Sun-Dual Characterizations of the Translation Group of ℝ Citation Jackson, Frances Yvonne (1998) Sun-Dual Characterizations of the Translation Group of ℝ. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3y5x-kg66. https://resolver.caltech.edu/CaltechTHESIS:11212019-103601328 Abstract Let E be a Banach space. The mapping t → T ( t ) of ℝ (real numbers) into L b ( E ), the Banach algebra of all bounded linear operators on E , is called a strongly continuous group or a C₀ -group, if G = { T ( t ) : t ∈ ℝ} defines a group representation of (ℝ, +) into the multiplicative group of L b ( E ), and if ∀ f ∈ E , [equation; see abstract in scanned thesis for details]. For example, if E = C₀ (ℝ), the function space which consists of all continuous, complex functions that vanish at infinity, then (∀ t ∈ ℝ) (∀ f ∈ C₀ (ℝ)), the function T ( t ) f ( x ) = f ( x + t ), x ∈ ℝ, defines a strongly continuous group, since each f ∈ E is uniformly continuous; this group is called the translation group . If we now consider E = B (ℝ), the space of bounded, continuous complex functions on ℝ, then although the translation group on E is not strongly continuous, it is strongly continuous on the subspace BUC (ℝ) of E , which consists of bounded, uniformly continuous functions. BUC (ℝ) is the largest subspace of E on which the translation group is strongly continuous. The adjoint family of a C₀ -group defined on a Banach space E , need not be strongly continuous on the Banach dual E* of E . Let E ⊙ (pronounced E -sun) be the largest linear subspace of E* relative to which the adjoint family is a C₀ -group: [equation; see abstract in scanned thesis for details]. E ⊙ is called the sun-dual or sun-space of E . If E = C₀ (ℝ), then it follows from a well-known result of A. Plessner that E ⊙ = L ¹(ℝ) ([Ple]). This research paper contains a characterization of the sun-dual of BUC (ℝ) and of the subspace W AP (ℝ) of BUC (ℝ), which consists of weakly almost periodic functions on ℝ. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Luxemburg, W. A. J. Thesis Committee: Luxemburg, W. A. J. (chair) Kechris, Alexander S. Makarov, Nikolai G. Wolff, Thomas H. Defense Date: 11 June 1997 Other Numbering System: Other Numbering System Name Other Numbering System ID UMI 9809231 Record Number: CaltechTHESIS:11212019-103601328 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11212019-103601328 DOI: 10.7907/3y5x-kg66 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 13588 Collection: CaltechTHESIS Deposited By: Mel Ray Deposited On: 21 Nov 2019 20:57 Last Modified: 16 Apr 2021 22:12 Thesis Files Preview PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

Sun-dual characterizations o f th e Translation Group o f R Thesis by- Frances Y. Jackson In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1998 (Submitted June 11, 1997) R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . ii © 1998 Frances Y. Jackson All Rights Reserved R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . A cknow ledgem ents A number of friends made it possible for me to complete this work; I intend not to neglect any of them. First, I would like to thank my lifelong friend J. Christ; he has been an absolute saviour. In all the years th a t I have known him , he has always encouraged me in whatever I sought to undertake, and has—on more than one occasion—prevented me from plunging into u tter despair. He is faithful and steadfast and I told myself years ago th at if ever I had an opportunity to publicly honor him, I would. Thank you J. Christ. I would also like to thank my friend of eighteen years, John N. McDonald, a professor of mathematics at Arizona State University. John is one of the best kept secrets in mathematics in America. Prior to my attending Caltech, I had two good m ath ideas in a span of five years: John is responsible for both of them. Many thanks Johnnie! Many thanks also to ‘the captain of the Caltech ship’: Wilhelmus Anthonius Josephus Luxemburg, a.k.a. W im—my advisor (and I hope, my friend). Wim is patient, kind, funny, knowledgeable, a superior mathematician, and humble with regard to his mathematical ability, etc. etc. It has been an honor to study with him. On numerous occasions I have tested this m an’s patience beyond what anyone should have to endure; but instead of tossing me out of his office— which is what he should have done—he invited me in, sat me down, and often taught me for hours at a time. We do not have the technology to measure my gratitude toward Wim; I hope he remains my mentor and friend for as long as I live. God bless you Wim!!! Last but not least, I thank my friend and fellow graduate student, Ilia Binder; he tirelessly taught me the intricacies of ET^X so that I could type my thesis. Thank you Bindie. R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . A bstract Let E be a Banach space. The mapping t —►T (t) of R. (real numbers) into Ct,(E), the Banach algebra of all bounded linear operators on E , is called a strongly continuous group or a Co-group, if G = (T(f) : t E R} defines a group representation of (R, +) into the multiplicative group of Cb(E), and if V / 6 E, l i m ||7 W - / ||= 0 . For example, if E = Co(R), the function space which consists of all continuous, complex functions th a t vanish at infinity, then (V£ E R) (V/ E Co(R)), the function T ( t ) f ( x ) = f { x + 1), x E R, defines a strongly continuous group, since each / E E is uniformly continuous; this group is called the translation group. If we now consider E = B {R), the space of bounded, continuous complex functions on R, then although the translation group on E is not strongly continuous, it is strongly continuous on the subspace B U C(R) of E , which consists of bounded , uniformly continuous functions. BUC(WL) is the largest subspace of E on which the translation group is strongly continuous. The adjoint family of a Co-group defined on a Banach space E, need not be strongly continuous on the Banach dual E* of E . Let E ° (pronounced E - suil) be the largest linear subspace of E* relative to which the adjoint family is a Co-group: E ° is called the sun-dual or sun-space of E. If E = Co(R), then it follows from a well-known result of A. Plessner that E ° = L l {R) ([Pie]). This research paper contains a characterization of the sun-dual of B U C (R) and of the subspace W AP(R) of BU C {R), which consists of weakly almost periodic functions on R. R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . V C ontents A cknow ledgem ents iii A b stract iv 1 In trod u ction 1 2 V ector la ttic e term inology 6 2.1 D efinitions...................................................................................................... 6 2.2 Properties of vector l a t t i c e s ....................................................................... 7 3 A characterization o f th e Banach d u al o f BUC (R) 11 4 T ranslation invariant linear functionals on BUC (R) 14 5 E xam ples o f elem ents in F ix 23 6 fi ad m its a sem itopological sem igroup structure 27 7 C onvolutions 36 8 A characterization o f B U C 0 (R) 40 9 W eakly alm ost periodic functions o n R 46 B ibliography R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 54 1 C hapter 1 Introduction Let R be the field of real numbers. Suppose G = {T(t) : t GR} is a family of maps such th at for each t G R, T(t) translates R by t, i.e., T( t ) x = x + 1, x G R. Then G is called a regular representation of the additive group of R. Suppose X is a real or complex linear function space which has the property th at if f ( x ) G X , then f ( x + 1) € X . Then we can extend the family G to a group of operators defined on X in such a way that the action of G on X is given by translation: V /€ l, T ( t ) f ( x ) = f ( x + t). Note th at G now has the following properties: (1) T(0) = / , where I is the identity operator, (2) Vs, t G R, T{s + t ) = T(s) T ( t ) = T(t) T(s). Let C{X) be the group of all linear operators that act on X . Since R is a group, then Vt g R, the inverse (T(t))~l exists in C(X) and (3) (rw)-1= n-ty Observe that G is a group representation of (R, +) into the multiplicative group of C{X). Suppose that (X, || • ||) is now a Banach space of functions defined on R, and th at G is now a family of bounded, linear operators on X . If in addition to properties (1-3) above, G also satisfies, (4) V / G X, limt_ 0 ||T ( t ) f - / || = 0, then G is called a strongly continuous translation group or a translation Co-group. For example, if X = C0(R), the space of continuous functions which vanish at infinity, R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 2 and the norm on X is the supremum norm, then IIT { t ) f - / || = sup |/( x + 1) - f ( x ) | -*• 0 as t -> 0, (1.1) x € fl since each / £ Co(R) is uniformly continuous. Let B ( R) be the space of bounded, continuous, complex-valued functions on R. The function g(x) = sin(er ) is a bounded, continuous function on R, but it is not uniformly continuous. For if Xk = log(7rfc) and x'k = log(7rfc + 1), then \xk — x'k\ —»■0, but \g(xk) —g{p^k)\ — 1- ^ follows that the quantity \\T(t)g —g||, does not tend to zero as t —>0. We can conclude from the above example that a function / £ B {R) satisfies properties (1-4) if and only if / is uniformly continuous. Consider the subspace B U C (R ) of B (R) consisting of all bounded, uniformly continuous, complex-valued functions on R; then B(R) D BUC(R) D Co(R); moreover, B U C (R) is the largest subspace of B(R) for which the translation group is a Co-group. The adjoint family of a Co-group defined on a Banach space X , need not be a Co-group on the dual space X* of X . To see this, consider again the space Co(R). The dual Cq(R) of Co(R) is identified with the space M (R), which consists of all bounded, regular, complex Borel measures on R, with norm equal to total variation over R (Riesz representation theorem). If G is the translation group and 6X is the point measure at x £ R, then 8X £ Cq(R). The action of the dual family (T* (t) : t £ R} on 8X is given by the dual relation: (/, 6x+t) = 6X) = (/, T*(t)8x), f £ C0(R), so T*(t)8x = 8x+t\ from whence it follows, ||T*(£)&,; —5r || = 2 when t ^ 0. Let X ° (pronounced X-sun) be the collection of all bounded, linear functionals R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 3 in X* with the following property: X Q = {fi 6 X * : lim | £—0 | - p\\ = 0}. Then X ° is a closed, linear subspace of X * ([Phi]). For example, consider the space Co(R); it is well-known that C^f(R.) is identified with the space L l (R). T hat is, /j. (= C q(R) exhibits ‘the strong continuity property’ if and only if /i is absolutely con­ tinuous with respect to Lebesgue measure ([Pie]). Note th a t the ‘sufficiency portion’ of this result follows from the fact th at Lebesgue integrable functions are L ^meancontinuous, i.e., if f € L1(R), then m / \ f ( x + t ) - f ( x ) \ d x = 0. J —OO The primary focus of this work is to characterize B U C e (R); Chapters 2-4 and Chapters 6 and 8 are devoted to this objective. Chapter 2 is divided into two parts; the first part contains definitions and theorems in Banach lattice theory and the second part contains an application of this theory to the space B U C (R). It is via lattice theory th at we determine an im portant property of the norm on BUC*(M) th at enables us to provide the first of three characterizations of BUC*(R): BUC*(R) = B U C °{R) ® (B U C °(R ))d, where (B U C °(R ))d is the disjoint complement of B U C Q(R) (Definition (v)b). We utilize the above representation in the proof of our m ain result, Theorem A. In C hapter 3 we employ the theory of commutative Banach algebras to determine a second characterization of BUC*(R): Let Q be the maximal ideal space of BU C(R) and let C*(Q.) be the Banach dual of the space of continuous functions defined on fl; then C*(f2) is identified with the space M(Q), which consists of bounded, regular, R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 4 Borel measures on Cl, and BUC*(R) = C*(Cl) = M(Cl). C hapter 4 is pivotal; we not only give a third characterization of BUC*{R): BUC*(R) = C0X(R) © Q ( R ) , but we use this characterization to show the existence of nonzero linear functionals <f>G BUC*{R) th at vanish on Co(R), and which have the property: V< 6 R, = <f>. (1.2) We call any <j>with property (1-2) translation invariant and we denote F ix as the col­ lection of all such (j). Note th a t F ix C BU CQ(R). The end of Chapter 4, specifically Lemma 4.7, contains a ‘preliminary description’ of B U C Q(R) in terms of {Fi x ) , the band generated by F ix (Definition (v)b and Remark 4.8), and the sun- dual C®(R): B U C °{R) = (C0Q(R) © {Fix}) © (C^(R) © {F i x ) ) d n B U C °{R). This Lemma is crucial to our work; thus a great deal of effort has gone into proving it. In C hapter 5 we digress a bit, by giving examples of elements in F ix. Chapter 7 may appear to the reader to be ju st as disconnected from our primary focus as Chapter 5. Afterall, we define a ‘convolution operation’ on M(Cl) which renders it a commutative Banach algebra with unit, and we show M(Cl), now regarded as a ‘measure algebra,’ contains B U C °{R ) as a closed algebraic ideal, but M(Cl) ■=£■ B U C Q(R). Although none of this information is used in proving Theorem A, all of the ideas in this chapter will be incorporated in the proof of Theorem B. In Chapters 6 and 8 we ‘gather the tools’ to prove (C®(R) © { F i x ) ) d fi BUC@(R) = {0}. Once we have shown this, we conclude B U C Q(R) = C®(R) © {Fi x} (Theorem A). In the final chapter we prove Theorem B—our ‘peripheral goal’. We obtain a R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 5 representation of the sun-dual, relative to the translation group G, of the space of weakly almost periodic functions on R (WM-P(R)). To do so, we rely on several theorems and definitions due to W.F. Eberlein ([Ebl] and [Eb2]); yet unlike Eberlein, we avoid any discussions of W AP(R) in the context of abstract ergodic theory. There are several points that we make before we prove Theorem B. The first is th at Cb(R) C W AP(R) C BU C{R). And the second is th at there exists a unique, translation invariant, linear functional on WAP(W); we designate this functional U00. If F = {m 6 F ix : ||m|| = 1}, then it turns out th at the restriction of the elements in F to W A P (R) is U ^. From this and other observations (for complete details, see Chapter 9), we obtain a represen­ tation of VFAP°(R), using the same ideas that we used to obtain a characterization of B U C Q(R). R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 6 C hapter 2 2.1 Vector lattice term inology D efinitions The following definitions were taken from [LuZ] and [Zan]. z) A vector lattice S is called Archimedean if (V 0 < u, v G S) and (Vrz, n = 1 , 2 , . . . ) , nv < u =>v = 0. ii) a) The indexed subset {xa : a G {a}} in S is said to be directed upwards, if (Va, /? G {a}) (3aro € {<*}) such that x a < x aa and xp < xao. This is denoted by x a f. If x a | and x = supa x a, then we write x a f x. 6) 5 has order continuous norm p if 0 < xa | x =>• p(xa —x) —►0. c) S has (r-order continuous norm if the definition in (b) holds for increasing sequences. Hi) a) S is Dedekind complete if every nonempty subset which is bounded from above (below) has a supremum (infimum). 6) 5 is < j -Dedekind complete if every nonempty at most countable subset which is bounded from above has a supremum. c) S is order separable if every nonempty subset D possessing a supremum d con­ tains an at most countable subset which possesses d as a supremum. d) S is super Dedekind complete if it is Dedekind complete and order separable. iv) An order ideal U in S is a linear subspace with the property th at if x G U and y G S, then \y\ < |x| implies that y G U. Observe th at every order ideal U in S has the property : x G U => |x| G U. R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 7 v ) a) A band is an order ideal B such that if u = sup{D : D C B } exists, then u E B . Equivalently, B is a band if and only if every positive, upward directed system (ur) C B has the property that if u = supr (tir ) exists in S, then u € 5 . 6) Dd = { / E S : V<7 E D ,in f(|/|, |<7|) = 0}. Dd is called the disjoint complement of the set D. c) Any band B in S such that S = B © B d, is called a projection band. d) The band generated by a set D C S, which we denote {£>}, is the intersection of all bands th at contain D. 2.2 Properties of vector lattices D efinition 2.1. For the remainder of this chapter, (X , p) is a Banach lattice ( i.e., X is a Banach space with the property that ifx, y E X and |x| < \y\, then ||x|| < ||y ||/ The indexed subset {ur : r E { r } } in X is said to be p-Cauchy i f (ie > 0) (Btq E { r } ) such that p(uTl —Ur2) < e whenever uTl > u ^ and D efinition 2.2. > uTo (fZanJ ). ( X , p ) satisfies the p-Cauchy condition i f the following equivalent conditions hold: i) Every order bounded increasing sequence in X is p-Cauchy. ii) Every order bounded upwards directed system in X is p-Cauchy. T heorem 2.3. The following conditions on (X, p) are equivalent: i) p is order continuous. ii) p is a-order continuous and X is Dedekind a-complete. In fact, X is super Dedekind complete if these equivalent conditions hold (fZanJ). T heorem 2.4. The norm p is order continuous if and only i f p is <r-order continuous and X satisfies the p-Cauchy condition ([ZanJ). T heorem 2.5. I f S is a Dedekind complete vector lattice, then the following hold: i) I f A \ and A<i are bands and A \ -L A^ , then Ax © A± is a band in S. ii) Every band B in S is a projection band, i.e., S = B © B d ([LuZ] ). R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 8 It follows from Theorem 2.3 and the definition of super Dedekind completeness, th at if X has order continuous norm p , then X is Dedekind complete. Thus, by Theorem 2.5, if B is a band in X , then X = B ® B d. (2.1) Let BUC*(R) be the Banach dual of BU C(R). We claim th at the Banach lattice (BUC*(W), || • ||*) has order continuous norm. To see this, first observe that the space (BUC(R), || • ||) is an abstract M - space, i.e., BUC(R) is a Banach lattice with the property th at ||/ + p|| = m ax{||/||, ||p||} whenever inf(/, g) = 0. T hat BUC(R) is an M-space follows from the expressions 2 sup(/, g) = ( / + g) 4- \ f —g | and 2 inf(/, g) = ( f + g) — \ f — g\, and from the fact that BUC(WL) is equipped with the supremum norm. Therefore, BUC*(R) is an abstract L-space ([Kal]), which imphes th at the norm on BUC*(R) has the following property: (V 0 < p. , u € BUC*(W)), II/* + H I* = IH U + I M I - (2-2) We shall show BUC* (R) has cr-order continuous norm and satisfies the p-Cauchy condition; we can then conclude th at it has order continuous norm (Theorem 2.4). To show (BUC*(JR.), || * ||*) satisfies the p-Cauchy condition (with p = || - ||*), suppose 0 < (f>n, <t>£ BUC*(R) such th at 0 < <pn t 0; or equivalently, given 0 < / € B U C (R), suppose 0 < 4>n(f ) T 4>(f)- Then <t>n is a norm-Cauchy sequence. Indeed, since BUC*(M) is a Banach lattice, then 0 < ||0n||* T ||0||«- This imphes th at the sequence (||0n||») is an increasing, bounded sequence of real numbers; thus, limn(||0n||*) exists. If n > m , then ||0 n ||. = 110m + 0 n - 0 m ||. = ||0 m ||« + ||0 n ~ 0 m ||. , by relation (2.2), so (0n) is Cauchy and the p-Cauchy condition is satisfied in BUC*(WL). To show that BUC*(R) is cr-order continuous, note th at since (0n) is a Cauchy R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 9 sequence and BU C*(R) is a Banach space, then for some 0 < 0o E BUC*(HL), lim ||0n 0oII*= 0. T I— +OO Thus, for any 0 < / € L, the following inequality holds: K ( /) - 0 o ( /) |< ||0 n - 0 o lU II/||- But this imphes th a t 0 < 0n t 0o, and since we also have 0 < 0„ t 0, then 0 = 0oTherefore, || - 1|» is cr-order continuous. Let G = {T(t) : t 6 R} be the translation Co-group defined on BUC(M). We estabhsh im portant properties of G (and its dual G*) th at we will find useful in later chapters. D efinition 2.6. Let V and W be vector lattices. i) A linear map ft of V onto W is called a lattice isomorphism, if it is a bijection and if V/, g E V , inf(/, g) = 0 => mf(ftf,ftg) = 0. Equivalently, tt is a lattice isomorphism i f it is a bijection and i f ftsup(f,g) = sup(7r/, Ttg). ii) -k is said to be positive i f ft f > 0 whenever f > 0 ([LuZ]). It fohows from Definition 2.6, that every lattice homomorphism is positive. For if / > 0, then we can write 0 = inf(/, 0), which imphes th at 0 = inf(rr/, 0) or th at ftf > 0 . T h eo re m 2.7. The 1-1 positive linear mapping ft of V onto W is a lattice isomor­ phism if and only ifft~ l is positive ([LuZ]). Recall that for each t E R , the operator T(t) E G is defined on BUC(M) by translation, i.e., T ( t ) f ( x ) = f { x + t), x € R. Then by Definition 2.6, G is a family of lattice isomorphisms. Therefore, so is the inverse family { T(t ) ~1}. And since the dual of any positive operator is itself positive, R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 10 then it follows by (T(t)-'r = (r*(i))-1, and by Theorem. 2.7, that the dual family G* is also a family of lattice isomorphisms. We end this chapter by stating the following result due to [Pag]. T h eo re m 2.8. Suppose E is a Banach lattice with the property that its Banach dual, E*. has order continuous norm p. I f H = {U(t) : t > 0} is a Co-semigroup of positive linear operators on E , then the sun-dual E ° o f E is a band in E*. Since B U C*(K.) has order continuous norm || • [|„ and since the group G is a family of lattice isomorphisms, then by Theorem 2.8, BUC°(JRL) is a bandin BUC*(R). Thus, by equation (2.1): BUC*(R) = B U C °{R ) ® (££/'C0 (R))'f. R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . (2.3) 11 C hapter 3 A characterization o f th e Banach dual o f BUC(R) To obtain a characterization of BUC*(R) we shall employ the Gelfand theory of commutative Banach algebras. Let A be a commutative Banach algebra with unit e (i.e., e{a) = 1, a € A), and let Ad be the set of all maximal ideals in A. Recall that if M € Ad, then the Banach algebra A /M is a division algebra that equals C, the set of all complex numbers ([Ru]). If a E A, we denote by a the complex function defined on M by d(M ) = a{M ) = a + M, and we put A = {a : a 6 A}. Clearly, A C C(Ad), where C{Ad) is the space of all continuous complex functions on Ad. The mapping a ■—►a is called the Gelfand map­ ping, and the weakest topology on Ad relative to which every function a is continuous is called the Gelfand topology ([Ru]). We call the topological space Ad the maximal ideal space. Recall that a multiplicative functional h on A is a nonzero element in the dual A* of A, which satisfies: h(ab) — h(a) h(b) and h{e) = 1. It is well-known th at there is a 1-1 correspondence between the elements of Ad and the set of all multiplicative functionals on A. This correspondence enables us to regard Ad as a. subset of A*. Moreover, since each multiplicative functional has norm 1, then Ad is also a subset of the unit ball £?i in A*. Recall that B \ is a compact Hausdorff space with respect to the weak*-topology, which is the weak topology generated by all functions Fa defined on B \ by Fa(h) = h(a). Observe th a t Fa restricted to Ad is R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 12 precisely a; th at is, Fa(h) = h(a) = a(h) (/i € -M). Then the topology which M. has as a subset of S i is exactly its Gelfand topology, so we can regard A i as a subspace of Si- In fact, it is a weak*-closed subspace of SiT h eo re m 3.1. A 4 is a compact ffausdorff space ([Ru])). D efinition 3.2. Let A be a Banach algebra urith unit. 1. A is called a B*~ algebra i f it has an involution , i.e., i f there exists a mapping o f A into itself with the following properties: • (a + 6)* = a* + b* • (otb)* = ab* (a E C) • (ab)* = b*a* • a** = a • lla *a ll = IM I2 (note that the last property implies that ||a*|| = ||a||,). 2. i f A ' is another B*-algebra and i f p is an isomorphism o f A onto A ', then p is called a * -isomorphism if p(a*) = (p(a))*. Note, if X is any topological space, then C( X) , the space of continuous complex functions on X , is a commutative S*-algebra with an involution defined by complex conjugation: f *(x) = f ( x ) , f 6 C{X). T h eo re m 3.3. (Gelfand-Naimark). I f A is a commutative B*-algebra, then the Gelfand mapping a*-* a is an isometric *-isomorphism g o f A onto C(.M). Thus, JS) = (a)' = g(a') = ( ? ) , R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . (3.1) 13 or equivalently, (fe(a)) = h(am), aeA, h e M ([GelV]). One can easily verify th at under the pointwise operations of addition, scalar mul­ tiplication, and multiplication, BUC(W) equipped with sup-norm, is a commutative Banach algebra with unit e. The mapping f f , where / G BUC{R) and / is the complex conjugate of f, is easily seen to be an involution that satisfies Definition 3.2: thus, BUC(M) is a 5 ‘-algebra (in fact, it is a C*-algebra). Let fi be the maximal ideal space of BUC(M.). By Theorem 3.3, BUC(M) is isometrically ^-isomorphic to C(Q). Consequently, if / G BUC(R) is real-valued, then by equation (3.1), (f) =(7r)=(7)=f, so th at the conjugate of / is equal to / , i.e., / is real-valued. Hence, we identify the real-valued elements in BUC{R) with the real-valued elements in C(Cl). We can now identify B U C*(R) with the space C*(f2); recall that C*(fl) is identified with the space M(Cl), which consists of all bounded, regular, complex Borel measures on f2, with norm equal to total variation (by a complex Borel measure we mean a complex-valued, countably additive function defined on the <r -algebra of Borel subsets). Remark 3.4- Note that the space fi contains R as a weak*-dense subspace. The embedding of R into Cl is defined by the point-evaluation map x ►-» gx of R into BUC*(R), where gx denotes the linear function gx( f ) = f{x), ( / € B U C {R)). Q can now be identified as the weak*-closure of R in the weak*-compact unit ball of B£/C*(R) (see Chapter 6 for complete details). R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 14 C hapter 4 Translation invariant linear functionals on BUC(R) Let Q be the maximal ideal space of BUC(R). If / E Co(R) and g E BUC(M), then f g E Co(R) ; that is, Cb(R) is a (norm-closed) algebraic ideal in BUC(M). Consequently, 3 0 < h E Q, with ||h|| = 1 , such th a t Co(R) C kerh ([Ru]). Stated precisely, if C„HR) = { S 6 BUCT( R ) : s(Co(R)) = 0}, then h G Cq-(R). Recall th at the dual Cq (R) of Co(R) is the space M(R), which consists of all bounded, regular, complex Borel measures on R, with norm equal to total variation over R. We assert th at h can not be represented by any nonzero A E M (R). For if this representation were possible, then by the Riesz representation theorem, Kf) = f fd\, JR f e C0(R), and |A|(R) = ||/i||. But Co(R) C kerh, so the fact th at h ( f ) = f fd X = 0, JR / G C0(R), and the fact th at A is a countably additive set function, imply 0 = IA| (R) = \\h\\ = 1 . Evidently, associated with this h G BUC*(R), there is a measure p G C*(Q) with the property th a t supp(fi) C (Q \ R) (supp(p) is the support of p). R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 15 Lemma 4 .1 . Cq-(R) is a band in BUC*(R), so BUC*(R) = Cg-(R) © (C(f (R))d. In fact, (C'o'(R))rf = C0*(R) (all equalities are isometric isomorphisms); thus, B U C \ R) = Cq-(R) © Cq(R). (4.1) Proof. In C hapter 2 we determined th at BUC*(R) has order continuous norm; thus. BUC*{R) is Dedekind complete. It follows that every norm-closed order ideal in BUC*(R) is a band ([And]); consequently, BUC*{R) = C0X(R) © (C0x (R))d (Theorem 2.5). (4.2) If M is a closed subspace of a Banach space E , then the dual M* of M is given by M* = £ 7 ( M ) X ([Ru]). Thus, C0*(R) = BUC*(R)/(C q (R)), s o by (4.2), (C0x (R))rf = Q (R ). □ D efinition 4.2. Let 0 ^ 0 € BUC*(R) and let G* = (T*(t) : t 6 R} be the dual family of the translation group G defined on BU C(R). Then <f>is said to be translation invariant i f Vt e R, T*{t)(f) = 0 . Remark 4-3. We determined at the end of Chapter 2, that G and G* are families of lattice isomorphisms. If 0 is translation invariant, then it follows from T*(t)(*+) = 7” (t)(sup(*, 0)) = sup<r («)*, 0) = sup(& 0) = $+, that the positive part of 0 , the negative part of 0 , and the variation of 0 are also translation invariant. R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 16 Lem m a 4.4. If 0 ^ 0 E BUC*(M.) is translation invariant, then 0(Co(R)) = 0. Proof. Since (7C(R), the space of continuous, complex functions with compact sup­ port, is dense in Co(R), it suffices to show that (f> vanishes on V([a,b]), the space of continuous, complex functions which vanish outside of a closed, bounded interval [a, b]. We first establish the result for positive, translation invariant linear functionals; the result for complex (f>will then follow from Remark 4.3 and the inequality W /)I< IW I). f e B U C ( R) ([Zan])- Let 0 < / E V\a, b] and (p > 0. Choose c > (b —a) so th at the translates /( - -|- kc), k = 0 , 1, 2, . . . , n, have disjoint support. Then n 0 n n < ( £ / ( ■ + *<:).