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Some algebras of linear transformations which are not semi-simple Langlands, Robert Phelan 1958

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i SOME ALGEBRAS OF LINEAR TRANSFORMATIONS WHICH ARE NOT SMI-SIMPLE. by ROBERT PHELAN LANGLANDS B.A.,. University of B r i t i s h Columbia, 1957 A THESIS SUBMITTED IN PARTIAL. FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics l e accept t h i s thesis as conforming to the required standard THE. UNIVERSITY OF BRITISH COLUMBIA August, 1958 11 ABSTRACT In t h i s thesis two problems concerning l i n e a r trans-formations are discussed. Both problems involve l i n e a r transformations which are not i n some sense semi-simple; otherwise they are unrelated* In part I we present a proof of the theorem that a l i n e a r transformation, of a f i n i t e dimensional vector space over a f i e l d , which has the property that the irreducible factors of i t s minimal polynomial are separable i s the sum of a semi-simple l i n e a r transformation and a nilpotent l i n e a r transformation,, which commute with the o r i g i n a l transform-ation and are polynomials i n the o r i g i n a l transformation.. We present an example to show that such a decomposition i s not always possible. In parts II. and I I I we obtain some representation theorems f o r closed algebras of l i n e a r transformations on a Banach space which are generated by spectral operators. Since such an algebra i s the direc t sum of i t s r a d i c a l and a space of continuous functions i t s r a d i c a l can be invest-igated more readi l y than the r a d i c a l of an arbitrary non-semi- simple commutative Banach algebra* In part II we remark that the reduction theory f o r rings of operators allows one to reduce the problem of representing a spectral operator, T, on a Hilbert space to the problem of representing a quasi-nilpotent operator.. When T i s of type m+1 and has a"slmple" spectrum i t i s -LXX quite easy to obtain an e x p l i c i t representation of T, In part III we consider spectral operators on a Banach space. We impose quite stringent conditions, hoping that the theorems obtained f o r these special cases w i l l serve as a model f o r more general theorems. The knowledge obtained at l e a s t delimits the p o s s i b i l i t i e s * We assume that T i s of type m+1 and has a "simple" spectrum* One other condition, which i s s a t i s f i e d i f the space, X, on which T acts i s separable,; i s imposed-* We are then able to obtain a representation of X as a function space. These function spaces are modelled on the analogy of the Orliez spaces. We are also able to obtain a representation of the not necessarily semi-simple algebra generated by T and i t s associated projections as an algebra of functions. In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representative. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Mathematics The University of B r i t i s h Columbia, Vancouver 8, Canada. Date />fy B L E Q E C 0 RT E N T S Semir-simple linear, transformations and nilpotent l i n e a r transformations. Spectral operators on a H i l b e r t space. Spectral, operators on a Banach space. 1 Part I The following theorem was given i n a lecture by R. Ree; I t i s a generalization of one due to Chevalley"'', who proves i t f o r a perfect f i e l d , , and arbitrary T. Theorem;; Let Y be a f i n i t e dimensional vector space over a f i e l d , , and l e t T be a l i n e a r transformation on V, I f every irreducible factor of the minimal polynomial of T i s separable then T can be written as the sum. of a semi-simple l i n e a r trans-formation, S, and a nilpotent l i n e a r transformation, N, both of which commute with T. Furthermore the decomposition i f unique, and S and N can be expressed as polynomials i n T. A polynomial i s