Author: Anthony Narkawicz
Institution: Virginia Tech
Short HTML abstract:

This paper characterizes the first cohomology group H^1(G,M) where M is
a Banach space (with norm ||.||) that is also a left CG-module such that
the elements of G act on M as continuous complex-linear transformations.
Of particular interest is the topology on this group induced by the norm
topology on M. The first result is that H^1(G,CG) imbeds in H^1(G,M)
whenever CG is contained in M which is in turn contained in L^p(G) for
some p. This shows immediately that if H^1(G,M)=0, then G has exactly 1
end. Secondly, it is shown that H^1(G,M) is not Hausdorff if and only if
there exist f_i in M with norm 1 (||f_i||=1) for all i with the property
that ||gf_i-f_i||->0 as i goes to infinity for every g in G. This is
then used to show that if ||.|| and M satisfy certain properties and if
G satisfies a "strong Folner condition," then H^1(G,M) is not Hausdorff.
The second half of the paper gives several applications of these
theorems focusing on the free abelian group on n generators. Of
particular interest is the case that M is the reduced group C^* algebra
of G.

Status: Submitted to ^Colloquium Mathematicum.^