■» = (Z,T (kc)f,4> ) = fc=0 fc=0 = (rc + l ) ( / » ; fc=0 thus, 0 < (n + l)</>(/) < ||0 || sup f ( x ) . Dividing the above inequality by (n -1- 1) and letting n —* oo, shows that <p(f) = 0. For real-valued f E V[a, 6 ], write: / = f +—f ~ • Observe th at 0 < / +, f ~ E V[a, 6 ] (since V[a,b] is a vector lattice). Now repeat the argiunent above, with f + and f~ taking the place of / . T heorem 4.5. □ (Markov-Kakutani) Let K be a nonempty, compact convex subset of a topological vector space X , and let F be a commuting family o f continuous linear mappings defined on X such that Y T E T , T ( K ) C K . Then 3p (=. K such that T (p ) = p for each T E T ([Mar], [Ka2]). R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 17 Lem m a 4.6. There exist nonzero, translation invariant <j>6 BUC*{R). Proof. Consider the set F = {<f>6 B U C *{R ): 0(Co(R)) = 0, <f>{e) = 1 , ||^|| = 1 }. Then F is a nonempty, convex, weak*-compact subset of the unit ball in BUC*(R). Furthermore, if 0 € F and f € BUC(R), then (f,T*{t)<f>) = (T { t ) f , <f>) shows, . ( V / € C 0 (R)) (VfceR), T*(t)<(>{f) = 0, • r*(4)0(e) = 1 and ||T*(<)0|| = 1. Thus, V£ 6 R: T*{t){F) C F . And since the dual family G* = : t £ R} is a commuting family of weak*-continuous linear operators, then we can apply Theorem 4.5, with F = G*, to conclude that F has a fixed point. □ We shall denote F ix as the collection of all translation invariant <p 6 BUC*{R). Observe th at F ix is a norm-closed, vector sublattice of BUC*{R), and BUCe {R) 3 F ir . Lem m a 4.7. Suppose {F ix } is the band generated by the set F ix and M = {F ix } © C®(R). Then 1. M is a direct sum, 2. M is a band in BU C*(R). Thus, BUC*(R) = M © M d and B U C Q{R) = M © (M d n BU C°{R)). Proof. By Lemma 4.4 , Go"(R) 3 {Fix); and since Cq~(R) is a band (Lemma 4.1), then Gq'(R) 3 {F ix }. But (C,f (R))rf = Gq(R) 3 Cq)(R), so M is a direct srnn. R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 18 To prove part (2), we first define an isometric isomorphic mapping of C®(K.) into BUC*(R). We then show th at the image of CJf (R) under this mapping is a band in B U C \R ) and B U C °(R ); hence, C?(R) is a band in BUC*{R) and B U C Q(R). Now we argue as follows: Since {F ix} and C^f(R) are disjoint bands in BUC*(R) by part ( 1 ), then M = CJf(R) © {F ix } is a band in BUC*(R) (Theorem 2.5). Therefore, BUC*(R) = M © M d and B U C °(R ) = M © {Md fl B U C °{R )). We remind the reader th at C®(R) = Z/1 (R.) ([Phi]). Hence, if fi E C®(R), then there exists a unique / E L 1 (R) such th at dfi = / dx (Radon-Nikodym theorem). Define fj. p., where p(g) = j fg d tx , g £ BU C (R). Jr Since / is unique, then the map 0 is well-defined. A trivial verification shows that p and 0 are bounded and linear. 0 is also 1 - 1 ; to see this, suppose || - 1|, is the norm on BUC*{R) and || • || 0 is the norm on Cq(R). If fiXl /j 2 £ C®(R)» then (fix —^ 2 ) £ C'o’W ? so 3h £ L L(R) such th at d(fix ~ P2) = h dx. We have ||/* i - P2II0 = su p | p€C0 (R) p d(fi 1 - ^ 2) | < JR llpll<l sup ae B U C (R ) IMI< 1 I JR g d(fix - ^ 2 ) I = ||Ai - P2W* = sup I / gh dx\ < 96BUC(R) J R llffll<l [ \h\ dx = \fix - ^ 2 ](R) = ||a*i ~ JR. This shows \\fix —/x2||o = ||Ai —A2 II* ; it also shows 0 is 1 - 1 . R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 1[0 * 19 We assert that 0 (C ?(R )) C BU Ce (R) C B17C*(R). Indeed, (9 ,T *(t)p) = (T (t)g,p) = [ g(x + t) f{ x ) dx = JRf g{x) f { x - t ) dx. JR Hence, d(T*(t)p) = f ( x —t) dx, so ||T*(t)/2 - a*|U = SUP I / ( / ( * ) ~ / ( * ~ *)) 0(*) <te\ sgBUC(R) IMI< 1 JR < f \ f{ x ) ~ f { x - t ) \ d x = \\T*{t)g,-fj\\0. JR Letting £ —* 0, we see th at p. G BU CQ(R) and that 0 is an isometric isomorphism. Since 0(C®(R) is the isometric isomorphic image of the Banach space C q)(K.). it is a norm-closed linear subspace. We claim that it is also an order ideal; that is. 0(C®(K)) has the following property: (a) A € BUC*{R), v 6 e (C 0Q(R)), and 0 < |A| < \v\=> A G 0 (C ? (R )). The first step is to show (a) holds for positive elements in B U C*(R) and ©(C®(R.)): we assume 0 < A < u, and show A G 0 (C ^ (R )). Note th at 0 < \ < u = > ( V 0 < g e BUC{R)), 0 < A(^) < u(g). Since v is identified with an element v G C®(IR.) which has the property that du = f dx R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 20 for some unique / E T L(R), then 0 < A(g) < v(g) = f g f dx. JR (4.3) If we restrict A to Co(R), then by the Riesz representation theorem, there is a positive, bounded, regular Borel measure r whose representation as a functional on Co(R) is A(p) = f p d r , p E Co(R). JR The inequality in (4.3), together with the above representation of A, imply that 0 ^ A(p) = f p d r < J jR p f dx, p E (7o(R). JR It follows th at r < i / . Thus, dr = h dv for some h € L l (v), so dr = h du = h f dx < dv. This last inequality implies that 0 < h < 1 ; thus, h f E T 1 (R). If we define dp = h f dx, then it is clear p E Cg^R), and ©(A4) ^ ) = K g ) = [ 9 dp = f g h f dx = A(g), g E B U C {R). Jr J r. Therefore, A E ©(Cq^R)). The second step is to show (a) holds for complex-valued elements in BUC*(R) and @(C®(R)). Since 0 < A+ < \v\ (similarly for A- ), we can assume 0 < A < \v\. W ith this assumption in place, it suffices to show (b) V e 0(C®(R)) =* \v\ € ©(C0° (R )); we can then apply the above argument to conclude that A E ©(C®(R)). Observe th at v E C®(R) \v\ E C®(R). Indeed, Cg(R) has order continuous norm, and Vt E R, T ( t ) is a positive operator (Chapter 2); it follows by Theorem 2.8 that C®(R) is a band in Cg(R). Thus, \v\ E C®(R). R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 21 We claim that 0 is a lattice isomorphism, i.e., 0 has the following property: M = 0(H) = [0WI = |i>|, which is precisely the property we need to show that (b) holds. To show the above equality, we use the following result of F. Riesz ([Zan]): (Vr 6 BUC*(M)) (V 0 < <7 € BUC(H&)), \r\(g) = sup {|r(h )|}. heBcrc(tt) (4.4) W<3 We will use (4.4) to show sup \ f h f dx\ = \u\(g) = \u\{g) = f g \f\ dx. heBUCQSt) J R Jr \h \< g Suppose gn is a sequence of functions with compact support, and 0 < gn t <7 - Then sup =BUC(R) h€S£/C(R) | / h f dx\ < / g \f\ dx = lim I gn\f\ dx, JR JR n —oo \h\<9 and lim ( f gn |/ | dx) = lim ( | i % n ) ) - n —*oo n —*oo However, by (4.4), lim (|£|( 0 n)) = sup( sup n —*°° n heBCSC(R) h f d x |) < | JR sup hG B U C ( R) | / h f d x |, JR so |0 (i/)| = 0 (|^ |), i-e., 0 is a lattice isomorphism. Consequently, |i/| E 0(C®(R.)), condition (b) is satisfied, and 0(C®(R)) is a norm-closed order ideal in BUC*($l) and B U C °{R). R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . □ 22 Remark 4 . 8 . Lemma 4.7 plays a crucial role in our characterization of BUC°(SL). While the representation of BUCe (SL) as B U C °( R) = M © (M d n B U C °{ R)) is im portant, it is still a preliminary characterization. In Chapter 8 we show M d n B U C Q(R) = {0}. In part ( 1 ) of Lemma 4.7, we use the fact that Cq(R) = (Q f (R))d; this implies that C 0*(R) is a band in BUC*(SL), since (Co^R))4* is a band. Actually, to show that Cq(R) is a band, we use an approach similar to the method used to embed C®(R) in BUC*{R), but with a slight variation. Instead of using the fact th at C®(R) = L l (R), we make use of the characterization of Cq(R) provided by the Riesz representation theorem . We end this chapter by offering the reader a more precise description of the struc­ ture of {F ix }. Let AFiX denote the order ideal generated by the set F ix . That is, v A Fix if for suitable m t- € F ix and a* € R, we have M < |a i m i | + - . - + K r a n | i[L u Z \). Note th a t A Fix is the smallest ideal which contains F ix . Clearly, u 6 Aptx if for some m. € F ix , u <§; m; moreover, {Apix} = {F ix }. Also, if 0 < fi € {Afix}, then /x = supa ua for some upward directed system (ua : a € {a}) contained in A Fix ([LuZ]). It follows th at if 0 < fi £ {Fix}, then ft = sup va = sup f a dm a, a a where m a € F ix and f a € L l(ma). R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 23 C hapter 5 Exam ples o f elem ents in F ix In Lemma 4.6 we proved th at