separable i f i t s formal derivative i s not zero. A l i n e a r transformation i s semi-simple i f the Irreducible factors of i t s minimal polynomial occur to the f i r s t power only. The following i s an alternative proof of t h i s theorem.. We remark f i r s t that a semi-local ring i s a Noetherian rin g with a f i n i t e number of maximal ideals and a l o c a l ring i s 2 a Noetherian ri n g with a unique maximal i d e a l . Proof; Let T be the given l i n e a r transformation and l e t C\ be the algebra generated by T and the i d e n t i t y . Since every i d e a l i s a subspace,. as a ring (X sat i s f i e s , both chain conditions. I t follows from the theory of rings with minimum condition that has only a f i n i t e number of maximal i d e a l s . Thus <X i s semi-local since i t i s cer t a i n l y commutative. <X i s complete since the 1 Claude Che valley,, A New Kind of Relationship Between Matrices, : American Journal of Mathematics, v o l . 65,. 1943, p. 523. 2 D. G. Northcott, Ideal Theory.. Cambridge University Press, 1953. 2 r a d i c a l i s nilpotent and the topology, therefore, discrete. Thus (X±s the d i r e c t sum of complete l o c a l r i n g s f Let Ot*Z>Q[.and T T>»T.. C K c i s also an algebra over ^ since Olcontains the i d e n t i t y . I f we can show that the scalar multiples of the i d e n t i t y i n Ol^ are contained i n a f i e l d "Li - 0\- such that = * where T|v i s the r a d i c a l of O l v ; then i t follows that 0(. cis the vector space dir e c t sum ofV-andI] ( * Let L cbe an arbitrary maximal f i e l d of 0( vwhich contains the scalar multiples of the i d e n t i t y . Cohen has shown that. Q^l-^. i s algebraic and inseparable over ^ J^"'^c\^ * This implies that the image of~Tc modulo M c i s i n "^c^d so^v^i ^Oi^since CK . i s generated by X and i t s i d e n t i t y . Let "T> S.^nL , S< 6 V r N t , and set 5 - , H-- ^ a N ; ; then ~T> 2> + N . Since N i s c l e a r l y nilpotent we have only to show that S i s semi-simple. Let pi be the minimal polynomial of S.l. since S< € t»it. pt i s i r r e d u c i b l e . Let \.<^} be the set of d i s t i n c t polynomials occurring among the ft , and set V - ^ ^ > then p i s the minimal polynomial of • For p^SV 2Lce j>(.&y o and f o r a n y , s,1) i » ^ S c ) so that c^^=Qimplies c^S^-o which implies that <^  i s d i v i s i b l e by G . ;; so that ^ i s d i v i s i b l e by >^ . Thus S i s semi-simple. 3 Let T? be the r a d i c a l of a semi-local ring,, i . e . the i n t e r -section of the maximal i d e a l s . A sequence fa„\is said to be Cauchy i f f o r any K there i s an M such that f o r any n>Mr m>M c^-a^OV* , the Kth power of the r a d i c a l . A sequenceconverges to a. i f f o r any K there i s an M such that f o r any n>M ^-onTi* . The given ring i s complete i f every Cauchv sequence converges. The l i m i t of a sequence i s unique since Q\( Tl*-=>co) . Cf.., f o r example, Claude Chevalley,. On the Theory of Local Rings. Annals of Mathematics, v o l . 44, 1943, p. 692.. 4 Ibid.,, p.. 693. 5 I. S. Cohen,, On the Structure and Ideal Theory of Complete  Local Rings.. Transactions of the A. M.. S., v o l . 59, 1946, p. 72. 3 5 and H are cer t a i n l y polynomials i n T so we have only to show that they are unique. Suppose~T= S'^ V^ , T S ' = VT ,."TW.-N'T» Let OCbe the algebra determined by T , C , U x, and the i d e n t i t y . As above l e t OOr 2«>(X^ and S- S>'- 2*m( , S ^  X«>S C, S'=Z»£» Since S and S> are semi-simple and CT^-is a l o c a l r i n g , $ c and ^ have the same irreducible minimal polynomial . But, by Taylor's expansion Since i s separable \>< i s not nilpotent and y> 0 A H t ^  V has an inverse. Therefore o , and ^--^ , . We give an example of a l i n e a r transformation for which the theorem i s not v a l i d . Let ^ be an imperfect f i e l d and l e t c* be an element of ^  which does not have a pth root. Let be the algebra / ^ p . ^ , i s the algebra of polynomials over v^> . Let "T be the l i n e a r transformation of t h i s algebra de-fined by mu l t i p l i c a t i o n by the image of , the minimal poly-nomial of T i s ^-^-u.