Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications

Let G be a compact group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group and A(G × H) be the Fourier algebra of G × H. We prove that the group von Neumann algebra VN(G × H) = A(G × H)^* has the weak uniform A(G × H)^** factorization property of level b(G). As a corollary we show that A(G × H) is strongly Arens irregular, and the topological centre of UC ₂(G × H)^* is equal to the Fourier–Stieltjes algebra B(G × H).