The algebra generated by idempotents in a Fourier-Stieltjes algebra

TL;DR

This paper investigates the algebra generated by idempotents in the Fourier-Stieltjes algebra of a locally compact group, introducing the idempotent compactification and characterizing isomorphisms between such algebras.

Contribution

It introduces the idempotent compactification of a group and characterizes when two such algebras are completely isometrically isomorphic.

Findings

The algebra B_I(G) is a regular Banach algebra with computable spectrum G^I.

B_I(G) is isometrically isomorphic to B_I(H) iff G/G_e= H/H_e.

Examples illustrate the theoretical results.

Abstract

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.