8 - Appendix
Published online by Cambridge University Press: 29 September 2009
Summary
Introduction
This appendix contains a section on bounded group cohomology and one on its relation with the ℓ1-group algebra cohomology. Though there is currently no link between bounded group cohomology and that of the reduced group or von Neumann group algebras it is an obvious question to ask if the subjects are related. Given that bounded group cohomology is a topic unknown to most operator algebraists, it seemed worth introducing it in Section 8.2 and linking it with Hochschild cohomology in Section 8.3. There is a list of problems in Section 8.4.
Bounded Group Cohomology
Remarks. Bounded group cohomology was related to corresponding geometrical and topological ideas for manifolds by Gromov in (1982) [Grom] following work of Hirsch and Thurston (1975) [HiT]. Earlier Johnson (1972) [J3] had used bounded cohomology of groups to show that H2(ℓ1(G), ℓ1(G)) ≠ 0 for G the free group on two generators. Bounded group cohomology is defined and a few of its properties are given in these notes. The theory is only developed as far as its current relevance to the Hochschild cohomology of Banach algebras warrants. For further details of the theory see the paper by Gromov [Grom] (beware there are errors), the survey by Ivanov [Iv1] and the paper by Grigorchuk [Gri2]. The authors are indebted to Professor Grigorchuk for the preprint [Gri2], which is recommended reading.
An elementary concrete approach is taken to the bounded cohomology of groups analogous to the Hochschild cohomology discussion.
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- Hochschild Cohomology of Von Neumann Algebras , pp. 171 - 181Publisher: Cambridge University PressPrint publication year: 1995