^ Suppose ~~T- S->H where S i s semi-simple and M i s nilpotent. S. i s determined by mu l t i p l i c a t i o n by an element of <X since i t commutes with T;' we ide n t i f y i t with t h i s element of OC. Then >^^ ~<* = o . But T- S i s i n the radical.. So SY* - *~T ^ - i s i n the pth power of the r a d i c a l of . Therefore"^?-ot ±s i n the pth power of the r a d i c a l . This i s however impossible since the r a d i c a l i s the p r i n c i p a l i d e a l determined by T^~4.. Part I I We f i r s t state some de f i n i t i o n s and theorems^ Spectral Measure: Let B be a Boolean algebra of subsets of a set P. A spectral measure i n the Banach space, i s a bounded homo-morphism, E , of ^> into a Boolean algebra of projections on )£ such that . Then the following conditions are s a t i s f i e d : E l O - , * ^ - E LoYi El<n) E icr, u cr^ - t(.<r,) + tA<r,} e<.<r^  and f o r a l l cre"Pj, uf imiu M where M i s a positive constant. Spectrum of an Operator T: I f T i s a bounded operator on a Banach space, the spectrum, a ( l ) , of T i s the set of a l l complex )\ such t h a t T - ^ I does not have an everywhere defined and con-tinuous inverse. It i s not hard to show that cr EH i s a closed and bounded set. Spectral Operator of class (B, f ): A bounded operator,T , fromX to X i s c a l l e d a spectral operator of class C8,r) i f 1)13 i s a Boolean algebra of subsets of the complex plane V . i l ) T i s a t o t a l l i n e a r manifold i n X*, that i s x-x* o f o r a l l x*t r implies x^o. i i i ) there i s a spectral measure E with domain^B such that T t ^ ) - - Eco-)l ( a L l \ E c<nX ) a- f o r any a t 1^* . wherec(T> ti<rtX) denotes the spectrum of T considered as an operator on the range of Eva-). iv.) f o r any x * t f , x t X , y»Acr) - x'cVc^x i s a countably additive 1 These are taken from: Nelson Dunford ? Spectral Operators. P a c i f i c Journal of Mathematics, Sept., 1954. 5 set function on'B . A bounded operator i s ca l l e d a spectral operator of class f i f i t i s a spectral operator of class 0&,r ) where^ i s the set of a l l Borel subsets of the complex plane. I f T i s not speci-f i e d i t i s c a l l e d a spectral operator. This i s the class to be discussed i n the following. £ i s c a l l e d the resolution of the id e n t i t y f o r the spectral operatorT . The following two classes of spectral operators are of pa r t i c u l a r i n t e r e s t : a) Scalar-type operator; An operator S i s of scalar type i f i t p i s a spectral operator and s a t i s f i e s the equation b) Quasi-nilpotent operator; An operator N i s quasi-nilpotent i f It i s not d i f f i c u l t to show that <riM^ {°\ ? Then i f f o r any Borel subset a of the complex plane we set I CM -- I v o to and o > o d <r and r= X" i t i s easy to see that U i s a spectral operator. The following theorems are true. Theorem; The resolution of the id e n t i t y f o r the spectral operator i s unique. Theorems An operator I i s a spectral operator of class T i f 2. The integral appearing i n t h i s d e f i n i t i o n i s defined i n the reference given i n the preceding footnote. £ I f * * ° , the inverse of IH->IV - i v i s v-sf 2Tt!l VX"NV" Wdoes not exist f o r i f i t did, then 1-which i s a contradiction. 