  We study the closed algebra B_I(G) generated by the idempotents in the
Fourier-Stieltjes algebra of a locally compact group G. We show that it is a
regular Banach algebra with computable spectrum G^I, which we call the
idempotent compactification of G. For any locally compact groups G and H, we
show that B_I(G) is completely isometrically isomorphic to B_I(H) exactly when
G/G_e= H/H_e, where G_e and H_e are the connected components of the identities.
We compute some examples to illustrate out results.
