Banach Algebras on Semigroups and on their Compactifications H. G. Dales, A. T.-M. Lau, and D. Strauss Author address: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom E-mail address: [email protected] Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada E-mail address: [email protected] Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom E-mail address: [email protected] Contents Chapter 1. Introduction 1 Chapter 2. Banach algebras and their second duals 15 Chapter 3. Semigroups 37 Chapter 4. Semigroup algebras 59 Chapter 5. Stone{Cechˇ compactifications 75 Chapter 6. The semigroup (βS; 2) 79 Chapter 7. Second duals of semigroup algebras 99 Chapter 8. Related spaces and compactifications 121 Chapter 9. Amenability for semigroups 133 Chapter 10. Amenability of semigroup algebras 147 Chapter 11. Amenability and weak amenability for certain Banach algebras 167 Chapter 12. Topological centres 175 Chapter 13. Open problems 199 Bibliography 201 Index 209 Index of Symbols 213 v Abstract Let S be a (discrete) semigroup, and let ` 1(S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` 1(S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are `forbidden values' for this constant. The second dual of ` 1(S) is the Banach algebra M(βS) of measures on the Stone{Cechˇ compactification βS of S, where M(βS) and βS are taken with the first Arens product 2. We shall show that S is finite whenever M(βS) is amenable, and we shall discuss when M(βS) is weakly amenable. We shall show that the second dual of L1(G), for G a locally compact group, is weakly amenable if and only if G is finite. We shall also discuss left-invariant means on S as elements of the space M(βS), and determine their supports. We shall show that, for each weakly cancellative and nearly right cancellative semigroup S, the topological centre of M(βS) is just ` 1(S), and so ` 1(S) is strongly Arens irregular; indeed, we shall considerably strengthen this result by showing that, for such semigroups S, there are two-element subsets of βS n S that are determining for the topological centre; for more general semigroups S, there are finite subsets of βS nS with this property. We have partial results on the radical of the algebras ` 1(βS) and M(βS). We shall also discuss analogous results for related spaces such as W AP (S) and LUC(G). 2000 Mathematics Subject Classification. Primary 43A10, 43A20; secondary 46J10. Key words and phrases. Banach algebras, second duals, dual Banach algebras, Arens products, Arens regular, topological centre, strongly Arens irregular, intro- verted subspace, amenable, weakly amenable, approximately amenable, amenability constant, ultrafilter Stone{Cechˇ compactification, invariant mean, semigroup, Rees semigroup, semigroup algebra, Munn algebra, semi-character, character, cancella- tive semigroup, radical, minimal ideal, idempotent, locally compact group, group algebra, measure algebra, bounded approximate identity, diagonal. vi CHAPTER 1 Introduction Our aim in this memoir is to study the algebraic structure of some Banach algebras which are defined on semigroups and on their com- pactifications. In particular we shall study the semigroup algebra ` 1(S) of a semigroup S and its second dual algebra; this includes the impor- tant special case in which S is a group. Here ` 1(S) is taken with the convolution product ? and the second dual ` 1(S)00 is taken with respect to the first and second Arens products, 2 and 3 ; these sec- ond dual algebras are identified with Banach algebras (M(βS); 2) and (M(βS); 3), which are, respectively, the right and left topological semi- groups of measures defined on βS, the Stone{Cechˇ compactification of S. We shall also study the closed subalgebras ` 1(βS) of M(βS) and some related Banach algebras. Much of our work depends on knowledge of the properties of the semigroup (βS; 2 ), and, in particular, of (βN; 2 ). We wish to stress that (βN; 2 ) is a deep, subtle, and significant mathematical object, with a distinguished history and about which there are challenging open questions; we hope to introduce the power of this semigroup to those primarily interested in Banach algebras. Indeed the questions that we ask