6 and only i f i t i s the sum of a scalar type operator,^,of class r and a generalized nilpotent operator, commuting with )^ » Furthermore t h i s decomposition i s unique, and 5 and "T have the same spectrum and the same resolution of the i d e n t i t y . The uniformly closed algebra generated by a set of operators, -ft, i s the smallest algebra containing JC\ which i s closed i n the metric topology defined by Theorem: Let E. be the resolution of the i d e n t i t y of the spectral operator ^ . Let Ot,be the uniformly closed algebra generated by the , or a Borel subset of the plane; l e t 0(, be the uniformly closed 'algebra generated by the E ^ and T , and l e t X be the r a d i c a l of Ot, » Then and 0\ i s isomorphic and homeomorphic to E"Sa<r(~r» where t^fe-o^ i s the algebra of a l l Borel measurable functions, \ , on <r(.T> such that <^.^ =i Ma-i s f i n i t e . We now discuss the representation of spectral operators, cpn-sidering f i r s t spectral operators on a Hilbert space since these 'x{ 4 have special properties. I t i s known that there exists an operator A with a continuous inverse such that /V't^/K i s se l f - a d j o i n t f o r any 6 . We suppose then that the ^ c < n are s e l f - a d j o i n t . 4 John Wermer, Commuting Spectral Measures on Hilbert Space f P a c i f i c Journal of Mathematics, Sept., 1954. 7 Then,, i f S i s the scalar-type operator occurring, i n the de-composition of T , 5 i s normal. -The notion^ of a dire c t i n t e g r a l w i l l he used to obtain a representation of the space "tj and the operators i n Ot, . Suppose X i s a l o c a l l y compact space and v a positive measure defined on Z> We s h a l l be concerned only with compact subsets, ~L , of the complex plane so that v may be interpreted as a countably add-i t i v e set function on the Borel subsets of ^ . With each point ^ £ 1. , l e t there be associated a Hilbert space . A v -measurable f i e l d of Hilbert spaces i s a l i n e a r spaced of functions, x , with domain \ such that i ) x « ) itjd) f o r a l l £ . i i ) For a l l XfcT, \\*1-1>U i s a v -measurable function, i i i ) I f y i s a function on 1 such that v l<> t~^V> f o r a n < * £ and <^0 - ^ ( O , \/iO) i s V -measurable f o r a l l x t "V , t h e n y t T . -iv) There i s a sequence ^Ov. i n T s u c h that f o r each Ce*2>. i s a fundamental set i n "^cf) , i . e. Cx, 10,^1=0 f o r a l l n implies a-o . The set of a l l elements, x r o f T such that ^ui-GttMv^e© forms a Hi l b e r t space,: ^  r ^  i^^) , with the inner product U „ X ^ 3 Cx.l^, x.t^ Art 5 The necessary theorems and de f i n i t i o n s are to be found i n : Jaqques Dixmier, Les Algdbres d 1 Qpe"rateurs dans l'Espaee H i l b e r t i e n , P a r i s , Gauthier-Villars, 1957, Ch. I I . The theory i s sketched i n : L. H. Loomis, An Introduction to Abstract Harmonic  A n a l y s i s f D. van Nostrand, 1953. 6 For a precise d e f i n i t i o n of the l a t t e r , see: N. Bourbaki,,, Elements de Mathematique., Livre IV, Integration, Paris, Hermann et C i e , 1952, p. 50. 8 An operator, R t from"^ in t o " ^ i s said to "be decomposable i f i t can be written i n the form where ~R i s an operator i n (.3) . In pa r t i c u l a r i f ^(.i) i s a multiple of the id e n t i t y f o r almost a l l ~i then~^ i s said to be diagonalisable. I f i s a uniformly closed, commutative algebra of operators on a Hilbert space, M. , which i s closed under the operation"^- > nR y then ( subject to certain more technical r e s t r i c t i o n s ) K i s isomorphic to a direc t integral.and the elements of J are r e -presented by diagonalisable operators. We assume now that the 4 spectral operator "T has the property that there exists such a vector x that the closure of the space (X, v ( i , e. the closure of the set of vectors of the form i s ^ty and that"! i s of type mi,, ,i.e• N m v L 0 • Lemma I; ^ i s separable. Proof; L e t T - S>N . Then S i s the l i m i t , i n the uniform topology, of operators of the form Thus <X i s the algebra generated by the tv<0 and N , so any element of °^ i s the uniform l i m i t of operators of the form CX i s isomorphic and homeomoirphic to the algebra E^vo-m) . Let (Xa be the subalgebra of CX which i s mapped by the given i s o -morphism onto the continuous functions i n t^<Ai>). We s h a l l show i n the next lemma that 0( i i s separable. Then the uniform closure, 9 "£j„ of the set of vectors of the form i s separable. We must show that V'"^-• ^ ^ s s u f f i c i e n t to show that tvB-N N\ f o r any Borel subset,, a r of the complex plane. 