[Return to main list of publications]## Simplicial cohomology of augmentation ideals in l^{1}(G)

#### Authors

Yemon Choi

#### Keywords

bounded cohomology, simplicial cohomology, Banach algebras, commutative transitive

#### Status

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

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

#### Reviews

[ Math Reviews | Zentralblatt (summary) ]

#### Abstract

Let G be a discrete group. We give a decomposition theorem for the Hochschild cohomology of l^{1}(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 l^{1}(G) is an isomorphism.

#### Updates/comments

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 **Ban**_{1}.

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