## 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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### Other editions - View all

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

### Common terms and phrases

aap(T abelian group algebra homomorphism analytic function approximate identity arbitrary automatic continuity Banach A-bimodule Banach space bounded approximate identity bounded operator C*-algebra Chapter closed codimension cohomology commutative Banach algebras compact operator continuous linear converges Corollary Dales decomposable operators decomposition defined Definition denote derivation disc dual element Eschmeier example Exercise exists finite-dimensional follows Fredholm operator functional calculus G is amenable Gelfand group algebras group G Haar measure hence Hilbert space ideal injective invariant subspace isometric isomorphism Laursen and Neumann left A-module Lemma Let G linear map linear operator linear subspace Ll(G locally compact group Math multiplication non-trivial non-zero norm open set polynomial proof of Theorem Proposition prove quotient result semi-regular semisimple sequence Show spectrum subalgebra subgroup subnormal operator super-decomposable suppose surjective SVEP at A.0 theory topology unital Banach algebra unitary representation vector weak XT(F ZT(F