Extensions and isomorphisms for the generalized Fourier algebras of a locally compact group

It is shown that for amenable groups, all finite-dimensional extensions of Ap(G) algebras split strongly. Furthermore, each extension of Ap(G) which splits algebraically also splits strongly. We also show that if G is an almost connected locally compact group, or a subgroup of GLn(V) (V being a finite-dimensional vector space), and if for a fixed p∈(1,∞), all finite-dimensional singular extensions of Ap(G) split strongly, then G is amenable. Continuous order isomorphisms for the pointwise order of Ap(G) algebras, are characterized as weighted composition maps. Similarly, order isomorphisms for the pointwise order of Bp(G) algebras, are characterized as ∗-algebra isomorphisms followed by multiplication by an invertible positive multiplier. In addition, it is shown that for amenable groups, an order isomorphism for the pointwise order between Ap(G) algebras that preserve cozero sets is necessarily continuous, and hence induces an algebra isomorphism.