## Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles: Basic Representation Theory of Groups and AlgebrasThis is an all-encompassing and exhaustive exposition of the theory of infinite-dimensional Unitary Representations of Locally Compact Groups and its generalization to representations of Banach algebras. The presentation is detailed, accessible, and self-contained (except for some elementary knowledge in algebra, topology, and abstract measure theory). In the later chapters the reader is brought to the frontiers of present-day knowledge in the area of Mackey normal subgroup analysisand its generalization to the context of Banach *-Algebraic Bundles. |

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### Contents

1 | |

41 | |

63 | |

Chapter III Locally Compact Groups | 163 |

Chapter IV Algebraic Representation Theory | 265 |

Chapter V Locally Convex Representations and Banach Algebras | 323 |

Chapter VI CAlgebras and Their Representations | 377 |

Chapter VII The Topology of the Space of Representations | 539 |

Appendix A The StoneWeierstrass Theorems | 613 |

Appendix B Unbounded Operators in Hilbert Space | 619 |

Appendix C The Existence of Continuous CrossSections of Banach Bundles | 633 |

Bibliography | 643 |

Name Index | 721 |

725 | |

739 | |

### Common terms and phrases

Abelian approximate unit arbitrary assume Banach bundle Banach space Borel measure bounded C*-algebra closure commutative Banach algebra compact Hausdorff space compact subset completely reducible complex Borel measure convergence Corollary countable defined Definition denote dense direct sum element example exists extension finite finite-dimensional follows Gelfand group algebra group G Haar measure Hausdorff space hence Hermitian Hilbert space homomorphism implies integral involution irreducible representation isometry isomorphic Ker(T Lemma Let G linear space linear subspace locally compact group locally compact Hausdorff locally convex representation locally u-measurable Math multiplication neighborhood non-degenerate representation non-void non-zero normed algebra open subset operator set pointwise positive linear functional Prim(A Proof properties Proposition Prove regional topology regular Borel measure regular complex Borel Remark representation of G representation theory respect satisfying sequence spectral measure Suppose Theorem topological group topological space two-sided ideal u-almost unique unitarily equivalent unitary representation vector