7 c But i s i n the weak closure of OU ; thus NT x i s i n the weak closure of OUN\ ; and, therefore,; i n the weak closure of t^,. But the weak and strong closure of "tj. are i d e n t i c a l . Thus Lemma 2: I f f i s a compact subset of the plane, the set of re a l of complex valued continuous functions on f i s separable i n the uniform topology. Proof: I t i s s u f f i c i e n t to prove the lemma f o r r e a l valued functions. Let the plane be described by a Cartesian coordinate system. Let 5 be the set of functions with domain p which can be written i n the form with «*«^  r a t i o n a l and with a l l but a f i n i t e number of the equal to zero. I f we can show that 5 separates points the lemma w i l l follow by the Stone-Weiers'trass theorem. But i f Yx-^x,,\pand T i r <.-*V,NM d i s t i n c t points of ? either x*x, or y, ; s o ? ? v are separated by ^ = * or $cx, M^-y . 8 c Then there i s a bounded measure v on the spectrum of ^  and an isomorphism of A with a direc t i n t e g r a l , \ I t ^ t W . The 7 Dixmier, op. e i t . , app. I. 8 Ibid.., Ch. I, } 7, Prop. 4; Ch. I I , $ 6, Th. 1. The proof of theorem 1 has to be s l i g h t l y modified f o r the situ a t i o n above. operators i n 01 correspond to the diagonalizable operators on $ ty.VuVs) „ Also^, i f we take the uniform closure, ^  , of the algebra generated byM,U*, and Clothe mapping, ^  ? defined on by ^ ^ ~-^S) t i f ^  i s i n 1^  , can be assumed to be a *-homomorph-ism of I into f the algebra of bounded l i n e a r transform-ations on V) . •' Lemma 3r t^ Ci") i s f i n i t e dimensional f o r almost a l l ^  . Proof: Let be the operator corresponding to the function \ t t^.xovo) ^  The vectors Z l * ^ * ( * are dense i n and therefore {uft,...,«MQUs a fundamental set i n " l ^ f o r almost a l l £ . Therefore i s f i n i t e dimensional f o r almost a l l ^ . Replacing "( by a 0 -dimensional space i f i t i s not f i n i t e o dimensional,; we can assume that a l l the a r e f i n i t e dimens-i o n a l . Lemma 4: Let yvv«o -Ct«.»r )^X) } then yv i s a basic measure oncrK), the spectrum of $ , i . e. yusVofor a Borel,subset of cr cv> implies o . v Proofr " yvv-vo implies tv^x.-o^ so that Since vectors of the form ZLTQ^ N<* are-dense i n ^ , twi=o . Then we assume that the direc t integral i s taken with respect to w • Set 11 i s Borel measurable and yv - e s s e n t i a l l y bounded f o r J 9 Ibid., Ch. II, : } 2, prop. 6. 10 Ibid., Ch. I I , \l, prop. 8. 11 Ibid., Ch. II,, £ 3, prop. 2. i r except on a set of measure zero. Then with ^M^(f„t?v = • ' f W y l Since A'.^ i s non-negative definite and yv - e s s e n t i a l l y bounded "b^^-a'V^ i s a bounded measurable function, being, by the proof of the Stone-Weierstrass theorem,, the uniform l i m i t , except on a set of measure zero,of a sequence of measurable functions. Then Let G-K , i , . . . , be the set on which Vi K*i^= o but w% o . then o"^  i s measurable, o\ n <rK>= § i f ,<j(S^ - vj . has measure zero, and ^= ZleE.to-^'Cj - 2» » "f^ K ; Theorem; Let denote the set of a l l measurable functions .. , with domain eK such that ^<Bi«*i-tt/fti«4itt>ty i s f i n i t e • Then i s a Hilbert space with the inner product v -^^."^V** , and " i ^ i s isomorphic to "^*« Proof: I f * ^  .f% t \l ,. then * CO I f \, ^  <- ^ ^ , then 12 Let I^ Ovbe a Gauchy sequence. Let tf* be the subset of crKon which det^Vi>-L ( treating AW as a k x. k matrix). Then For i f At"^ i s singular then ( u ^ , . . '. , i s l i n e a r l y dependent. But "21 " t N l 4 o implies »>,,NKMxi$=o which implies 0 . Similarly <*\r ^  r . . . =a» • Since N *<.V»W-- ^ j . 