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Cohomology of commutative Banach algebras and

`1-semigroup algebras

Yemon Choi

Thesis submitted for the degree of

Doctor of Philosophy

NEWCASTLE

UN IVERS ITY OF

School of Mathematics and Statistics

University of Newcastle upon Tyne

Newcastle upon Tyne

United Kingdom

May 2006

For APS,

comrade and confidante in bygone times,

and to whom I shall always be indebted.

Audere est facere.

Acknowledgements

First and foremost I would like to thank my supervisor Michael White: for all the

encouragement and guidance he has given me over the last three years, for constant

generosity with his time and mathematical insight, and for being so good-humoured

about my lax interpretations of the phrase “by tomorrow”.

Each person should render unto Caesar what is Caesar’s; this would not have

been possible for me without the financial support provided by EPSRC. I am also

grateful to the Product Development department of the National Extension College,

Cambridge, for providing a year outside the bubble, and for showing me more about

Wider Key Skills than I really wanted to know.

The production of this thesis has benefited from a wealth of other people’s freely

available TEXpertise. Particular thanks are due to Paul Taylor for his diagrams.sty

macros, without which what follows would be several pages shorter and significantly

less readable.

Life as a PhD student would have been much more arduous without the help

and cheer provided by staff and fellow students in the School of Mathematics and

Statistics. Special thanks to the 5-a-siders for putting up with my lack of first (or

second) touch, and to the Jesmond Academics for reintroducing me to the simple

pleasures of sacrificing on f7.

I am deeply grateful to my family, who have been ever-supportive of my stud-

ies and who have tolerated many garbled attempts at explanation over the years.

Lastly, thanks to friends past and present for their patience, and for making this all

worthwhile.

“What is this thing, anyway?” said the Dean, inspecting

the implement in his hands.

“It’s called a shovel,” said the Senior Wrangler. “I’ve

seen the gardeners use them. You stick the sharp end in

the ground. Then it gets a bit technical.”

– from Reaper Man by Terry Pratchett

Contents

1 Preliminaries 2

1.1 General notation and terminology . . . . . . . . . . . . . . . . . . . . 2

1.2 A word on units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Homological algebra in normed settings . . . . . . . . . . . . . . . . . 5

1.4 Hochschild (co)homology . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 The “Hodge decomposition”: basic definitions . . . . . . . . . . . . . . 12

1.6 Harrison and Lie (co)homology . . . . . . . . . . . . . . . . . . . . . . 16

1.7 Semigroups and convolution algebras . . . . . . . . . . . . . . . . . . . 19

2 Aspects of simplicial triviality 21

2.1 Simplicial triviality and related homological conditions . . . . . . . . . 21

2.2 Augmentation ideals of discrete groups and their cohomology . . . . . 23

2.2.1 Introduction and terminology . . . . . . . . . . . . . . . . . . . 23

2.2.2 Disintegration over stabilisers . . . . . . . . . . . . . . . . . . . 26

2.2.3 Corollaries and remarks . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Simplicial triviality for commutative Banach algebras . . . . . . . . . . 32

2.3.1 Acyclic base-change: motivation from ring theory . . . . . . . . 32

2.3.2 Acyclic base-change: weaker versions for Banach modules . . . 33

2.3.3 An application of base change . . . . . . . . . . . . . . . . . . . 36

3 On the first order simplicial (co)homology 38

3.1 Harrison (co)homology as a derived functor . . . . . . . . . . . . . . . 38

3.2 A “baby Künneth formula” . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 The norms on Z1(Aω,A ′ ω) and Z

1(`1(Z∞+ ), ` 1(Z∞+ )

′) . . . . . . . . . . . 48

4 (Co)homology of `1(Zk+) with symmetric coefficients 54

4.1 Hochschild homology via TorA . . . . . . . . . . . . . . . . . . . . . . 54

Page i

CONTENTS

4.2 `1-homology for Z+-sets . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Surgery arguments . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.2 Resolutions of some Z+-sets . . . . . . . . . . . . . . . . . . . . 68

