INTRODUCTION

In this chapter we introduce various well known semigroups from the open right half plane H into particular Banach algebras. We discuss the power semigroups in a separable C*-algebra, the fractional integral and backwards heat semigroups in L1(ℝ+), and the Gaussian and Poisson semigroups in L1(ℝn). While doing this we shall develop notation that is used in subsequent chapters. The discussion is very detailed throughout the chapter, and is designed to introduce and motivate following chapters dealing with more abstract results for analytic semigroups. For example we are concerned with the asymptotic behaviour of ∥al + iy∥ as |y| tends to infinity, but not with the infinitesimal generators of our semigroups even though they are important. We shall discuss generators in a different context in Chapter 6.

C*-ALGEBRAS

The functional calculus for a positive hermitian element in a C*-algebra that is derived from the commutative Gelfand-Naimark Theorem enables us to construct very well behaved semigroups in C*-algebras. We shall briefly discuss the case of a commutative C*-algebra before we state and prove our main result on semigroups in a C*-algebra. The commutative Gelfand-Naimark Theorem (see, for example, Bonsall and Duncan [1973]) enables us to identify the commutative C*-algebra with C∘ (Ω), which is the C*-algebra of continuous complex valued functions vanishing at infinity on on the locally compact Hausdorff space Ω. It is easy to check that C∘ (Ω) has a countable bounded approximate identity if and only if Ω is σ-compact (that is, Ω is a countable union of compact subsets of itself).