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Simplicial cohomology of augmentation ideals in l1(G)


Yemon Choi


bounded cohomology, simplicial cohomology, Banach algebras, commutative transitive


Appeared as Proc. Edinburgh Math. Soc. 53 (2010), no. 1, 97--109.

Preprint version available at arXiv 0711.3669 (last updated Aug. '08)


[ Math Reviews | Zentralblatt (summary) ]


Let G be a discrete group. We give a decomposition theorem for the Hochschild cohomology of l1(G) with coefficients in certain G-modules. Using this we show that if G is commutative-transitive, the canonical inclusion of bounded cohomology of G into simplicial cohomology of l1(G) is an isomorphism.


This paper is taken from Chapter 2 of my PhD thesis, but with some modifications in the presentation. It was inspired by a preprint of Pourabbas and White, together with a sudden realisation that all the proper centralizers I could think of in the free group on two generators were cyclic groups. (Staff and students in Newcastle who worked in geometric group theory, after getting over my ignorance of the literature, then explained to me that commutative-transitivity was well known and occurred for other interesting examples.)

The arXiv version used the notion of a chain complex in Ban being "1-split". As was pointed out by the referee, it is more accurate to instead call the notion in question "1-contractible", and this amended terminology is thus the one appearing in the final paper.

Although we don't deal in the paper with simplicial objects in Ban, the proof of Theorem 4.1 is really exploiting the canonical simplicial resolution of a Banach module over a unital Banach algebra, a construction which naturally lives in the contractive category Ban1.

Yemon Choi
Last modified: Mon Nov 22 13:09:51 CT 2010