3. Applications of the integral representation theorems

Transcription

1 3. pplications of the integral representation theorems Objekttyp: Chapter Zeitschrift: L'Enseignement Mathématique Band (Jahr): 25 (1979) Heft 1-2: L'ENSEIGNEMENT MTHÉMTIQUE PDF erstellt am: Nutzungsbedingungen Die ETH-Bibliothek ist nbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder usdrucke aus diesem ngebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen ngebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss lle ngaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-ngebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses ngebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 Since (x*) -> = xn it follows that (x*) -> q and hence This implies so that u is a positive measure and this complètes the proof. If has an identity, as is the case in Raikov's original version, the above proof can be somewhat simplified. Theorem 3 (Raikov). Let be a complex commutative Banach algebra with an identity e and with an isometric involution which is sym metric.if/is a continuous positive functional on, then there exists a unique positive Radon measure \i on such that II II = II /II and Proof. s above we know that From Theorem 1 that u < / and there exists a complex Radon measure jjl on such Hence ^ < / = / (e) = fi (1) < \\fi so that fi (l)=\\ fl \\ which is enough to imply that /( is positive. The uniqueness of \i follows as in the Remark to Theorem pplications of the integral representation theorems pplication 1 (Bochner's Theorem). Let Gbea locally compact abelian group and let G dénote the (locally compact) character group. Dénote

3 Haar measure on G by dt and the group algebra (with convolution as multiplication) by L 1 (G). Définition : function p eu (G) is said to be positive definite provided for every FeL 1 (G) Using the natural involution F->FofL* (G) defined by F (t) = F(-t)9 we can rewrite the définition of positive definiteness as follows :pe is positive definite provided LOOL 00 (G) For a positive definite function p define the mapping P il 1 as follows: (G) -> C Therefore P is a continuous positive linear functional on the symmetric involution algebra L 1 (G) such that \\P = \\p W^. By the discussion preceding Theorem 2 and the fact that L 1 (G) has an approximate identity, we know that where F is the Fourier-Gelfand transform. By Theorem 2 there exists a unique positive Radon measure fi on (L 1 (G)) such that It is classical that (L 1 (G)) is homeomorphic to the locally compact character group G under the correspondence F<r+y where Therefore using the same symbol for the measure induced on G by ju under this identification, we obtain which implies

4 Since this holds for ail FeL 1 (G) we conclude that This is, of course, the famous Bochner characterization of positive definite functions ([s], [6], [7], [10], [11], [16]). pplication 2 ( theorem of Schoenberg-Eberlein). complex valuedfunction xp defined on the locally compact abelian group G is called a Fonder-Stieltjes transform if there exists a bounded Radon measure ity defined on the dual group G such that Définition : measurable complex-valued function \j/ on G satisfies the Schoenberg condition provided (a) xf/ is integrable on every compact set (b) there exists a constant M such that for every FeK(G), where K(G) is the set of continuous functions on G with compact support. The following theorem is due to Schoenberg [17] for the case G = R and to Eberlein [9] in the gênerai case. Theorem. measurable complex-valued xj/ has a représentation as a Fourier-Stieltjes transform if and only if xjj satisfies the Schoenberg condition. Proof. It is immédiate that if is a Fourier-Stieltjes transform then xj/ satisfies the Schoenberg condition for the constant M = llv^. If now xjj satisfies the Schoenberg condition define L :L1 (G) -? C as usual by

5 By hypothesis From Theorem 1 there exists a bounded Radon measure v# defined on G such that Therefore it follows that and hence which complètes the proof. pplication 3 (Positive definite functions on abelian semigroups). Let S be an abelian semigroup, i.e. a set equipped with a composition law denoted by + such that the commutative and associative laws are valid. We shall assume the existence of a neutral élément 0. By Z 1 (S) we shall dénote the real commutative Banach algebra of functions / :S-» R with the property that Multiplication in Z 1 (S) is defined by convolution; viz., By a character on S we shall mean a function y : S -> [? 1, 1] which satisfies (i) y (0) = 1 (ii) y(s +t) = y (s) y (t) for ail s, tes. The set S of ail characters is an abelian semigroup under pointwise multiplication. If we endow S with the topology of pointwise convergence, then S is a compact Hausdorff space. By (Z 1 (S)) we shall dénote the

6 compact subset of Z 1 (S)' consisting of ail continuous homomorphisms from Z 1 (S) onto R endowed with the weak * topology. If y g S then the mapping defined by the relations is a non-trivial continuous homorphism from Z 1 (S) onto R, so that F y e (Z 1 (S)). Conversely for every Fe (S)) there exists yes with It is easily verified that the mapping is a homeomorphism of S onto (Z 1 (S)). We shall identify S with (Z 1 (S)) via this homeomorphism and consider the Gelfand transform/of/e Z 1 as a continuous function on S via (S) bounded function (p : S -> R is called positive definite if In the sequel we shall assume that the spectral radius formula is valid in the real Banach algebra Z 1 (S); i.e., we shall assume that for every fe Z 1 (S). One can easily prove that the spectral radius formula is true for the simple functions s a9 s a + e b9 Às a and s a * s b9 where for each ag S the function s a is defined by the relations

7 In the gênerai context of real Banach algebras, of course, the spectral radius formula need not hold. The following theorem was demonstrated without the assumption 1 l1l of the spectral radius formula for (S). The theorem was obtained by Berg-Christensen-Ressel [3] using the Krein-Milman Theorem along with an interesting characterization of the extrême points of the convex compact set of normalized positive definite functions. / Theorem. ssume that the spectral radius formula is valid for each ell (S). If ç? is a positive definite function on S, then there exists a unique positive Radon measure jâ on S such that Proof. The functional L:l1 (S) -> R defined by is positive, i.e. L (/ */) > 0. By our assumption of the validity of the spectral radius formula we hâve Exactly as in Theorem 1 measure \.i on S such that and Theorem 3 there exists a positive Radon Hence and therefore we conclude that The uniqueness of \i is a conséquence of the Stone-Weierstrass Theorem : the algebra /l/ l (S) is dense in the algebra of ail real continuous functions on S.

8 pplication 4 (The spectral theorem for normal operators). Let 2tf be a Hilbert space and let j?(jf) dénote the Banach algebra of ail bounded linear operators on tf. Consider a subalgebra c < (tf) with the foliowing properties : (i) (ii) (iii) is commutative; is closed; If Te then T* e (iv) The identity operator belongs to. Let dénote the maximal idéal space of. Since each Te is normal it follows that II T II = II T IL for every Te. For each pair of vectors, rç ejf define a mapping L? :->Cby then we hâve Therefore by Theorem 1 there exists a bounded complex Radon measure and ii^ n on such that ^ < Ç? /i n application of the Gelfand-Neumark theorem establishes the uniqueness of the measure. The usual construction of a unique resolution of the identity on the Borel sets of can be made based on this formula. specialization of this formula to a single normal operator leads to the classical spectral theorem. We shall not give the détails hère since many excellent accounts exist (cf. Berberian [I], [2], Segal-Kunze [18]). n especially lucid présentation is given in Rudin [161. BIBLIOGRPHY [1] Berberian, S. Notes on spectral theory. Van Nostrand, Princeton, N.J., [2] Lectures infunctional analysis and operator theory. Springer Verlag, New York- Heidelberg-Berlin, 1974.