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

For a locally compact group G, let XG be one of the following introverted subspaces of VN(G): , the C∗-algebra of uniformly continuous functionals on A(G); , the space of weakly almost periodic functionals on A(G); or , 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 into , where (XG,XH) is one of the pairs , , or . When G is amenable, these extensions are characterized in terms of piecewise affine maps.