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Simplicial homology of strong semilattices of Banach algebras


Yemon Choi


simplicial homology, cohomology of Banach algebras, semilattices, Clifford semigroups, Plonka sums


Appeared as Houston J. Math. 36 (2010), no. 1, 237--260.

Preprint version available at arXiv math.FA/0609450 (last updated Feb. '08)


[ Math Review | Zentralblatt (summary) ]


Certain semigroups are known to admit a 'strong semilattice decomposition' into simpler pieces. We introduce a class of Banach algebras that generalise the l1-convolution algebras of such semigroups, and obtain a disintegration theorem for their simplicial homology. Using this we show that for any Clifford semigroup S of amenable groups, l1(S) is simplicially trivial: this generalises previous results of the author (Glasgow Math. Journal, 2006). Some other applications are presented.


See here for minor corrections.

This article is based on Chapter 5 of my PhD thesis, though it considers a slightly more general class of algebras. (Briefly, one is no longer confined to the case where the constituent algebras are unital.) There is also an extra application to the convolution algebras of normal bands, which was worked out after the thesis.

Although the main result is a considerable generalisation of the work in the GMJ paper, I personally view that earlier paper as an easier introduction to the ideas in this one. In particular, the key inductive/recursive argument is much less complicated in the GMJ paper, because one is only considering semilattice algebras rather than semilattices of algebras.

The final arXiv version differs from the version to be published in two important respects. Firstly, several expository asides were removed for the published version, to make the account more focused, and a lot of whitespace was removed to bring the page count down. Secondly, the published version uses an observation of the referee that, in several technical calculations, explicit mention of the transition homomorphisms can be suppressed by using the action of L on algL,A. (While this comes down to the same thing, since the proof that we have a well-defined action inevitably relies on calculation with the transition homomorphisms, the referee's suggestion makes the presentation a little shorter and more elegant.) I have decided not to incorporate these changes into the arXiv version, both to maintain some independence from the published version, and also because credit for the improved presentation really belongs with the referee. On the other hand, I have taken the opportunity to correct a few typos which would have been (probably) spotted regardless.

Thus, in summary, the arXiv version should be correct mathematically, and may be of interest for readers who want a more leisurely explanation of the ideas in and around the main proofs. On the other hand, the published version is the definitive one.

Yemon Choi
Last modified: Fri Aug 12 13:58:33 CST 2011