## Introduction to Banach Algebras, Operators, and Harmonic AnalysisThis work has arisen from lecture courses given by the authors on important topics within functional analysis. The authors, who are all leading researchers, give introductions to their subjects at a level ideal for beginning graduate students, and others interested in the subject. The collection has been carefully edited so as to form a coherent and accessible introduction to current research topics. The first chapter by Professor Dales introduces the general theory of Banach algebras, which serves as a background to the remaining material. Dr Willis then studies a centrally important Banach algebra, the group algebra of a locally compact group. The remaining chapters are devoted to Banach algebras of operators on Banach spaces: Professor Eschmeier gives all the background for the exciting topic of invariant subspaces of operators, and discusses some key open problems; Dr Laursen and Professor Aiena discuss local spectral theory for operators, leading into Fredholm theory. |

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### Contents

1 Definitions and examples | 3 |

2 Ideals and the spectrum | 12 |

3 Gelfand theory | 20 |

4 The functional calculus | 30 |

5 Automatic continuity of homomorphisms | 38 |

6 Modules and derivations | 48 |

7 Cohomology | 58 |

Harmonic analysis and amenability George Awills | 73 |

18 Invariant subspaces for subdecomposable operators | 171 |

19 Reflexivity of operator algebras | 178 |

Invariant subspaces for commuting contractions | 186 |

Appendix to Part III | 193 |

Local spectral theory Kjeld Bagger Lausen | 199 |

21 Basic notions from operator theory | 201 |

22 Classes of decomposable operators | 212 |

Duality theory | 226 |

8 Locally compact groups | 75 |

9 Group algebras and representations | 86 |

10 Convolution operators | 98 |

11 Amenable groups | 109 |

12 Harmonic analysis and automatic continuity | 121 |

Invariant subspaces JŐrg Eschmeier | 135 |

13 Compact operators | 137 |

14 Unitary dilations and the Hfunctional calculus | 143 |

15 Hyperinvariant subspaces | 154 |

16 Invariant subspaces for contractions | 160 |

17 Invariant subspaces for subnormal operators | 166 |

24 Preservation of spectra and index | 230 |

25Multipliers on commutative Banach algebras | 241 |

Appendix to Part IV | 254 |

Singlevalued extension property and Fredholm theory Pietro Aiena | 265 |

26 Semiregular operators | 267 |

27 The singlevalued extension property | 285 |

28 SVEP for semiFredholm operators | 298 |

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Introduction to Banach Algebras, Operators, and Harmonic Analysis Harold G. Dales No preview available - 2003 |

### Common terms and phrases

abelian amenable analytic function approximate identity arbitrary argument Banach algebra Banach space bounded C*-algebra called Chapter character Check Clearly closed commutative complex condition connected consequence consider contains continuous contraction converges Corollary Dales decomposable decomposition defined Definition denote dense derivation disc dual element equality equivalent Eschmeier establish example Exercise exists extension fact finite follows Fredholm functional calculus given hence Hilbert space holds homomorphism ideal implies injective invariant subspace isomorphism L¹(G Laursen Lemma Let G linear operator locally compact group Math means measure Moreover multiplication neighbourhood Neumann non-trivial norm normal Note obtain operator particular polynomial Proof Proposition prove question range representation respect result satisfies semi-regular sequence Show shown spectral spectrum subgroup subnormal subset suppose surjective SVEP Theorem theory topology unital unitary vector weak