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This book gives a coherent account of the theory of Banach spaces and Banach lattices, using the spaces C_0(K) of continuous functions on a locally compact space K as the main example. The study of C_0(K) has been an important area of functional analysis for many years. It gives several new constructions, some involving Boolean rings, of this space as well as many results on the Stonean space of Boolean rings. The book also discusses when Banach spaces of continuous functions are dual spaces and when they are bidual spaces.

### Chapter 1. Introduction

Abstract
In this chapter, we shall begin by introducing some basic notations. This will be followed by a discussion in $$\S$$ 1.4 of some topological concepts, including those of locally compact spaces and Stonean spaces. We shall later frequently refer to the Stone–Čech compactification β X of a completely regular space X, and this is introduced in $$\S$$ 1.5. In $$\S$$ 1.6, we shall prove Gleason’s theorem characterizing projective topological spaces as the Stonean spaces, and, in $$\S$$ 1.7, we shall also recall some basic theory of Boolean algebras, generalizing this slightly to cover Boolean rings; we shall discuss the Stone space of a Boolean ring and give various important examples of Boolean rings.
H. G. Dales, F. K. Dashiell, A. T.-M. Lau, D. Strauss

### Chapter 2. Banach Spaces and Banach Lattices

Abstract
We shall now give some background in the theory of normed and Banach spaces, including the key definitions of dual and bidual spaces and of an isomorphism and an isometric isomorphism between two normed spaces. In particular, we shall show how certain bidual spaces can be embedded in other Banach spaces. In $$\S$$ 2.3, we shall also recall some basic results and theorems concerning Banach lattices. We shall define complemented subspaces of a Banach space in $$\S$$ 2.4, and also we shall discuss, in $$\S$$ 2.5, the projective and injective objects in the category of Banach spaces and bounded operators. We shall conclude the chapter by discussing dentability and the Krein–Milman property for Banach spaces in $$\S$$ 2.6.
H. G. Dales, F. K. Dashiell, A. T.-M. Lau, D. Strauss

### Chapter 3. Banach Algebras and C ∗-Algebras

Abstract
This chapter will first give the basic background that we shall require concerning Banach algebras, C -algebras, and von Neumann algebras. In particular, in $$\S$$ 3.1, we shall discuss the bidual of a Banach algebra, taken with its Arens products. In $$\S$$ 3.3, we shall exhibit the Baire classes as examples of commutative C -algebras. We shall conclude the chapter in $$\S$$ 3.4 with a few remarks on the generalizations of some of our discussions concerning the commutative C -algebras C  0(K) to general (non-commutative) C -algebras; as we said, these generalizations will not be used within our main text.
H. G. Dales, F. K. Dashiell, A. T.-M. Lau, D. Strauss

### Chapter 4. Measures

Abstract
In this chapter, we shall study the (complex) Banach lattice M(K) consisting of all complex-valued, regular Borel measures on a locally compact space K and, in particular, the positive measures in M(K), which form the cone M(K)+. The Banach space M(K) is isometrically isomorphic to the dual of C  0(K). In $$\S$$4.2, we shall discuss the linear spaces of discrete measures and of continuous measures on K.
H. G. Dales, F. K. Dashiell, A. T.-M. Lau, D. Strauss

### Chapter 5. Hyper-Stonean Spaces

Abstract
We shall now define, in $$\S$$ 5.1, the compact spaces of most interest to us, the ‘hyper-Stonean spaces’, and characterize them in terms of the existence of category measures. For a locally compact space K and μ ∈ P(K), we shall discuss in $$\S$$ 5.2 the commutative C -algebra L (K, μ), which has been identified with the space C(Φ μ ). In particular, we shall describe Φ m , where m is Lebesgue measure on $$\mathbb{I}$$: in this case, Φ m is called $$\mathbb{H}$$, the hyper-Stonean space of the unit interval. We shall give a topological characterization of $$\mathbb{H}$$ in $$\S$$5.3.
H. G. Dales, F. K. Dashiell, A. T.-M. Lau, D. Strauss

### Chapter 6. The Banach Space C(K)

Abstract
The main aim of this chapter is to determine when a space of the form C(K) for a compact space K is a dual space or a bidual space,either isometrically or isomorphically. However, we shall first discuss when two spaces C(K) and C(L) are isomorphic and when they are isometrically isomorphic. Some results come from rather elementary considerations, but some require more sophisticated background.
H. G. Dales, F. K. Dashiell, A. T.-M. Lau, D. Strauss