We repeat 2. 26 as

1. Theorem [P. 2].Let (X, T) be a- topological Markov chain. There is a unique T-invariant probability m such that hm (T) ≥ hμ (T) for all T-invariant Borel probabilities μ. m is Markov and is supported by X.

This result was proved in LP. 2J without the knowledge that Shannon had included a similar theorem in his 1948 paper [s. W.]. In this Appendix we interpret the relevant parts of Shannon's paper to compare his theorem with 1. We will show that Shannon proved:

2. Theorem. Let (X, T) be an aperiodic topological Markov chain. There is a Markov probability m on X such that hm (T) ≥ hμ (T) for all (compatible)Markov probabilities μon; X

Comparing 2 with 1, we notice that in 2 μ is allowed to run through only Markov probabilities. This (insignificant) restriction is natural since at the time Shannon wrote, entropy had not been defined in general - in Is. W. J he defined it, for the first time, in some special cases. More significantly, there is no explicit uniqueness statement in 2.

All of Shannon's work is in the setting of (a model of) a communication system and in fact he proves 2 for systems (superficially) more general than aperiodic topological Markov chains. We consider Shannon's communication system and state his theorem for such a system. Then we show that this theorem is equivalent to 2, and prove it by Shannon's method.

INTRODUCTION

The theory of analytic (one parameter) semigroups t ↦ at from the open right half plane H into a Banach algebra is the main topic discussed in these notes. Several concrete elementary classical examples of such semigroups are defined, a general method of constructing such semigroups in a Banach algebra with a bounded approximate identity is given, and then relationships between the semigroup and the algebra are investigated. These notes form small sections in the theory of (one parameter) continuous semigroups and in the general theory of Banach algebras. They emphasize an approach that is standard to neither of these subjects. A study of Hille and Phillips [1974] reveals that the theory of Banach algebras has been used as a tool in the study of certain problems in continuous semigroups, but that semigroup theory has until recently (1979) not impinged on the theory of Banach algebras. These lecture notes are about this recent progress.

Throughout these notes we use ‘semigroup’ for ‘one parameter semigroup’ when discussing a homomorphism from an additive subsemigroup of ℂ into a Banach algebra, and we write our semigroups t ↦ at to emphasize the power law at+s = at · as and function property of the semigroup. In the standard works on semigroups much attention is given to strongly continuous semigroups and their generators (see Hille and Phillips [1974], Dunford and Schwartz [1958], and Reed and Simon [1972]). In these works the generator itself is important, plays a fundamental role, and is often an object of considerable mathematical interest (for example, it may be the Laplacian).

In this chapter we shall prove Theorems 3.1 and 3.15, and the various lemmas required in the proofs. In 4.1 we sketch the ideas behind the proofs, and after proving all the lemmas we prove 3.1 in 4.7 and 3.15 in 4.8. Throughout this chapter A will denote a Banach algebra with a countable bounded approximate identity bounded by d(≥1), X will denote a left Banach A-module satisfying ∥a.x∥ ≤ ∥a∥.∥x∥ for all a ∈ A and x ∈ X, and [A·x]− will denote the closed linear span of the set {a·x : a ∈ A, x ∈ X}. Taking d = 1 simplifies the calculations slightly. We assume that A does not have an identity.

SKETCH OF THE PROOF

The proof is a variation of Cohen's factorization, theorem (Cohen [1959]) with the analytic semigroup obtained as a limit of exponential semigroups in the unital Banach algebra A#. The variation is influenced by the proof of the Hille-Yoshida Theorem. If the algebra A had an identity, then the factorization results would be trivial as we could take at = 1 for all t ∈ H. Though our algebra does not have an identity, we shall use the case when there is an identity and an approximation to prove 3.1. We work in the algebra A# = A ⊕ ℂl obtained by adjoining an identity to A, and we regard X and Y as left and right Banach A#- modules by defining 1.w = w and u.1 = u for all w ∈ X and u ∈ Y.

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).

Throughout this chapter A will denote a Banach algebra with a countable bounded approximate identity, which will usually be bounded by 1. The main theorem of this chapter ensures that such an algebra contains a semigroup analytic in the open right half plane. This theorem is proved by an extension of Cohen's factorization theorem for a Banach algebra with a bounded approximate identity. In this chapter we shall discuss the properties of the semigroups that may be obtained by these methods, and shall investigate some applications of the existence of the semigroups. The proof of the existence of the semigroups is given in detail in Chapter 4, and is discussed there. The basic properties of the semigroup are stated in Theorem 3.1, and additional properties relating to derivations, multipliers and automorphisms are dealt with in Theorem 3.15.

Many of the properties of the semigroups given below are generalizations of those of the fractional integral semigroup in L1 (ℝ+), or the Gaussian and Poisson semigroups in L1 (ℝn). We should like the semigroups constructed in A to be as nice as possible: to have good growth, norm and spectral behaviour. In Chapter 5 it is shown that certain polynomial growth properties of the Gaussian and Poisson semigroups cannot hold in radical Banach algebras.

The semigroup properties of at are emphasised rather than the factorization properties, and all the semigroup properties that I know are included in Theorems 3.1 and 3.15. However in any given situation one is usually interested in only two or three properties of the semigroup, or of the module factor of x.