Norm one idempotent cb-multipliers with applications to the fourier algebra in the cb-multiplier norm

Abstract

For a locally compact group G, let A(G) be its Fourier algebra, let M cbA(G) denote the completely bounded multipliers of A(G), and let AMcb (G) stand for the closure of A(G) in M cbA(G). We characterize the norm one idempotents in M cbA(G): the indicator function of a set E ⊂ G is a norm one idempotent in M cbA(G) if and only if E is a coset of an open subgroup of G. As applications, we describe the closed ideals of AMcb (G) with an approximate identity bounded by 1, and we characterize those G for which AMcb (G) is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.) © Canadian Mathematical Society 2011.