## Hausdorffness and vanishing of the first
cohomology group H^{1}(*G*,*M*)

### Anthony Narkawicz

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 H^{1}(*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
H^{1}(*G*,*M*) imply
about *G*? Secondly, what are necessary and sufficient
conditions on *G*
and *M* so that H^{1}(*G*,*M*) is a Hausdorff space?
The results are as follows. If H^{1}(*G*,*M*)
vanishes and *M* is contained in L^{p}(*G*)
for some *p*, then *G* has exactly 1 end.
Also, H^{1}(*G*,*M*) is not Hausdorff if and
only if there is a sequence *f*_{i} in *M*
with norm 1 (||*f*_{i}|| = 1) for all
*i* such that ||(*g*-1)*f*_{i}|| --> 0
as *i* --> infinity for
every *g* in *G*. These results
give useful tools for studying the dimension of groups of the form
H^{1}(*G*,*M*). Specifically, we attempt to
examine the groups
H^{1}(**Z**^{n},C^{*}_{red}(**Z**^{n})).
The relationship between the C^{*}-algebra
C^{*}_{red}(*G*) and a specific algebra of
continuous functions allows us to examine these groups very closely.