GELFAND S PROOF OF WIENER S THEOREM

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1 GELFAND S PROOF OF WIENER S THEOREM S. H. KULKARNI 1. Introduction Te following teorem was proved by te famous matematician Norbert Wiener. Wiener s proof can be found in is book [5]. Teorem 1.1. (Wiener s Teorem) Let f be a periodic function on [ π, π]. Suppose f as an absolutely convergent Fourier series and f(t) 0 for all t [ π, π]. Ten 1/f also as absolutely convergent Fourier series. Gelfand gave a proof of tis teorem using te tecniques from Banac Algebras in is celebrated paper [1]. Tis was te first paper in wic teory of Banac algebras was developed systematically. Gelfand s proof is muc sorter tan te original proof of Wiener and ence it attracted te attention of Matematicians to te teory of Banac algebras. In tis article, 1 we present Gelfand s proof. In order to make tis article self contained, we also give some basic facts about Banac algebras required to understand Gelfand s proof. Tese are kept to te bare minimum. Interested reader can consult [3] for a deatailed and toroug treatment of Banac algebras. Standard results from Functional Analysis are assumed. Tese can be found in any introductory text on Functional Analysis, for example [2] or [4]. Any one wo as done one course in Functional Analysis sould be able to understand tis proof. Bot Wiener as well as Gelfand ave been illustrious matematicians wit epocmaking contributions to various fields. Teir work as also influenced developments of several brances of matematics. Interested reader can find te biograpical information about tese matematicians in te following web cites. (1) Biograpy of Wiener (2) Biograpy of Gelfand 2. preliminaries In tis section, we give some basic definitions and examples. 1 Tis is an expanded version of a talk given at te Worksop on Fourier Analysis in Department of Matematics, Indian Institute of Tecnology Madras, on July 10,

2 2 S. H. KULKARNI Definition 2.1. A complex algebra A is a ring tat is also a complex vector space suc tat (αa)b = α(ab) = a(αb) for all a, b A A is called commutative if ab = ba for all a, b A. We sall assume tat A as a unit element 1 satisfying 1a = a = a1 for all a A. We need te following concepts. Definition 2.2. Let A be a complex algebra wit unit 1. Let a A. If tere exists b A suc tat ab = 1 = ba, ten a is said to be invertible and b is called inverse of A. It is easy to prove tat suc an inverse, if exists, is unique. A is said to be a division algebra if every nonzero element in A is invertible. A subset I of A is called an ideal if it is a subspace as well as a two-sided ideal in te sense of ring teory. Tis means tat if a, b I, α, β C, c A, ten αa + βb, ac, ca I. If I is an ideal and x A, ten te set x + I := {x + a : a I} is called a coset. Te set of all suc cosets is denoted by A/I. Tere is a natural way of defining algebraic operarions on A/I in suc a way tat it becomes a complex algebra, known as quotient algebra. An ideal is called a proper ideal if it is different from A. A maximal ideal is a proper ideal not properly contained in any proper ideal. Tus M is a maximal ideal if and only if M A and wenever M I A and I is an ideal, we ave I = M or I = A. It can be proved easily tat M is a maximal ideal if and only if A/M is a division algebra. A multiplicative linear functional on A is a function φ : A C satisfying te following: φ(αa + βb) = αφ(a) + βφ(b) for all a, b A and α, β C φ(ab) = φ(a)φ(b) for all a, b A. It can be proved easily tat if φ is a nonzero multiplicative linear functional, ten te null space N(φ) := {a A : φ(a) = 0} is a maximal ideal in A. Definition 2.3. Banac algebras Let A be a complex algebra. An algebra norm on A is a function. : R satisfying: (1) a 0 for all a A and a = 0 if and only if a = 0. (2) αa = α a for all a A and α R (3) a + b a + b for all a, b A. (4) ab a b for all a, b A. A complex normed algebra is a complex algebra A wit an algebra norm defined on it. A Banac algebra is a complete normed algebra.

3 GELFAND S PROOF OF WIENER S THEOREM 3 We sall assume tat A is unital, tat is A as unit 1 wit 1 = 1. Next we give two examples of Banac algebras. Many more examples can be found in [3]. Example 2.4. Let X be a compact Hausdorff space, and let C(X) denote te set of all complex valued continuous functions. Ten C(X) is a commutative Banac algebra under pointwise operations and te sup norm given by f := sup{ f(x) : x X}, f C(X) Example 2.5. Wiener Algebra Let W be te set of all complex valued functions on [ π, π] wit absolutely convergent Fourier series, tat is, functions of te form f(t) = c n exp(int),, t [ π, π] wit f := n c n <. W is a complex Banac algebra. Tus to prove Wiener s teorem we need to prove te following: If f W and f(t) 0 for all t [ π, π], ten f is invertible. 3. Proofs Teorem 3.1. Let A be a complex Banac algebra wit unit 1. a A and λ C wit a < λ, ten λ a is invertible and (λ a) 1 1 λ a Proof. Consider te infinite series n=0. Since a / λ < 1, tis λn+1 is an absolutely convergent series. Hence by completeness of A, it converges. Let b denote te sum and b n := 1 + a an denote te λ λ 2 λ n+1 partial sum of te series. Ten for eac n, (λ a)b n = 1 (a/λ) n+1. Taking limits as n, we get (λ a)b = 1. Similarly, b(λ a)b = 1. Tis sows tat λ a is invertible. Next, (λ a) 1 = b an λ a n n+1 λ 1 n+1 λ a n=0 Corollary 3.2. Suppose a A is invertible and b A is suc tat a b < 1. Ten b is also invertible and a 1 a 1 b 1 a 1 2 a b 1 a b a 1 Tis means tat te set of all invertible elements in A is an open set in A and te map a a 1 defined on tis set is continuous. a n n=0 If

