## Topological AlgebrasThis book consists of nine chapters. Chapter 1 is devoted to algebraic preliminaries. Chapter 2 deals with some of the basic definition and results concerning topological groups, topological linear spaces and topological algebras. Chapter 3 considered some generalizations of the norm. Chapter 4 is concerned with a generalization of the notion of convexity called p-convexity. In Chapter 5 some differential and integral analysis involving vector valued functions is developed. Chapter 6 is concerned with spectral analysis and applications. The Gelfand representation theory is the subject-matter of Chapter 7. Chapter 8 deals with commutative topological algebras. Finally in Chapter 9 an exposition of the norm uniqueness theorems of Gelfand and Johnson (extended to p-Banach algebras) is given. |

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### Contents

1 | |

Chapter 2 Topological Preliminaries | 73 |

Chapter 3 Some Types of Topological Algebras | 100 |

Chapter 4 Locally PseudoConvex Spaces and Algebras | 174 |

Chapter 5 Some Analysis | 222 |

Chapter 6 Spectral Analysis in Topological Algebras | 262 |

Chapter 7 Gelfand Representation Theory | 296 |

Chapter 8 Commutative Topological Algebras | 353 |

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a.sm alge assume Banach algebra bijection bounded C-net called choose clearly closed Co(S compact Hausdorff completely regular completing the proof complexification continuous convex COROLLARY defined DEFINITION denotes division algebra epimorphism follows formally real function f Further Gelfand algebra Hausdorff space Hence holomorphic homeomorphism hypermaximal ideal idempotent inequality invertible elements invertible iff isomorphism LEMMA locally compact locally sm maximal ideal maximal regular mazimal normed algebra nucleus obtain open neighbourhood open set p-admissible p-Banach p-convex p-normed p-seminormed algebra p-topology particular polynomial pre-boundary PROPOSITION prove pseudo-convex Q algebra quarter-norm real algebra regular bi-ideal relative unity respy satisfies seminormed sequence Similarly spectrally Gelfand strictly real subalgebra subset subspace Suppose theorem Topological Algebras topology unique weak topology whence Write zero divisor