about Banach algebras are often resolved by inspecting the properties of this semigroup, and sometimes require new results about it. So we also hope that those primarily interested in topolog- ical semigroups will be stimulated by the somewhat broader questions, arising from Banach algebra theory, that we raise about (βS; 2 ). In brief, we aspire to interest specialists in both Banach algebra theory and in topological semigroups in our work. For this reason we have tried to incorporate general background from each of these theories in our exposition in an attempt to make the work accessible to both communities. This paper is partially a sequel to the earlier memoir [21]. (For a correction to [21], see p. 198 of the present work.) Notation We recall some notation that will be used throughout; for further details of all terms used, see [19] and [21]. We shall use elementary properties of ordinal 1 2 1. INTRODUCTION and cardinal numbers as given in [19, Chapter 1.1], for example. The minimum infinite ordinal is ! and the minimum uncountable ordinal is !1; these ordinals are also cardinals, and are denoted by @0 and @1, respectively, in this case. The cofinality of an ordinal α is cof α; a cardinal α is regular if cof α = α. The Continuum Hypothesis (CH) @0 is the assertion that the continuum c = 2 is equal to @1; this hy- pothesis is independent of the usual axioms ZFC of set theory. Results that are claimed only in the theory ZFC + CH are denoted by `(CH)'. Let S be a set. The cardinality S is denoted by jSj, the family of all subsets of S is P(S), and the family of all finite subsets of S is Pf (S). Let κ be a cardinal. Then [S]κ = fT ⊂ S : jT j = κg and [S]<κ = fT ⊂ S : jT j < κg : The characteristic function of a subset T of S is denoted by χT ; we set δs = χfsg for s 2 S. We set N = f1; 2;::: g, Z = f0; ±1; ±2;::: g, and Z+ = f0; 1; 2;::: g. + The sets f1; : : : ; ng and f0; 1; : : : ; ng are denoted by Nn and Zn , respect- ively. The set of rational numbers is Q, I = [0; 1], and the unit circle and open unit disc in the complex plane C are denoted by T and D, respectively. The complex conjugate of z 2 C is denoted by z. Algebras Let A be an algebra (always over the complex field, C). The product map is mA :(a; b) 7! ab ; A × A ! A: The opposite algebra to A is denoted by Aop; this algebra has the product (a; b) 7! ba; A × A ! A. In the case where A does not have an identity, the algebra formed by adjoining an identity to A is A# (and A# = A # if A has an identity); the identity of A or A is often denoted by eA. The centre of A is Z(A) = fa 2 A : ab = ba (b 2 A)g : An idempotent in A is an element p such that p2 = p; the family of idempotents in A is denoted by I(A). For p; q 2 I(A), set p ≤ q if pq = qp = p, so that (I(A); ≤) is a partially ordered set; a minimal idempotent in A is a minimal element of the set (I(A) n f0g; ≤). 1. INTRODUCTION 3 Let I be an ideal in an algebra A, and let B be a subalgebra of A such that A = B ⊕ I as a linear space. Then A is the semi-direct product of B and I, written A = B n I. Let I be an ideal in an algebra A, and suppose that I has an identity eI . Then we remark that eI 2 Z(A). Indeed, for each a 2 A we have aeI ; eI a 2 I, and so eI (aeI ) = aeI and (eI a)eI = eI a. Thus eI 2 Z(A). Let A be an algebra. We denote by RA, NA, and QA the (Jacobson) radical of A, the set of nilpotent elements of A, and the set of quasi-nilpotent elements of A, respectively; the sets RA, NA, and QA are defined in [19], but RA is denoted by rad A in [19]. We recall that RA is defined to be the intersection of the maximal modular left ideals of A, that RA is an ideal in A, and that A is defined to be semisimple if RA = f0g. We always have the trivial inclusions: RA ⊂ QA;NA ⊂ QA : In general, we have RA 6⊂ NA and NA 6⊂ RA; further, neither QA nor NA is necessarily closed under either addition or multiplication in A. For an ideal I in A, we have RI = I \ RA; in the case where A=I is semisimple, we have RI = RA. A nilpotent element a 2 A has index n if n = minfk 2 N : ak = 0g; a subset S of A is nil if each element of S is nilpotent; the radical RA contains each left or right ideal which is nil. Let A be an algebra.