Hausdorffness and vanishing of the first
cohomology group H1(G,M)
Bekka and Valette (among others) have showed us that there are very
deep connections between the group G,
the Banach space M (on which G
acts continuously and linearly), and topological properties of the
cohomology group H1(G,M).
Applications of this study have been used
in electrical engineering. This paper answers the following two
questions. First, what does the vanishing of the group
about G? Secondly, what are necessary and sufficient
conditions on G
and M so that H1(G,M) is a Hausdorff space?
The results are as follows. If H1(G,M)
vanishes and M is contained in Lp(G)
for some p, then G has exactly 1 end.
Also, H1(G,M) is not Hausdorff if and
only if there is a sequence fi in M
with norm 1 (||fi|| = 1) for all
i such that ||(g-1)fi|| --> 0
as i --> infinity for
every g in G. These results
give useful tools for studying the dimension of groups of the form
H1(G,M). Specifically, we attempt to
examine the groups
The relationship between the C*-algebra
C*red(G) and a specific algebra of
continuous functions allows us to examine these groups very closely.