nonzero, translation invariant linear functionals exist in BUC*{R), and we designated the collection of all such functionals F ix. In this chapter we construct examples of elements in F ix. We proceed as follows: (Vx, y £ R) (V/ e BUC(M.)), consider W ) l = - f f ( x + s )d s . y Jo (5.1) It is obvious that Uy is linear on BUC{R). Moreover, m m )\ < g [ l/(* + *)l ds < ll/ll, (5.2) shows th at ||t/y|| < 1 . If e is the unit in BUC(M), then equation (5.1) shows that \Uye(x)\ = 1,soin fact [|C/^|| = 1. Hence, Uy is a bounded, positive linear functional on B U C {R ). Let x be a fixed element in R. We claim the following: If we regard Uy( f ( x )) as a function of y, then 1- Uy(f{x )) e B U C (R ) if / e BUC{R), and 2- Uy(f(x )) 6 C 0 (R) if f e Cb(R). To show (1) and (2), we first transform equation (5.1) by making the substitution s = u y, so that Uy{ f{x ))= f f ( x + u y )d u . R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . (5.3) 24 Let / be an element in BC/C(R). Then (Ve > 0) (3£ > 0) such th at for every y, y' E R with \y — y '\< 8 , we have \f{y) —/(y 7)! < e. If 0 < u < 1, then |(x -+- uy) - (x + uy')| = |u| |y - y'\ < [y - y'\ < 8 , so by the uniform continuity of / , we have |/ ( x + uy) - /( x -I- uy') | < e. It follows th at Uy( f( x ) ) E BI7C(R) (note that Uy(f(x )) is jointly uniformly continu­ ous in x and y). If / 6 Co(R), then we claim that lim / / ( x + uy) du = 0 . °°y o Suppose this were not the case; then (3yn € R) (3e > 0) such that lim yn = oo => | [ / ( x + uyn) du\ > e. n~*°° Jo (5.4) However, since / E Co(R), then / ( x + uyn) —> 0 as yn —*■ oo, so by dominated convergence, lim / o / ( x + uy„) du = 0, which is a contradiction of (5.4). Therefore, Umy^ 00 U y(f(x)) = 0, so Uy( f(x )), as a function of y, is an element in Co(R). In Chapter 3 we showed th at BU C (R) is *- isometrically isomorphic to the func­ tion space C(fl). Thus, if / E B t/C (R ), then we can continuously extend Uy(f(x )), as a function of y, to Q. Let C/Q( /( x ) ) , a E Q, denote the extension. T hen there R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 25 exists a net (yT) C R such that Ua{f{x)) = lim {/„, (/(* )). Clearly, Ua is linear on BUC(1SL). And since UVt > 0 and UyTe(x) = 1 , Vr, then Ua > 0 and Ua(e(x)) = 1 . It follows that ||f/a || = 1. Lem m a 5.1. Let f G C'o(R) and let f be its corresponding function in C(Q). Then V a 6 ( f i \ R), / ( a ) = 0. Proof. By hypothesis, there exists a net (f{) C R such th at f ( a ) = lim; f( ti) . Hence, (Ve > 0) (31q G {/}), such that l/(a)-/ft)l<§, *><o- Since / G Co(R), then there exists a compact set K such that l/(*)l<§, Let A = {I : I > Iq} and B — {I : ti £ K } . Observe that both B and A fl B are nonempty. Indeed, compact sets in R with respect to the usual topology, are also compact in R with respect to the weak*- topology (recall th at fi D R as a weak*-dense subspace). It follows th at K is compact in 12, which means that the complement of K . Q \ K , is an open set which contains S2\R; that is, Q \ K is an open neighborhood which contains a. And since t[ a , then there are plenty of I € {1} for which tL £ K . Ergo, B and A n B are nonempty. Let d G A fl B; then \f(a )\ < |/( o ) - f ( t d)\ + \ f( td)\ < § + § = * □ R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 26 We determined in (2) th at Uy( f ( x )) G Cq(R) if / G Co(R). Consequently, we can apply Lemma 5.1 to the extension UQ(f(x)): & (/(* )) = 0, / 6 Co(R), a e n \ R . We claim th a t 17a is translation invariant, i.e., VCgR, T*(t)Ua = Ua. Indeed, (Vt G R) (V/ G BU C {R)), \U ,{T {t)f(z)) - U y(f(x))\ = |£ /„ [r(i)/(l) - / M ] | = Letting \y\ —*• oo shows th at Uy[T (t)f(x ) —f(x)] G Co(R); thus, Ua\T {t)f{x) —/(a^)] = 0, by Lemma 5.1. We have (/, Ua) = (T {t)f, Ua) = (f,T*{t)U a), which shows Ua G F ix. In C hapter 9 we characterize the sun-dual of the space of weakly almost periodic functions on R (W A P(R )). The ideas of this chapter are central to th at characteri­ zation. R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 27 C hapter 6 Q , adm its a sem itop ological sem igroup structure Let 12 be the maximal ideal space of BU C (R). In this chapter we focus on the map cp : (x, t) — ►(x + t) o f R x R - ^ R . It is clear that ^ is a jointly continuous, binary operation on R. It is also clear that this operation renders the additive group of R a topological group. Ideally, we would like to extend ip over 12 x 12 so that it is jointly continuous, since this would eliminate the necessity of this chapter, which would in turn simplify the proof of our main theorem (Theorem A). As we shall see presently, there exists a binary operation ip on 12 th at is both commutative and associative, which renders 12 an ‘additive’ semigroup; the additive group of R is a subgroup of 12 with respect to this ‘additive semigroup structure.’ Unfortunately, 12 is neither a topological group nor a topological semigroup. However, it is a semitopological semigroup, i.e., 12 is a Hausdorff semigroup equipped with a binary operation that is only continuous in each variable separately. Theorem A can still be proved, notwithstanding the lack of joint continuity of <p over 12 x 12 , by employing the weaker condition that ip be separately continuous in each variable. How can we extend ip so that it is even separately continuous? We accomplish this in the following manner. First, we define a uniformity ZYr on R in which the completion of (R,ZYr) is (12, ZYr- ) (ZYr. will be defined directly). Having done this, we provide a definition of uniform continuity with respect to uniform spaces, and point out that for fixed t 6 R, ip{-, t) is uniformly continuous on R according to this definition. We then use the fact that 12 D R as a weak*-dense subspace, in order to extend (p(-,t) to a unique, uniformly continuous function defined on 12. Finally, for R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 28 fixed a € fi, we show th at <p(a, •) is uniformly continuous on R; ergo, <p(a, •) extends continuously to Q. D e fin itio n 6 . 1 . A uniformity fo r a set X is a nonempty family S o f subsets o f X x X such that • i f A € S , then A D {(x, x) : x E X } • i f A e S , then {(x ,y ) : (y,x) £ X } € S • i f A E .S, then {(x, z) : 3y such that (x, y), (y, z) E A } € S • i f A ,B € S , then ( A f l B ) 6 S • i f A e S and A C B C {X x X ), then B € S . Let B denote the unit ball in BUC*(R). We define a uniformity on B, which in turn induces a uniformity on R: (V/ € BU C{R)) (Ve > 0), define U u = { ( ^ x ) 6 B x B : \(f,ip) - ( f,x ) \ <e}Then the sets = { (^ ,x ) : max | (fi,ij}} ~ ( fu x ) I < e}> f i t B U C ( R), form a uniformity U on B . We refer to U as the weak*-uniformity on B. The topology Tu of the uniformity U or the uniform topology, is the weak*-topology on B. R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 29 Note th at via the embedding x i—►Sx of R into B given by <W ) = / ( * ) , f e B U C ( R ), U induces a uniformity on R; we shall designate this uniformity Us- T hat is, V E Us if (Ve > 0) (V<7i E BUC{R)), V = {(Sx,6y) E B x B : max | (&-,£*) - (&,<$„)I < e} = l< t < m {(x,y) E R X R : max \gi(x) - # (y )| < e}. Recall that each Hausdorff uniform space is uniformly isomorphic to a dense sub­ space of a complete Hausdorff uniform space ([Kel]). Accordingly, define (R*,£/r.) as the completion of (R, Us) ■ Since B is compact with respect to the weak*-topology on BUC*{R), then the weak*-closure of (R,ZYr) in B is compact, i.e., (R*,£/r-) is compact. It follows im m ed ia te ly th at since 0, D R as a weak*-dense subspace, then Q = R*; moreover, Us- consists of fin ite intersections of sets of the form: Wh = {(a, /?) 6 fl x f l : |/ ( a ) - /(/J)| < e}. D efinition. 6.2. Let {X\,U\) and {X 2 M 2) be uniform spaces and let g : (X i,U i) —►(X 2,U 2)Then g is uniformly continuous i f (VW E Uf) (3V € U\) such that (x,y) E V => (g(x),g(y)) E W. T h e o re m 6.3. Let g be a function whose domain is a subset A of a uniform space (X \,U i) and whose values lie in a complete Hausdorff uniform space (X 2 , W2 )- I f g is uniformly continuous on A, then there is a unique uniformly continuous extension g of g whose domain is the closure of A ([Kel]). R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 30 One can easily verify the following: 1. For fixed f 6 1 , the map ¥?(-,£) : (R, £ / r ) —* ( R , £ / r ) given by <p(x, t) = {x 4 - 1), is uniformly continuous according to Definition 6.2. 2. Let / 6 BUC(1BL) and let IAm be the metric uniformity on the complex numbers C, i.e., V 6 Um if V has the form : V = {(rr, y) € C x C : \x —y\ < e}. T hen / : (R.,£/r ) —*• satisfies Definition 6.2. According to properties ( 1 ) and (2) above, we may apply Theorem 6.3 to the map <£>(-, t) and to each function / € BUC{R), to claim that each has a unique, continuous extension to all of Cl. We also claim th at if a 6 (fl \ R) and t € R, then cp(a, t ) £ (fi \ R). Suppose this is not the case, i.e., suppose <p(a, t) = b € R. Then b = <^(a, £) = lim(ui + 1), where (u i) is a net in R such that U[ ™ a. It follows that a = lim; {u{) = (b — t) 6 l . which contradicts the fact that a € (D \ R). L e m m a 6.4. The map ip has the following properties: i ) V q G f2 , <p(a, •) : (R, U&) —* (Q,Wr.) satisfies Definition 6.2. ii) Let f be the continuous extension to Q o f any f 6 BU C{R). Then as a function o f t e R, f(<p(c*,t)) e BU C{R). R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 31 Therefore, (p(a,t) and f(<p(a,t)) extend continuously to fl. Hi) Va, /?, u € fl, (а) cp(a, 0 ) = a , (б ) <p(a,0) =<p(0,a), (c) <p(a, <?