4.2.3 An example with infinitely many junctions . . . . . . . . . . . 75

4.3 Return to Hochschild homology . . . . . . . . . . . . . . . . . . . . . . 77

5 Semilattices of algebras 78

5.1 Definitions and preliminaries . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 Transfer along semilattice homomorphisms . . . . . . . . . . . . . . . . 84

5.3 L-normalised chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4 Statement of the main normalisation theorem . . . . . . . . . . . . . . 92

5.5 π is null-homotopic for finitely generated F . . . . . . . . . . . . . . . 93

5.6 The main splitting theorem . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6.1 Formulating the inductive step . . . . . . . . . . . . . . . . . . 97

5.6.2 Proof of the inductive step (Proposition 5.6.3) . . . . . . . . . 100

5.7 `1-algebras of Clifford semigroups . . . . . . . . . . . . . . . . . . . . . 105

5.8 Future generalisations? . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Homological smoothness for CBAs 108

6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 A definition in terms of lifting problems . . . . . . . . . . . . . . . . . 109

6.3 Examples of smooth and nonsmooth CBAs . . . . . . . . . . . . . . . 118

6.4 Smoothness and HarH2 (quantitative aspects) . . . . . . . . . . . . . 119

6.5 Smoothness and H1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.6 Remarks and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7 The third cohomology of some Beurling algebras 133

7.1 Statement of main result . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2 Constructing cocycles: preliminaries . . . . . . . . . . . . . . . . . . . 134

7.3 Constructing cocycles: key estimates and final example . . . . . . . . . 138

A Normalising over a contractible subalgebra 144

B Biflatness implies simplicial triviality 148

C Another proof that H1(` 1(Z+), `1(Z+)) is an `1-space 150

Page ii

Abstract

In this thesis we investigate the higher cohomology groups of various classes of Banach

algebras, focusing on the `1-convolution algebras of commutative semigroups.

The properties of amenability and weak amenability have been studied by many

authors. A theme of the work presented here is that for commutative Banach al-

gebras, vanishing conditions on simplicial cohomology – which provide a spectrum

of intermediate notions between weak and full amenability – yield partial results on

cohomology with symmetric coefficients.

The strongest such vanishing condition is simplicial trivality. We show that the

augmentation ideals of `1-group algebras are simplicially trivial for a wide class of

groups: this class includes all torsion-free, finitely-generated word-hyperbolic groups,

but not the direct product F2×F2. Using very different arguments we show that the ` 1-

convolution algebras of commutative Clifford semigroups are simplicially trivial. The

proof requires an inductive normalisation argument not covered by existing results in

the literature.

Recent results on the simplicial cohomology of `1(Zk+) are used to obtain vanish-

ing results for the cohomology of `1(Zk+) in degrees 3 and above for a restricted class

of symmetric coefficient modules. In doing so we briefly investigate topological and

algebraic properties of the first simplicial homology group of `1(Zk+). In contrast,

examples are given of weighted convolution algebras on Z whose third simplicial co-

homology groups are non-Hausdorff (and in particular are non-zero).

We also investigate a natural extension of the cohomological notion of a smooth

commutative ring to the setting of commutative Banach algebras, giving examples to

show that the notion is perhaps too restrictive in a functional-analytic setting.

Introduction

Hochschild cohomology has proved a useful tool in studying commutative rings and

group rings. The analogous theory for Banach algebras has been studied by many au-

thors. However, there have been several technical obstacles in the Banach world that

make computations more difficult (and the theory more subtle) than in the “purely

algebraic” setting. For example, while the polynomial ring C[z] in one variable has

homological dimension 1, that of its `1-completion `1(Z+) has nontrivial cohomology

in dimension 2.

In this thesis we aim to compute the cohomology groups of various classes of

Banach algebras. For the most part we restrict our attention to the `1-convol