Cohomology of commutative Banach algebras and â€؛ ~choiy1 â€؛ pubmath â€؛ thesiscopy â€؛ Cohomology

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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