(/?,a/)) = <p(<p(a,0 ),u;); £/zzzs, (fl, (p) is an abelian semigroup. Proof. We will prove (z) and (zz) simultaneously. We must show (VW e ZYr-) (3V e ZYr ) such that (t,*7) € V =» (^(a,£),(/?(a,t')) E W. Recall th a t each W 6 ZYr- is of the form: W = {(a,/?) E fl x fl : max |/i(a ) - /*(/?)| < e}, l< i< p where each is the extension of an / t- € BZ7C(R). Note th a t in each variable separately, the function f ( x -ft) is a bounded, uniformly continuous function on R.. Consequently, for some real-valued net (u[), we have Va e fl, M<f(a, £)) = lim fi{ut - f 1). Let (D, >:) be the directed set of the net (ui). Then f(u [ 4-1) E BU C (R ), for every I € D. Thus, (Ve > 0) (3d € D) (35 > 0) such th at \t.- t'\< 6 = > \fi( u d + £) - fi(u d -F t ' ) | < 1 < z < p. The set V = {(x, y) G R x R : max \fi(ud + x ) - fi(ud + y)| < J } , i<i<p R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 32 is clearly a set in U r . It follows that (t, t') E V => \fi{<p{a, £)) - /*(<?(«, f ) ) | < e, 1 < i < p. Thus, f ) ) E W and f{<p(a, t)) E BU C{R). This proves (i) and (ii). To show (iii)a), let (t[) be a real-valued net such that t.[ ^ a. Then <p(a, 0 ) = limv?(£f, 0 ) = lim(fy -f 0 ) = a; this proves (a).To prove (6 ), we will show th at if / € BUC(R) and / isthe unique extension of / to all of Cl, then a )). Now we suppose ip(a,f3) 7^ ¥?(/?, a ). Since fl is a Hausdorff space, 3g E C(Q) such th a t g(<p(a,0)) ^ g(<p((3,a)). Let h be the restriction of g to R. Since the restriction of Ur- to R is Ur, and since Ur is the weakest uniformity on R with respect to which all the / E B U C (R ) axe uniformly continuous in the sense of Definition 6.2, then h E B U C (R ). But this implies h(<p(a,/?)) = h(<p({3,a))- moreover, h = g. Thus, we obtain a contradiction of the hypothesis on g. Let / E BU C(R); then Vx, £ E R, f ( x + t) = f ( t + x). In each variable separately, f ( x + 1) E B U C (R ), so we can extend / to Cl: f(<p(x,a)) = lim f ( x + t[) = lim f ( t l + x )= f(< p (a ,x )). By part (ii), Vx E R, f(tp (a ,x)) E BUC(R); hence, we can extend f(<p(a,x)) to Cl. It follows th at f(<p(a,/3)) = f(<p(0, a)). This shows (6 ); a similar argument shows (c). Let E = C(f2). Consider the map a □ T (a ) of Cl —►C b ( E , E ) , where C b ( E ,E ) is the Banach algebra consisting of all bounded, linear operators from E into E . For R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 33 each a £ fl, let T (a) : E —►E be given by T (« )/M = f(<p(a,uj)), uj € fl. (6.1) We end this chapter by determining certain properties of the family (T (a ) : a e fl} and the dual family (X*(a) : a € fl}, th at we will use in the proof of our main theorem (Theorem A). L em m a 6.5. Let H be the family of operators that satisfies equation (6.1). Then 1. V a , p z Q , T(a)T(P) = T(0)T(a) = T(<p(a,P)). Therefore, H is a commuting family. 2. Va e ft, Proof. ||X(a)|| = 1. By equation (6 . 1 ), (Va,/? 6 Cl) (V/ 6 E ) , T(a)T(j3)f(u) = r ( a ) ( /( v (0,u/))) = f(<p(a,<p(p,uj))). By part (m)c) of Lemma 6.4, /« « ,# ,» ) ) ) = T( fi yr(a)nu) and ! ■ « « , « ) =■/(*>(*>(<«,/*),«)) - r ( a ) /( v (/},u>)) = T(a)T(J3)fM, uj € { 2. This proves (1). To show (2), note that (Va € fl) (V/ 6 IS), ||T(a)/|| = sup{|/»>(a,w)|} < sup{|/(u,)|} = ||/||. wen wen R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 34 Hence, ||X,(ct) || < 1 . If e is the unit in. E , then ||TT(Qr)e|| = 1 ; thus, ||T (a)|| = 1 . □ For each a E Cl, there exists a unique T*(a) E £b(E*, E*) which satisfies (V / E E ) (V^ £ E*), the dual relation (T (a )f, //) = ( /, T*(a)p). The action of T*(a) on an arbitrary n E E* is given by = ( T { a ) f, p.) = f f{<p(a,u))dp(u) = f Jn Jn •))>- Thus, r ( a M - ) = ^ ‘ lK - ) ) . (6 -2 ) Remark 6.6. We summarize the ideas in this chapter. • Note th a t we can regard the translation group on B U C (R ) as a subgroup of the fa m ily of operators outlined in Lemma 6.5. • We consider the uniformity £/r on R because it is the only uniformity on R in which fl is the completion of R; it is also the weakest uniformity on R in which each / E BU C{R) is uniformly continuous in the sense of Definition 6.2. • In Chapter 2 we determined that B U C (R) is *-isometrically isomorphic to the space C(Q) of all continuous, complex functions on Cl. In this chapter we verify this isometry in another way; that is, the fact that f 2 is the completion of (R,ZVr) also illustrates the isometry between B U C (R) and C(Cl). • Note th a t the uniformity £/r is weaker than the natural uniformity on R given by the metric d(x, y) = \x~ y \, rr, y E R, in spite of the fact the uniform topology of R, Ttfo, coincides with the usual topology of R. However, R is not complete with respect to tur . R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 35 • We asserted th at <p is not a group operation on Cl; to show this, we need the following: 1. Suppose G is a compact Hausdorff group. Then G is a topological group if and only if the group operation on G is separately continuous ([Ell]). 2. Let G be a locally compact abelian group. Suppose a is a continuous homomorphism defined on G, with the property th at |a(^)| = 1 , g € G. If in addition a satisfies the equation a ( h g '1) = a(h )(a(g ) ) ' 1 (h, g 6 G), th en a maps G onto the multiplicative group of complex numbers of ab­ solute value 1 . Define F(G) as the group of all such homomorphisms on G: C(G) is called the group of characters on G. If G = (R, then every character a (x) on R is of the form a(x) = etyx, and T(R) is homeomorphic and isomorphic to R under the correspondence etyx 3. y ([Loo]). G is a compact abelian topological group if and only if T(G) is discrete ([Loo]). Now, we argue as follows: Suppose ip is a group operation on fl. Then, by ( 1 ), fl is a compact abelian topological group; by (3), this implies that T(fi) is discrete. Since ( R ,+ ) Q (Cl,cp), then T(R) C T(fl). But T(R) = R, so r( fi) induces the discrete topology on R. However, this contradicts the fact that the topology on R , as a subspace of Cl, is the usual topology. The above argument not only shows that <p is not a group operation on fl, it also shows th at there is no extension g of the map (x, t) —►(x 4 - £) of R x R —*• (R, +), which renders (fl, g) a topological group. R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 36 C hapter 7 C onvolutions Let E — C(Q), th e space of continuous functions on fi. We remind the reader that E* = M (Q), the space of all bounded, regular, complex Borel measures on Q. Let £ Af(fi); in this chapter we define their convolution. But first, T heorem 7.1. ([Bej]). Let X and Y be locally compact Hausdorff spaces; let f be a bounded, separately continuous function on X x Y , and let M ( X x Y ) be the space o f bounded Borel measures on X x Y . I f A € M (X x Y ), then f is X- measurable. Let / £ E and let p be the semigroup operation defined on Q (Chapter 6 ). ByLemma 6.4, the composition function f(p (a ,0 )), a, fi E Q, is a bounded, separately continuous function on fl x ft. If € E*, then Theorem 7.1 asserts th at f o p is p.1 x //2-integrable. Thus, by Fubini: f /(<?(«>/?)) d(jii x fi2)(«,/?) = [ //(¥>(<*,/?)) dp,x{a)dp,2((3) = Joxo Jo Jo f [ Jo Jo d fii(a). (7.1) Given p.\, p2 £ M(f2), consider the mapping f -* [ [ f (&(.<** 0)) dpi(a) dfi2(/3), J o Jo of E —» C. The mapping defines a bounded linear functional $ on E; hence, by the Riesz representation theorem, 3 v E E* such that * (/) = D efinition 7.2. f fd v = ( f,v ) . Jo We define the convolution of fii and fi2 as the linear functional R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 37 v — (ii * fi2 given by [ f ( f,fi\* fi2 ) = P)) dfii(a) dfi2(l3) = Ja Ja f f f{<p{a,P)) dfi2(P) dnx{a) = Ja Ja feE . Lem ma 7.3. (7.2) Let h i , (i2 £ E*. Then 1- 11^1 * H I < IIHI IIHI 2. hi * fi2 = fi2 * Hi and (h i * Hi) * Pz = Pi * (p 2 * P-z) 3 . ( h i + P 2 ) * y-3 = (p-i * H + (^2 * /*3) 4- (E*, *, +) is a commutative Banach algebra with unit. Proof. The proof of ( 1 — 3) follows from equation (7.2). That (E*. *, + ) is a commutative Banach algebra follows from (1 —3). We show that (E*. *, -F) has a unit. Let 50 be the Dirac-5 measure concentrated at 0. Then (V/ (E E) (V/? £ Q), w (< p (-,p ))) = m o , p ) ) = m . If Eg is the Borel ^--algebra on Q and A € Eg, then <*o(A) = 1 if 0 € A 0 ifO £A. Clearly, 5q £ (E*, *, + ). Moreover, Vh £ E*, ( / , 60 * h ) = 7n f [ f(<P(<*, P)) d 60( a) dn( P) = JaJ n 3 ) ),« o > M fi) = f / ( * > ( 0, 0 )) M ./n 0) = / / ( « < W ) = ( /> ) . vn which proves th a t 50 is a unit. R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . □ 38 We can conclude from equation (7.2) that ( f , y l * y 2) = ( / , [ T*(0)fii dy2(/3)). Ja Indeed, by equation (6.1), T * (0 )y i(-) = yi((p~l (~, (3)). It follows th a t (/, [ T m n x dti2m = f fd{ f Ja Ja Ja dfi2m = f f f{u ) dfii(tp~l (u, p ) ) dy2{P) = Ja Ja [ [ /(¥>(“ >/?)) dpi(a) d y 2((3) = ( f , y i * p 2). Ja Ja Remark 7-4- Let M (R) be the Banach space of all complex Borel measures A on R, with ||A|| = |A|(R). Then the mapping Co(R) —*•C given by f-+ [ f f i ^ + y) dy{x) d\{y), A , / i 6 M ( R ) , Jr Jr is a bounded, linear functional on Co(R). Thus, by the Riesz representation theorem, there exists a unique element u = y * A which satisfies: f f d (y * A) = J rf J r[ f ( x + y) dy(x) dX(y). ./r But (R ,+) is abelian, so by Fubini’s theorem, / R/ d{y * A) = / R/ d { \ * y ). Ergo, properties (1 —3) of Lemma 7.3 are satisfied, and since 6q E (M (R ), *, + ), then it follows that (ilf(R), *, -+-) is a closed subalgebra of (M(f2), *, +). T h eo rem 7.5. I f u E B U C Q(W) and y E E* = M (f2), then (u * y ) E BUC®(R), i.e., BC/C°(R) is a closed algebraic ideal in (£"*, *, +). Proof. We first show , (V/ € £ ) (Va 6 12), (a) ( /, T*(a)(y * y)) = ( /, ^ ( a ) * / * y). R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 39 The above equality follows from </,r ( * ) v * /•>= (/, Jnf T*G9) dWW= (/, ./n[ r ( a ) T-(J3)v dptf)) = (T (o 0 /f [ T*(0)v d e W ) = ( T ( a ) f ,v * f i ) = ( f,T * ( a ) ( v * fi) ) . Jsi If £ € R, then T*{t)(u * fi) — v * fi — T*{t)v * fi — v * fi, (7-3) by (a). Applying Lemma 7.3 -3 to the right-hand side of equation (7.3) yields: T*(t)i/ * (i — u * fj. = (T*(t)u — Therefore, ||T * ( t) ( ^ * / * ) - « ' * /«ll < I I - HI M l ; and since the right-hand side of the above inequality tends to zero as t —*• 0 , the result follows. □ Remark 7.6. Note th at 6q (/, St) = B U C Q(M). This follows from the equations So) = </, r(t)So) ( / € BTO(R)), which show that T*(t)6o = St, t 6 R. Hence, ||T*(£)<5o —<5o|| = \\Sq — 6t \\ = 2 if t ^ 0 . It follows th at BU CQ{R) ^ BUC*{R). R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 40 C hapter 8 A characterization o f BU C °(R ) In this chapter we characterize BUC°(W ) (Theorem A); we will need the following results. D efin itio n 8.1. Let (X, E, p) be a finite measure space and let (E ,p ) be a Banach space. a) A function f : X —* E is called p- measurable i f there exists a sequence of simple functions /„ such that lim Pifn - / ) = 0 n —*00 p.a.e. b) f : X —* E is weakly p-measurable if for each x* (E E*, the scalar function x * f is p-measurable. c) A p-measurable function f : X —> E is Bochner p-integrable i f there exists a sequence of simple functions f n such that ttm Q f f U - =0 ([DWj). We state the following well-known results. T h e o re m 8.2. (Pettis) A function f : X —►E is p-measurable i f and only if i) 3W € 53 lL(W) = 0 such that f ( X \ W ) is norm-separable, and ii) f is weakly p-measurable ([Pet]). T h e o re m 8.3. A p-measurable function f : X —*■ E is Bochner integrable i f and R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 41 only if f \ \f \ W < OO Jx ([Boc]). Let 0 < / i € B U C Q(R). Define F : (R, | - 1) - » {BUCe (R), || * ||) by F(t;/z) = T*(t)fi. Then F is a bounded, uniformly continuous function on (R, | - 1), since y E B U C Q(R). Thus, (Ve > 0) (35 > 0) such that \\F (t;y) —F(tf;fi)\\ < e whenever |t —tf\ < 8. We claim th at F is a uniformly continuous mapping (in the sense of Definition 6 .2 ) of (R,£/r) into (BU C Q(R), || *||). Indeed, if g(t) = ||F (t:/i)||, then g(t) E BUC(HL), and \g(t) — gif')] < ||F (t;y) —F (^;y)\\ < e. It follows that if V = {(a:, y) E R x R : |#(x) —</(y)| < e}, then V E Wr, and \t - t'\ < 8 => (4,0 e V ^ ||F(t;y) - F(t';/i)|| < e. This proves our claim; we can therefore extend F continuously to Q (Theorem 6.3). Let Hb be the <x-algebra of Borel subsets of fi and let F (a ; y) be the extension of F . Note th at for each m E F ix , the space (f2, Eg, m) is a finite measmre space. We claim that F is Bochner m-integrable. To see this, observe that Vrr** E B U C**(R), the func­ tion g on Cl given by g(a) = x**(F(a; y)), is a bounded, continuous, complex-valued function. Hence, g is m-measurable, so F is weakly m-measurable. Furthermore, the compactness of F(Q; fi), together with the fact that F(fl; y) C (BUC°(1SL), || • ||), imply that F(fi; y) is norm-separable. Thus, by Theorem 8 .2 , F is m-measurable. Consequently, ||F(o:;^)|| < 1; thus, f ll^ (< * ;/* )ll< M a ) < M f t ) ) ( I M I ) = M l < o o . Jo It follows by Theorem 8.3 th at F is Bochner m-integrable (in the above inequality we assume m(fi) = 1 (Theorem 4.6)). Let 0 < // E B U C e (R) and let A = co{F(a-,y) : a E fi}, where A is the convex hull of the set (F(a:;/i) : a E D}. Then A C B U C Q(R). Indeed, if y E A, then y = ]F?=i ^ ( a i ; y) with ^ L i ^ ' = 1 ^ 0- But F ( a ,;^ ) E B U C Q(R), R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 42 1 < i < p, so A C B U C °(R). In fact, BUC°(JRL) contains the norm-closure A of A , since BI7CQ(R) is a norm-closed subspace of BUC*(M). L em m a 8.4. (Vm e F ix ) (Vjj € BC/U°(R)), fim = Proof. Jn F(a;/i) dm(a ) € A. We will first show, (Va € fi) (Vy** € BUC**{R)), C/^F(a;/i) = ^ (F(a;/f),y**)dm(a) (8.1) Recall that there exist simple functions Pn sn = ^ Ci7nXEi,n u = 1 ,2 ,..., and a positive integer IV, such th at i=l f ||F(a;/x) - *£vn(a)||dm(a) < e, n > N. Thus, [( [ F(a; fj.) dm(a),y**)] - [ f (F(a; /i),y**) rfm(a)] = f /. V N - ^ C i j r X s ^ i a ) ) d m (a ),y* ’ f / { Y i ciJfXEijria) ~ F(a;/i),y* P I s It follows that II F ( a ;/i) dm(a),y**^/Sj - [ J (F(a;y.) ,y**) dm(a)^j || < 2 e||y**||. Letting e —> 0 , we see that equation (8.1) holds. Suppose pm £ A. Then by the Hahn-Banach theorem, 3y** € BUC**(R) such R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 43 that the following dual relations hold: (Pm,y**) = i and i(y,y**)) < 1, y e A. In particular, ({F(a; fi),y**}) < 1 , a € f i . Thus, = Q f (F (a ;/i) dm (ac),y")^ = 1= ( / (F (a ; n), y**) d m (a )j < J d m (a ) = m (fl) = . 1 Therefore, /Jm e A. □ Let M = C®(R.) © {Fix}. In Lemma 4.7 we determined that BU C e {R) = M © (AF* n £17C®(R)). L e m m a 8.5. I f 0 < fiG (M d fl B U C °(R )), then Vm 6 F ix, i n f } ^ , m} = 0 , i.e., ^ Proof. X m. We will apply the following (Birkhoff) identity: Let V be a vector lattice and let A, /?, r 6 V. Then | sup (A, r) - sup(P, r ) | + | inf(A, r) - inf(/?, r)| = |A - 0\ ([LuZ]). I_l G (M d fl B U C Q(R)) =» \i 1 C®(R) and X {F ix}. In particular, fi X F ix . But the operators T*(t), t € l , are lattice isomorphisms (Chapter 2), so T*(t)/J. X T *(t)m , m E F ix. Since T*(t)m = m, by the definition of F ix , then V£ e R, (F (t; /i) = T*(t)p) X m. (8.2) As previously mentioned, the function F (a ;n ), a € Q, fj. € BI7CQ(R.), is contin- R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 44 uous. Hence, |[F f c f i ) —F (a; ^)[| —*■0 for every net (£/) C R with U ^ a. (8.3) Using the Birkhoff identity, w ith A = F(fy;/z), (3 = F (a;fi), and r = m, yields: | inf(F (ti;fi),m ) - in f(F (a ;/i),m )| < |F f c f i ) - F ( a ; / z ) |. Since inf{F(ti; = 0, by equation (8.2), then | in f(F (a ;/i),m )| < |F (tf;/i) - F (a;n)\. But, the right-hand side of the above inequality tends to 0 as t/ a; therefore, (Vm £ F ix) (Va £ fi), F(a; fi) _L m . But, SO f i m □ -L m - L e m m a 8 . 6 . (Vm £ F ix ) (VO < / i £ B U C e (R)), fim < ^m , i.e., fim is absolutely continuous with respect to m . Proof. Note th at for each a £ fl, F (a ; fi) is a bounded linear operator on B U C7G(R); to reflect this property, we will write F (a )fi in place of F ( a ; p.). For each / £ C(Q), [ ( / F *(a)/G 8 ) d[i((3))dm(a) Jci Jn Observe th at (V/ £ BUC(R )) (Vx, t £ R), f ( x + t) = T ( t) f( x ) = T (x )f(t) implies R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 45 F*(a)f(/3) = F*(f3)f(a); thus, it follows by Fubini’s theorem, [ ( f F'(a)f({3) rf/x(/?))rfm(a) = / ( f F*(/?)/(a) dn{!3))dm{a) = Ja Ja Ja Ja [ ( f F *{0)f{a) dm(a))dfi(P). Ja Ja But, (F*((3)f,m) = ( / , F(P)m) — (/ , m ); therefore, </,Pm> = / ( f F*{0)f{a) dm(a))dfi(0) = ./n /*(Q) [ f( a ) dm (a) = v (Q )(f,m ). Ja This shows fim ^ m . □ We axe now in a position to complete the proof of our main theorem. T h eo re m A . B U C °(R) = M . Proof. Having determined that B U C °( R) = M $ M d f l B U C °(R ) (Lemma 4.7), we need only show M d D B U C Q(1BL) = {0}. First, assmne 0 < fi € M d f l B U C ° (R). On the one hand, ^ X m, by Lemma 8.5. But on the other hand, /^ , C m, by Lemma 8 .6 . Therefore, fim = 0. But if e is the unit in C(Q), then (e, fim) = I I dm(z) dfi{a) = m(Q)fi(Q) ^ 0 . Ja Ja Now consider complex fi 6 M d f l B U C e (M). Note th a t \fi\ is contained in M dn B U C Q(R); this follows from the fact that M dn B U C Q(R) is a band in BUC*{K.). Thus, we obtain a contradiction by applying the previous argument to | / i | . R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . □ 46 Chapter 9 W eakly alm ost periodic functions on R Our objective in this chapter is to characterize the sun-dual of the Banach space of weakly almost periodic functions on R. Several results due to [Ebl] (Definition 9.1 and Theorems 9.2-9.4) will help us accomplish this. D efin ition 9.1. Let B(R) be the space o f bounded, continuous, complex functions on R (with sup-norm). Then f £ B (R) is a weakly almost periodic function on R (W AP(R)), if the set of translates { T (t)f(x ) : t £ R} is a conditionally weakly compact subset of B{R). Because there is no descriptive representation of the linear functionals on £ (R ), it is not always simple to determine the weak almost periodicity of a given function. This is true, in spite of the fact th at in the weak topology of any Banach space £, you need only distinguish between compactness and