2003/10/18 by Anthony Narkawicz, Narkawicz, Anthony
Mathematics · #20J06 (Primary) 46L05 #43A15 (Secondary) #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA) #math.OA #msc:20J06 #msc:43A15 #msc:46L05
paper · pdf · doi:10.48550/arxiv.math/0310296
19 pages, 0 figures, sumitted to "Colloquium Mathematicum."
arxiv created 2003/10/18 · openalex publication_date 2003/10/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper characterizes the first cohomology group H1(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 H1(G,CG) imbeds in H1(G,M) whenever CG is contained in M which is in turn contained in Lp(G) for some p. This shows immediately that if H1(G,M)=0, then G has exactly 1 end. Secondly, it is shown that H1(G,M) is not Hausdorff if and only if there exist fi in M with norm 1 (||fi||=1) for all i with the property that ||gfi-fi||->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 H1(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.