Weak*-closedness of subspaces of Fourier-Stieltjes algebras and weak*-continuity of the restriction map

Abstract

Let G be a locally compact group and B(G) the Fourier-Stieltjes algebra of G. We study the problem of how weak8-closedness of some translation invariant subspaces of B(G) is related to the structure of G. Moreover, we prove that for a closed subgroup H of G, the restriction map from B(G) to B(H) is weak8-continuous only when H is open in G. INTRODUCTION Let G be a locally compact group, and let B(G) be the Fourier-Stieltjes algebra of G as defined by Eymard [8]. Recall that B(G) is the linear span of all continuous positive definite functions on G and can be identified with the Banach space dual of C*(G), the group C*-algebra of G. The space B(G), with the xlorm as dual of C*(G), is a commutative Banach *-algebra with pointwise multiplication and complex conjugation. The Fourier algebra A(G) of G is the closed *-subalgebra of B(G) generated by the functions in B(G) with compact support. In particular, A(G) is contained in Co(G), the algebra of complex valued continuous functions on G vanishing at infinity. As is well known A(G) is weak*-dense in B(G) if and only if G is amenable. In [3] translation invariant *-subalgebras A of B(G) were studied, and it was shown that if such A is weak*-closed and point separating, then it must contain A(G). However, apart from this, very little seems to be known about weak*-closed subspaces of B(G). The first purpose of this paper is to investigate the relation between weak*closedness of certain interesting norm-closed translation invariant subspaces of B(G) and the structure of G. Secondly, we solve the problem of whexl, for a closed subgroup H of G, the restriction map from B(G) to B(H) is weak*-continuous. A brief outline of the paper is as follows. In Section 2 we estsablish for almost connected locally compact groups G the relation between weclk*-closed.ness of Bo(G) = B(G) n Co(G) in E3(G) and the structure of G (Theorem 2.10). The key result is that for a connected Lie group G, Bo(G) is weak*-closed in B(G) if and only if G is a reductive Lie group with compact centre and Kazhdan's property (T). Received by the editors December 1S, 1995. 199l Mathematics Subject Classification. Primary 22D10, 43A30. Work supported by NATO collaborative research grant CRG 940184. (g)1997 American Mathematical Society