1 ^ 1 ^ f o r a fi x e d constant K and f o r almost a l l \ , \\ AT'UW ± 5. almost everywhere on cr^ . Then on ^ . v." Thus there exists a measurable V_d.efined on c-^such that 0 * V^ 1 ^ ^ ^ M ~ ' Y > < V Let U ^ v ^ ^ S N f o r ^ t <r * . Then U«S i s uniquely defined on o H f o r CuV--Ct^almost everywhere on xr* o> <r ~ • \ i s measurable and v_ < ^ ,"B*><\v^' v ~ S a „ ^ * > < > ^ > i s f i n i t e . and t , «• M^., vi^m, / cKY-' «• f o r a l l v*' >Vi and ^ <f-^., f ^ > <^ <. e, K f o r a l l >*l then ^ < ? U - f 3^-cro.\> e\ ^  f o r a l l n\' > K So ^ i s a Hilbert space. Map £i«\jf2»\NV} onto Cx^ K^ B ). .. ,x-j^v» This i s a norm-preserving transformation of a dense subset of ^ K onto a dense subset of "i^K , and may be extended to an isomorph-13 ism of ^ with \^ • We now investigate the form of the decomposable l i n e a r transformations on"^ K T i n p a r t i c u l a r , the form, of ^ • Theorem 2; Let be a measurable function on <r^ with range contained i n the set of k x k matrices;. thenH represents a l i n e a r transformation on , given by ( ^ s the coordinate vector of ^ with respect to the basis W ^ V • > ) i f and only i f WiH^t^W i s V- - e s s e n t i a l l y bounded. Proof; Considered as an operator i n the H i l b e r t space I) the norm oflMHs \\ i\<, sf^iAH . Thus the condition of the theorem amounts merely to specifying that the norm of Rt-V) considered as an operator on i s \^  - e s s e n t i a l l y bounded. This, however, i s known to be a necessary and s u f f i c i e n t condition that.'R"--pe-present a bounded l i n e a r transformation on . We note i n p a r t i c u l a r that the matrix of HIM i s just 0 " and the matrix of ivV) i s just V . o So i s i n a form resembling the Jordan canonical form. We remark that the reduction theory f o r rings of operators can be very often used to reduce questions concerning the re-presentation of spectral operators on a Hilbert space to questions concerning the representation of quasi-nilpotent operators. I t 12 Dixmier, op. c i t . , . pp. 159-160. 14 i s only necessary to combine the following theorems with those stated at the beginning of part I I . Since we wish only to indicate the type of general representation theorem which would be de-sirab l e i n the case of a Banach space, we do not attempt to de-fine the concepts introduced i n the statements of these theorems. Theorem: Let ^  be a complex Hilbert space, ^  a uniformly closed *-algebra of operators i n \ ,t. the spectrum of ^  f and -» a basic measure on £ . Assume that 1 i s i n the weak closure of and that the commutant,^' , of ^  i s the von Neumann algebra generated by i t s centre and a countable set of i t s elements. Then there exists a •» -measurable f i e l d , $-* "^ CV> , of complex, non-zero H i l b e r t spaces on Z t and an isomorphism of onto £ diLK) which transforms the isomorphism of Gelfand into the canonical isomorphism of L^Z^ onto the algebra of continu-ously diagonalizable operators. Theorem: An operator T t i s decomposable i f and only i f i t commutes with the diagonalizable operators. Theorem: a) I f T i s a decomposable operator, then I I T l l ^ ess. ^"--p. ^TLWU h) I f T,,TX are decomposable operators, then so i s T.T^ and T,T-^ivsrT,i-^T,.(A^ l o c a l l y almost everywhere. From the l a t t e r theorem we conclude that i f M i s a quasi-nilpotent decomposable operator then i s quasi-nilpotent l o c a l l y almost everywhere. For 13 Dixmier, op. c i t . , pp. 160-161, 169-,. 