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.