Operator amenability of ultrapowers of the Fourier algebra

Abstract

It has been shown by Matthew Daws that the group algebra of a discrete group is never ultra-amenable. We explore the weak analogue to this statement and demonstrate that if any commutative group algebra is ultra-weakly amenable, then the underlying group must necessarily be discrete. By showing that ultrapowers of complete maximal operator spaces are themselves maximal, we are able to demonstrate that the assumption of ultra-operator amenability of the Fourier algebra A(G) forces G to be discrete. By considering a wide class of discrete groups, we find sufficient evidence to make reasonable the conjecture that such a property may well force G to be finite. We conclude with consideration of another weak analogue, showing that ultra-weak operator amenability of A(G) already forces G to be discrete.