## Uniform Fréchet AlgebrasThe first part of this monograph is an elementary introduction to the theory of Fréchet algebras. Important examples of Fréchet algebras, which are among those considered, are the algebra of all holomorphic functions on a (hemicompact) reduced complex space, and the algebra of all continuous functions on a suitable topological space. The problem of finding analytic structure in the spectrum of a Fréchet algebra is the subject of the second part of the book. In particular, the author pays attention to function algebraic characterizations of certain Stein algebras (= algebras of holomorphic functions on Stein spaces) within the class of Fréchet algebras. |

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### Contents

1 | |

PART 2 GENERAL THEORY OF FRECHET ALGEBRAS | 57 |

PART 3 ANALYTIC STRUCTURE IN SPECTRA | 195 |

APPENDIX A Subharmonic functions Poisson integral | 331 |

APPENDIX B Functional analysis | 335 |

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admissible exhaustion algebra homomorphism algebraically isomorphic analytic structure analytic subset approximated uniformly arbitrary element Banach algebras Choose compact neighbourhood compact open topology compact set compact subset continuous function contradiction converges countable defines definition domain of holomorphy euclidean topology example exhaustion of M(A F-algebra finite follows Fréchet algebras Fréchet spaces Gelfand topology Gelfand transform Gleason Hausdorff space hemicompact space hence holomorphic functions injective k-space lemma Let A,X Let f Let G let p e locally compact spectrum M(An M(Hol Math maximal ideal maximum modulus morphic n-generated open mapping open neighbourhood open set open subset p e M(A peak point Proof reduced complex space reflexive uR-algebra remark resp Riemann domain Schwartz space seminorms Shilov boundary spectrum M(A Stein algebra Stein space subalgebra subset of M(A Suppose surjective theorem topological algebra topologically and algebraically