Recently, there has been interest in developing versions of these algebras forlocally compact groupoids, justification being that, just as in the group case,the algebras should play a useful role in the study of groupoid operatoralgebras. Versions of these algebras for the locally compact groupoid caseappear in three related theories: (1) a measured groupoid theory (J. Renault),(2) a Borel theory (A. Ramsay and M. Walter), and (3) a continuous theory (A.Paterson). The present paper is expository in character. For motivationalreasons, it starts with a description of the theory of B(G), A(G) in thelocally compact group case, before discussing these three realted theories.Some open questions are also raised.