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The algebra generated by idempotents in a Fourier-Stieltjes algebra

We study the closed algebra BI(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 GI, which we call the idempotent compactification of G. For any locally compact groups G and H, we show that BI(G) is completely isometrically isomorphic to BI(H) exactly when G/Ge= H/He, where Ge and He are the connected components of the identities. We compute some examples to illustrate our results.