4 4 S. H. KULKARNI Proof. Since 1 a 1 b = a 1 a a 1 b a 1 a b < 1, by te above Teorem 3.1, a 1 b and ence b is invertible. Furter, note tat a 1 b 1 = b 1 (a b)a 1 b 1 (a b) a 1. (*) Since b 1 b 1 a 1 + a 1, we obtain from (*) above, b 1 b 1 (a b) a 1 + a 1, tat is, b 1 a 1 1 (a b) a 1. Using tis estimate (*) becomes a 1 b 1 a 1 2 a b 1 a b a 1 Corollary 3.3. Let φ be a nonzero multiplicative linear functional on A. Ten φ = 1. Proof. Since φ is multiplicative, (φ(1)) 2 = φ(1 2 ) = φ(1) and since it is also nonzero, φ(1) = 1. Tis also means tat if a A is invertible, ten φ(a) 0. Now φ := sup{φ(x) : x A, x 1} φ(1) = 1. Next let x A and x 1. Ten for every λ C wit λ > 1, by te above Teorem 3.1, λ x is invertible. Hence 0 φ(λ x) = λ φ(x). Tus φ(x) 1. Tis implies tat φ 1 and completes te proof. Teorem 3.4. (Gelfand Mazur Teorem) Every complex Banac division algebra is isometrically isomorpic to C. Proof. Let A be a complex Banac division algebra. Let a A. We sall prove a = z (tat is z1) for some z C. Suppose a z 0 for all z C. Ten, since A is a division algebra, a z is invertible for all z C. In particular, a is invertible. By (a corollary of) Han-Banac teorem, tere exists a continiuos linear functional φ on A suc tat φ(a 1 ) 0. Now define f : C C by f(z) := φ((a z) 1 ), z C. Now let z C and consider f(z + ) f(z) = φ( (a (z + )) 1 (a z) 1 = φ((a (z + )) 1 φ((a z) 1 )) ) = φ((a z) 1 )(a (z + )) 1 ) Now using continuity of φ and of te map b b 1, (see Corollary 3.2) it f(z + ) f(z) can be sown tat for every z C, lim 0 exists and, in fact, equals φ((a z) 2 ). Tis sows tat f is an entire function. Also, for z > a, f(z) φ (a z) 1 1 by Teorem z a

5 GELFAND S PROOF OF WIENER S THEOREM Hence, f(z) 0 as z. Tis, in particular, implies tat f is bounded and ence by Louville s teorem, f is constant. Moreover, since f(z) 0 as z, tis constant must be 0. In particular, f(0) := φ(a 1 ) = 0, a contradiction. If I is a closed ideal in a normed algebra A, ten te quotient algebra A/I can be made into a normed algebra by defining x + I := inf{ x + y : y I}, x A Furter, if A is a Banac algebra, ten A/I is also a Banac algebra. Also, if M is a maximal ideal, ten M is a closed ideal and A/M is a Banac division algebra. Tis is used in te next corollary. Corollary 3.5. Let A be a complex commutative Banac algebra wit unit 1. If a A is not invertible, ten tere exists a nonzero multiplicative linear functional φ suc tat φ(a) = 0. Proof. Consider I = aa := {ab : b A}. Since a is not invertible, I is a proper ideal in A and is ence contained in a maximal ideal M. Ten A/M is a Banac division algebra and is ence isomorpic to C. Let tis isomorpism be ψ and define φ : A C by φ(x) := ψ(x + M), x A. Ten φ is te required multiplicative linear functional. Proposition 3.6. Let φ be a nonzero multiplicative linear functional on te Wiener algebra W. Ten tere exists t 0 [ π, π] suc tat φ(f) = f(t 0 ) for all f W Proof. Consider te function g defined by g(t) = exp(it), t [ π, π]. Ten g W and g = 1. Let λ = φ(g). Ten λ = φ(g) φ g = 1. (Note φ = 1 by Corollary 3.3.) Furter g is invertible and g 1 (t) = exp( it), t [ π, π]. Tus g 1 = 1. Also, since φ is a nonzero multiplicative linear functional, φ(1) = 1 and ence φ(g 1 ) = 1/λ. Tis implies tat 1/λ 1. Hence λ = 1. Terefore tere exists t 0 [ π, π] suc tat λ = exp(it 0 ). Now let f W. Ten f(t) = c n exp(int),, t [ π, π] Tus f = c ng n. Hence φ(f) = c n φ(g n ) = c n (φ(g)) n = c n λ n = c n exp(int 0 ) = f(t 0 ) Now we ave all tools required to prove Wiener s teorem. Recall te observation at te end of te last section tat to prove Wiener s teorem we need to prove te following: If f W and f(t) 0 for all

6 6 S. H. KULKARNI t [ π, π], ten f is invertible. Suppose suc f is not invertible. Ten by Corollary 3.5, tere exists a nonzero multiplicative linear functional φ on W suc tat φ(f) = 0. Next, by Proposition 3.6, tere exists t 0 [ π, π] suc tat φ(f) = f(t 0 ). Tus f(t 0 ) = 0, a contradiction. Tis is te essence of Gelfand s proof of Wiener s teorem. References [1] I.M. Gelfand, Normierte ringe Mat. Sbornik N. S.9(51), 3-24, [2] B.V. Limaye, Functional Analysis, New Age International, [3] T.W. Palmer, Banac algebras and te genreral teorey of *-algebras, first ed., vol. 1, Cambridge University Press, [4] A.E. Taylor and D.C. Lay Introduction to Functional Analysis, Jon Wiley, [5] N. Wiener, Te Fourier integral and certain of its applications, Cambridge University Press, Department of Matematics, Indian Institute of Tecnology - Madras, Cennai address: sk@iitm.ac.in