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

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