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

Abstract

For a locally compact group G, let X G be one of the following introverted subspaces of VN ( G ) : UCB ( G ˆ ) , the C ∗ -algebra of uniformly continuous functionals on A ( G ) ; W ( G ˆ ) , 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 ( X G , X H ) is one of the pairs ( UCB ( G ˆ ) , UCB ( H ˆ ) ) , ( W ( G ˆ ) , W ( H ˆ ) ) , or ( M ρ ∗ ( G ) , M ρ ∗ ( H ) ) . When G is amenable, these extensions are characterized in terms of piecewise affine maps.