Theory of Operator Algebras I pp 1-57 | Cite as

# Fundamentals of Banach Algebras and *C**-Algebras

Chapter

## Abstract

In this this first chapter, we lay the foundation for later discussion, giving elementary results in Banach algebras and

*C**-algebras. The first three sections are devoted to the general Banach algebras. The most important results in these sections are Theorem 2.5, Corollary 2.6, and Theorem 3.11, which are really fundamental in the theory of Banach algebras. Discussion of*C**-algebras starts from Section 4. As an object of the theory of operator algebras, a*C**-algebra is a uniformly closed self-adjoint algebra*A*of bounded linear operators on a Hilbert space ℌ. The major task of the theory of operator algebras is to find descriptions of the structure of {*A*,ℌ}. This problem splits into two problems:- (a)
Find descriptions of the algebraic structure of

*A*alone; - (b)
Given an algebra

*A*, find all possible pairs {*B*,ℜ} such that*B*is isomorphic to*A*as an abstract algebra.

## Keywords

Hilbert Space Banach Algebra Left Ideal Functional Calculus Closed Ideal
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## Copyright information

© Springer-Verlag New York Inc. 1979