217, 15 l o c a l l y almost everywhere,, and by induction l o c a l l y almost everywhere; thus f o r a l l m , l o c a l l y almost everywhere. Then l o c a l l y almost everywhere 16 Part I I I For a Banach space, however, there are no such reduction theorems. The special case investigated below indicates the form that such a general theorem might take i f a measure such as that constructed i n lemma 6 e x i s t s . The spaces which appear as the c a r r i e r s of the representation were modelled on the analogy of an O r l i c z space. Henceforth,; we shall, be concerned with the problem of representing a spectral operator defined on a Banach space. The notation w i l l be the same as that introduced at the be-ginning of part I I . l e f i r s t prove some lemmas. Lemma 1: There exists a unique maximal set A with the properties demanded of V i n the d e f i n i t i o n of a spectral type operator, and A i s a uniformly closed subspace of . Proof: bk. i s just the set of V t ^ * such that y^cc-v <*b.<r\«^  i s a countably additive set function f o r a l l ^  i X. . Since A i s cl e a r l y a l i n e a r space,, i t i s only necessary to show that & i s uniformly closed. Let And l e t {.^ \^ be a sequence of d i s j o i n t subsets i n ^  . Set <T=Ucr and pm--<0 °- „ Then so „^ t«.<r- p.O^ = ^ e i c - p ^ l ^ uniformly i n m and Therefore by the Moore theorem on interchange of l i m i t s Corollary: I f X i s r e f l e x i v e . Proof: Otherwise, by the Hahh-Banach theorem , k would not be a t o t a l l i n e a r manifold. Lemma 2,z Ux\\^ \ V C * M f o r any * Proof: The X -closure of i s c e r t a i n l y K . Take the X -closure of the intersection of A with the unit sphere of X and then take the s e t , ^ 1 , of positive multiples of t h i s set;. t h i s i s also X -closed f o r : A convex set i n X * i s "K- -closed i f and only i f i t s i n t e r -section with every positive multiple of the closed unit sphere of X* i s closed. Thus X'-X* , and X-closure of the intersection of the unit sphere of K'with A i s the unit sphere of )i . Therefore i f \\ x*tt-i and \«V»I \ - - \ \M\ there i s such ay't/s that \wp\^i and i s arbitrary; so that Lemma 3: I f X i s separable k contains a countable fundamental set of l i n e a r functionals on X , i . e. a set such that x > v o f o r a l l r\ and a fix e d xe X implies x = o . Proof: Let be a countable dense subset of X . Let x* be such that Wx'w^i and U ^ O M W U • Then ^*»\ i s fundamental. For suppose x+o and x, e X . Then f o r some n 1 L. H, Loomis, Abstract Harmonic Analysis, pp. 19-20. 2 For d e f i n i t i o n s and theorems see Dunford & Sehwarz, Spectral  Theory. ( to be published ), Ch. V, i n pa r t i c u l a r , § 3.11, § 5.7. 18 - - \\x- <»,M ~ \ u *«\\ - \\ x- <«»\ ' * & u x\i - J vi > o Lemma 4: Let vs and ^  be two countably additive set functions defined on the Borel subsets of a closed subset, ^ , of the complex plane. I f ^ a ^ f o r a l l continuous functions i n (p then yxN^  ^ v » Proof: ym,^-- \ = \ 4 « ( ^ - ^ * • I t i s o n l y n e c e s s a r > y t o s h o w that yi,$s<r} f o r closed subsets since then so that tv-y^ on open sets; then on the Borel sets since y\i and^ va-re countably additive. If* cr i s closed o-1 i s the union of a countable number of closed sets so that there i s a closed <= e- such that f o r arbitrary t>o • Let be continuous, V ^ - Q on o~, -WA - i on v , and V^*>\ M on ^  * Then a n d ; so Since t>«o i s arbitrary ^ v v ^ - ^ 0 • This shows that the measure, v\ , defined o n t f ^ b y the li n e a r functional \^  - , **t & and -V continuous, and the Riesz-Alexandroff theorem and the measure defined by v'C^^vj) are equal; f o r i f % i s continuous \ i s the uniform 19 l i m i t of f i n i t e l i n e a r combinations, Z l ^ i . ^ 0 ^ , of character-i s t i c functions of Borel sets and i s the uniform l i m i t of 2^ "<*^ -<-*tO so that Lemma 5 : Let be the countably additive measure defined by v * ^ * ^ * T L t c ^ N , ^ t A. . Let -s>Nr be the va r i a t i o n of t*>*»-y t "then there exists anx*£A such that v^-^-Proof: For convenience, set ^!,^ v * T n e r e i s a Borel measurable function, <^  , such that fo r a l l Borel subsets c" of