## A Course in Commutative Banach AlgebrasBanach algebras are Banach spaces equipped with a continuous multipli- tion. In roughterms,there arethree types ofthem:algebrasofboundedlinear operators on Banach spaces with composition and the operator norm, al- bras consisting of bounded continuous functions on topological spaces with pointwise product and the uniform norm, and algebrasof integrable functions on locally compact groups with convolution as multiplication. These all play a key role in modern analysis. Much of operator theory is best approached from a Banach algebra point of view and many questions in complex analysis (such as approximation by polynomials or rational functions in speci?c - mains) are best understood within the framework of Banach algebras. Also, the study of a locally compact Abelian group is closely related to the study 1 of the group algebra L (G). There exist a rich literature and excellent texts on each single class of Banach algebras, notably on uniform algebras and on operator algebras. This work is intended as a textbook which provides a thorough introduction to the theory of commutative Banach algebras and stresses the applications to commutative harmonic analysis while also touching on uniform algebras. In this sense and purpose the book resembles Larsen’s classical text [75] which shares many themes and has been a valuable resource. However, for advanced graduate students and researchers I have covered several topics which have not been published in books before, including some journal articles. |

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### Contents

1 | |

Gelfand Theory | 43 |

Functional Calculus Shilov Boundary and Applications | 139 |

Regularity and Related Properties | 193 |

Spectral Synthesis and Ideal Theory | 253 |

Appendix | 319 |

342 | |

349 | |

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algebra and let algebra with identity approximate identity Banach space belongs locally bounded approximate identity C*-algebra closed ideal closed subalgebra closed subset commutative Banach algebra compact Abelian group compact Hausdorff space compact subset continuous function Conversely Corollary coset Cq(X define denote dense Ditkin set element Exercise finite follows function f G A(A G CC(G Gelfand homomorphism Gelfand representation Gelfand topology Haar measure Hausdorff space hence holomorphic functional homomorphism hull-kernel topology idempotent implies injective invertible isometric isomorphism Lemma let f Let G locally compact Abelian locally compact group locally compact Hausdorff open neighbourhood open subset Proof Proposition prove regular commutative Banach satisfies Section semisimple commutative Banach set of synthesis Shilov boundary spectral extension property spectral set structure space subgroup of G subset of A(A supp Suppose tensor product uniform norm uniform norm property unique uniform norm