Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals

For a locally compact groupG, let XG be one of the following introverted subspaces of VN(G): UCB( G), the C∗-algebra of uniformly continuous functionals on A(G); W( G), the space of weakly almost periodic functionals on A(G); or M∗ρ(G), the C∗-algebra generated by the left regular representation on the measure algebra of G.We discuss the extension of homomorphisms of (reduced) Fourier–Stieltjes algebras on G and H to cb-norm preserving, weak∗–weak∗-continuous homomorphisms of X∗G into X∗H , where (XG,XH ) is one of the pairs (UCB( G),UCB( H)), (W( G),W( H)), or (M∗ρ(G),M∗ρ(H)). When G is amenable, these extensions are characterized in terms of piecewise affine maps. © 2008 Elsevier Inc. All rights reserved.