## Banach Algebras and the General Theory of *-Algebras: Volume 1, Algebras and Banach AlgebrasThis is the first volume of a two volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. This account emphasizes the role of *-algebraic structure and explores the algebraic results that underlie the theory of Banach algebras and *-algebras. The first volume, which contains previously unpublished results, is an independent, self-contained reference on Banach algebra theory. Each topic is treated in the maximum interesting generality within the framework of some class of complex algebras rather than topological algebras. Proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Volume II is forthcoming. |

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### Contents

Introduction to Normed Algebras Examples | 1 |

Algebras | 9 |

The Spectrum | 187 |

Commutative Algebras and Functional Calculus | 301 |

Ideals Representations and Radicals | 437 |

Approximate Identities and Factorization | 519 |

Automatic Continuity | 553 |

Structure Spaces | 619 |

Algebras with Minimal Ideals | 655 |

Bibliography | 715 |

783 | |

791 | |

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### Common terms and phrases

A/Aj algebra and let algebra norm algebra semi-norm algebraic tensor product approximate identity arbitrary Arens product Arens regular Banach space belongs bounded C*-algebra called closed ideal commutative Banach algebra compact operators completely regular condition converges Corollary defined denote dense direct sum double centralizer dual equation equivalent example finite finite-dimensional functional calculus gives group algebras Hausdorff Hence Hilbert space idempotent implies included injective invertible irreducible representation Jacobson radical kernel left ideal Lemma linear functional linear map Ll(G locally compact group matrix maximal modular ideal modular annihilator algebra multiplication neighborhood nilpotent non-zero nonempty normed algebra normed linear space Note obvious operator norm pointwise polynomial primitive ideal Proof Proposition Let prove quasi-invertible quotient reflexive satisfying semiprime semisimple sequence Sp(a spectral algebra spectral normed spectral radius spectral semi-norm spectral subalgebra spectrum subalgebra subset subspace Suppose tensor product Theorem Let theory two-sided unique unital algebra weakly

### Popular passages

Page 758 - Functors and categories of Banach spaces. Tensor products, operator ideals and functors on categories of Banach spaces.

Page 722 - FF Bonsall, A minimal property of the norm in some Banach algebras, J.

Page 753 - Concerning the second dual of the group algebra of a locally compact group, J. London Math. Soc. (2), 41 (1990), 445-460. [15] V. LOSERT and H. RINDELER, Assymptotically central functions and invariant extensions of Dirac measures, Proc. Conf. "Probability measures on groups" Oberwollfach, Springer Lecture Notes 1064, 368-378.

Page 717 - The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848.

Page 764 - Banach spaces of compact multipliers and their dual spaces

Page 738 - Weighted group algebra as an ideal in its second dual space, Proc. Amer. Math Soc.