conditional compactness ([Eb2] and [Smu]). Of course, if we consider / € B {R) which vanish at infinity, then not only does the Riesz representation theorem give a precise description of the linear functionals associated with these functions, but as a consequence of this theorem we obtain the result that in Co(R) weak convergence is equivalent to boundedness and pointwise convergence. If we apply this result to the set of Co(R)-translates, { f ( x + t) : t £ R}, we obtain: T h e o re m 9.2. C0(R) C W A P (R). The countable additivity and translation invariance of Lebesgue measure on R imply that every / £ W AP(R) is uniformly continuous. Thus, T h e o re m 9.3. I f f £ W A P (R), then f £ BU C(R). R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 47 T h e o re m 9.4. W A P (R ) is a closed, invariant C*-subalgebra o f B (R ) (and B U C (R )), with the involution operation given by complex conjugation (by invariant we mean f ( x ) G W A P (R ) =► Vf GR, T ( t ) f ( x ) = f ( x -ft) G W A P { R )). Consider Uy(f(x)) = - T f ( x + s ) ds , x , y e R, f e B U C ( R ) . y Jo In Chapter 5 we determined that |[C^,|[ = 1, and IimJ, _ 00 Uy{ f { x )) = 0 if / In fact, if / G W A P(R), then lim v _ 00 Uyf ( x ) exists uniformly in x, G Co(R). and is constant ([Ebl]). We outline a sketch of this fact: By Definition 9.1, for each / G W A P(R ), the set of translates A = [T( t ) f ( x ) : t G R}, is conditionally weakly compact. Then by [Eb2] and [Smu], B f = coA = closed convex hull of A in £ (R ), is weakly compact. Note th at B f is a convex subset of W A P (R), which is invariant under the translation group G = {T(t) : t G R}, i.e., V£ G R, T(t )(Bf ) C B f . Thus, by a fixed point theorem of [Mar] and [Ka2], there exists a j G f i / C W A P(R ) such th at T{t)g = g. Note that g is a constant, since the only fixed points of G are the constant functions. [Ebl] showed that g = lim ^oo Uy{f{x)), uniformly in x. To illustrate this we will need the following two facts: 1 . V / G W A P(R ), Uy(f) G Bf , 2. (V£ G R) (V/ G W AP(R)), linij,_oo Uy{ T ( t ) f ) = lin^^oo Uy(f). The proof of (1) requires uniform continuity to guarantee the uniform in x approxi­ mation of Uy( f ( x )) by Riemann sum s. That (2) is true follows by the inequality \\U ,{T(t)f) - U„tJ)II < 2 M L M . R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 48 Now we argue as follows: Since B f is a weakly closed, convex subset of the locally convex space W A P(R ), then B f is closed in the norm-topology of W AP(R) ([Ru]). Hence, g E B f implies, (V« >0) (3 £ Ci m w - ) e B f) such that ||S - £ 1=1 « T (t,)/(-)ll < t= l By fact number (2), 3yo E R such that \\Uy(T (t,)f) - W ) l l < | , »>»o, 1<«<P- Since t/y(c?) G Bg by (1), then Uy{g) = g, so lb - U ,U )II < [|W J n r ( t i ) / ) II + £ 1=1 c,-||0 -,(T (t,)/) - u ,U ) III < «, t= l y > y o, 1 5~ i < V- Thus, g = lim ,- ^ Uyf { x ) , uniformly in x. Define U o o if) = l i 111 [ ~ [ y-*°° y Jo f ( x + u ) d u \- Then U00 is a linear transformation of W A P {R) into itself; ||£/oo|| = 1; C/oo(C0 ( R )) = 0 , and by (2), U U m f ) = lim Uy( T( t ) f ) = Urn Uy(f ) = U ^ f ) . y-*oo y—*<x> Therefore, Uoo is a unique, translation invariant linear functional on W AP(R). Let 0 < m G F ix , with ||m|| = 1. Clearly, the restriction of all such m to HrAP(R) is the unique mean Uoo- We claim that WAP®(R) = C®(R) ® {Uoo}, where {Uoo} is the band generated by UooBefore we prove this, we give a synopsis of the pertinent information regarding R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 49 W A P(R ) and W AP*(R), th at we implement to support our claim. Let B q denote the unit ball in WAP*(R); define a uniformity Uq on B q with respect to functions in W A P(R), which in turn induces a uniformity U ^ on R (observe that if U is the uniformity on the unit ball B in BUC*(R), defined in terms of functions in BU C {R), then Uq is weaker than U, since BUC(R) D W A P (R) (Theorem 9.3)). Using the same approach as in Chapter 6 , let (R o * ,^ * ) be the completion of (R, Ux^). Then Ro* is a weak*-closed subspace of Bo; hence it is compact. If fio is the maximal ideal space of W AP(R), and R is embedded in Qq as a weak*-dense subspace, then Qo = Ro* (note that fi, the maximal ideal space of BUC(R), is contained in fio)Consider the map (x , t) (x + 1) of R x R —*■R. Then <p can be extended over flo x so that it is separately continuous in each variable (Lemma 6.4). Let C be the extension. If / € C(Do), then the composition of / and C, /(C), is a separately continuous function on f2o x Q0 (Lemma 6.4). Let ti,T 2 € C*(Qo). Recall that C*(Qo) is identified with M(Qo)—the space of bounded, regular, Borel measures on Qq. The linear functional defined on C(Qo) by f [ f (£(<*, 0)) dri(a)<to(j3), Jflo Jflo is boimded; thus, 3u € C*(fio) such that ( /, v) = 'f,(/). Let i/ = n * r2. Since /(C) is measurable with respect to the product measure x t 2 (Theorem 7.1), then (/, n * r 2) = [ /(C (a, /?)) d {n x r2) = ( /, r 2 * r x). JSloxSlo We define *r 2 as the convolution of T\ and r 2 (Definition 7.2); and we conclude that M (Q o, *, +) is a commutative Banach algebra with unit, which contains W A P °(R ) as a closed algebraic ideal. We now have everything in place to prove R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 50 T h eo rem B . Proof. W A P°(R ) = C ?(R ) © {Uoo}. We first summarize essential properties of W AP(W ) and WAP*(R.). The Banach lattice WCAP(R) is an abstract M-space, so WAP*(R) is an abstract L-space (Chapter 2) => WAP*(R) has order continuous norm (Chapter 2) =>■ WAP*(R) is Dedekind complete (D efinition (Hi)) => every band S C W AP*(R) is a projection band (Theorem 2.5(b)). By Theorem 2.8, W A P°(R ) is a band in WAP*(R); thus, U oo e WAPe (R) =» { U o o } c W AP°(R). Using the fact th a t Co(R) C W A P(R ), we embed the band C®(R) in WAP* (R), and assert that it also a band in W A P 0 (R) (Lemma 4.7). Consequently, the direct sum {Uoo} © Cg^R) is a band in W A P *(R) and W A P Q(R) (Theorem 2.5); hence, a la Lemma 4.7, if M = C 0 (R) © {Uoo}, then W AP*(R) = M © (M)d and W A P °(R) = M ® ((M)d fl W A P °(R )). It remains to show (M)d fl W A P°(R ) = {0}. Suppose 0 < r € (M)d fl WAP®(R). Consider the map G : ( R , H ) - * ( W A P 0 (R ),||.||), given by G (t)r = T*(t)r. Then Vt R, G(t) is a bounded linear operator on WAP®(R). Note th at as a function of t, G (t)r is a bounded, continuous function on R; thus, we can extend G to Q0. We claim that as a function defined on Qo- G has the following form: G (a)r = lim T *(ti)r = T*(a)r, R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 51 where the collection {T*(a) : a G Qo} is the dual of the family H = {T(a) : a G fio}, and where each element in H is defined on C(fio) by To see this, recall that (Vx G R) (V/ G W A P(R)), /(£ (a ,x )) = limi f ( x - f £*), where (t[) is a real-valued net such th at ti ^ a , and / is the extension of / to fio- It follows th a t since / G IFAP(R), then the net ( /( x -+- £*)) has a weakly convergent subnet ( / ( x + £;(„))). Hence, ( f , G { t l[v)) T ) = ( f , T ' { t l{t,))T) = < T (a )/,r) = { f ,T * (a )r). From Chapter 7 (“Convolutions” ) and the above discussion, if is the regular Borel measure on fio corresponding to Uoo, then f T*(a)r(-) d m ^ a ) = (r * m 00)(-) = JQo (moo * t)(-) = [ T*(a)moo(-) dr(a) = [ m OQ(-) d r(a ). «/fio ■/fio Thus, ( r * moo) -C moo. Furthermore, a similar argument to the one used in Lemma 8.5 shows, ( r * moo) JL mooConsequently, r * moo = 0. But (e, r * moo) 7^ 0, if e is the unit of IFA P(R ); ergo, (M )d n W A P 0 (R) = {0}. For complex r G (M )dn W A P°(R ), observe that since M )dfl W A P ® (R) is a band R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 52 in W AP*(R), then |r | E M ) d CI W A P °(R ). We therefore obtain a contradiction byapplying the above argument to |r |. □ Remark 9.5. For each fi € B U C °(R), let us again consider the map F : I'D “ * (B(7C°(R), || - 1|), defined by F(t)fi = F(t;fj.) = (Chapter 8 ). Recall that we can extend F(t)fi continuously to fl. If F(a)fi is the extension, we claim where Va E fl, T*(a) is the dual of the operator T(a), and V / E C (fl), T {a )m = /?)). Indeed, if m, mo E F ix with m (fl) = mo(fl) = I, and F(a)fi = T*(a)fj., then V/ EC(fi ), ( /, m> = ( / , / T*(a)m dm 0(a)) = ( / , m * m0) = Jn ( / ,m 0 *m ) = (/, f T*(a)m 0 dm{a)) = ( / , m 0). Ja Therefore, (/,m ) = ( / ,m 0), so th at m = mo, which implies that the set F ix has one element. However, in Chapter 5 we showed that F ix has plenty of elements; therefore, F( a) n Let A P(R) be the space of almost periodic functions on R, i.e., / E A P(R ) if the set of translates { /(x 1) : t E R} is conditionally compact. Then A P(R ) C W A P(R) C BUC{R) C B (R). Note th at AP(R) fl Co(R) = 0. If 6 (R) is the Bohr compactification of R, then R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n . 53 AP*(R) = Af(6 (R)), the space of bounded, regular, complex Borel measures on 6 (R). It was shown by [Neu] th at there exists a unique, translation invariant m ean k on AP(R). It follows immediately th at k = Uaa. Moreover, a proof analogous to the one that shows C^f(R) = L 1 (R), yields: A P °(R ) = L \ m « ). R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . 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