The First Cohomology Group H^1(G,M)
| dc.creator | Narkawicz, Anthony | |
| dc.date | 2003-10-18 | |
| dc.date.accessioned | 2026-07-07T05:02:04Z | |
| dc.date.available | 2026-07-07T05:02:04Z | |
| dc.description | 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. | |
| dc.description | 19 pages, 0 figures, sumitted to "Colloquium Mathematicum." | |
| dc.identifier | https://arxiv.org/abs/math/0310296 | |
| dc.identifier | http://arxiv.org/abs/math/0310296 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68913 | |
| dc.subject | Operator Algebras | |
| dc.subject | 20J06 (Primary) 46L05, 43A15 (Secondary) | |
| dc.title | The First Cohomology Group H^1(G,M) | |
| dc.type | text |