o-CO we show f i r s t that V C ^ * M = \ v -almost everywhere. Let <3a be the set on which \ ^ x>\ i and l e t <*"Kbe the set on which N G ^ O ^ - M - I , Then <V-o o-K , and SO VC«r^--So - S K ^ - Q and >)cvr^ =-o • Si m i l a r l y the set on which \«^ x>\ > > i s a ^  - n u l l set. Thus Vc^O^i » -almost everywhere. We may assume then that V^c^\= \ everywhere. Then 1 i s a bounded measurable function. I t i s clear that & reduces the adjoint, T * , of the operator Tj> . Let •«* . Then We s h a l l make the following assumptions: ( i ) There i s a vector K 0such that the closure of the space (X, i s 3t ( OX,is the uniformly closed algebra of operators generated by the and T ), ( i l ) T i s of type ^ \ , i . e . i f H i s 2U~ the nilpotent part o f T then N\X*M: o > ( i i i ) ^ contains a countable fundamental set.. Although ( i ) and ( i i ) are a r t i f i c i a l and one should be able to construct a satisfactory representation without them, i f (iii:).. i s not sa t i s f i e d the situation becomes much different'. I f X i s , f o r example, the set of bounded Borel measurable functions on the unit i n t e r v a l with the uniform topology and * the operation corresponding to mu l t i p l i c a t i o n by -S<^-* ^ then ( i i i . ) i s not s a t i s f i e d ('• we should need a countable set," ,, of countably additive functions defined on the Borel subsets so that f o r any point % and at least one n , y«.wCi)40). For t h i s \ a representation l i k e the one below i s not available or, of course, desirable. In the next lemma we do not impose conditions ( i ) and ( i i ) . Lemma 6: I f ( i i i ) above i s s a t i s f i e d and there i s a countable set \Kn\ such that the closed l i n e a r manifold determined by the set o ^<X»*,\lsX then there i s a basic measure on the spectrum of ~t ,. i . e... a measure such that i f .»«»=o then tt» = o . Proof: Let be a countable fundamental set i n ^  . Let x * be the element of A related to and.x._, as i n lemma 5. Set < r S ^ and set ^ = I 5^ ^ *>r,»,.V* Then v. i s positive and countably additive;; v«.<x) = o implies V*^-*^ ~ 0 s o " t h a" t N .* *^ - o s 0 that i^ x * iti,)<Kv>o) s o that \ n'C^o^VQ so that t u i ^ - o . Then 21 But vectors of the form H ^^"^V ^ O are dense i n by assumption so that E.w^ =o .. I f assumption (i). i s s a t i s f i e d we may take >>v--x«,y f o r rw>\ and yx..,, = u. . ( 1** ^ * ). We now assume ( i ) , : ( I I ) , and ( i i i ) hold. Let; >j * now have the same significance f o r the closed subspace determined by V ^ ^ ^ a n d NCn^ as ^* does f o r X- and ... v c=. y^*,^4* i s absolutely continuous with respect to v 0 , thus by the Radon-Nikodym theorem there i s a measur-able function Vc<oe> defined on o-ct^ such that f o r any Borel set <f .. Let Then ^ 1 1 ^ ^ H . f o r o ^ 3 j ^ % 2 Q ^ c \ ^ f .and X-1**1^*=. 2<*K W » We s h a l l be concerned only with X. . If*-V- r N*C«.C.\X» l e t »p map * onto the vector-valued function *OJ> . /X^.j'O. I f v/^ fe & ( Cv.*w)Cs. •= (v, ) l e t the derivative of ^ w i t h respect to. -o0 be. and l e t ^ map onto i ^ o ) . . j < ^ • We remark that ^Nmaps non-zero \|* onto non-zero functions and that i s well-defined;, f o r i f then T - e^vN^.^ G. s o that \ i « ^ s V ^ o and X i ^ A ^ s i n c e X ^ V ^ o on <rK. S i m i l a r l y X ^ y ^ r X v r o ^ - • • •--"XL»*^ a= c> . We also have Lemma 7: Let » t obe a countably additive, f i n i t e , , and positive 22' measure defined on the Borel subsets of a Borel subset, <r , of the complex plane, then Ci) \»rr, ^^ =-9 i n measure and ^  measurable imply i n measure. ( i i ) i f V\ i s measurable and Vs i s nowhereo on <r then A i s measurable.. Proof: Since ( i i ) i s clear we need only prove ( i ) . Set then so that . I f t>o and ^  >o are given, l e t t \ B b e such that ^ ^ o r ^ v s a nd l e t w»„be such that ^ l ^ \ ^ i ^ - M > 4 \<.S fornix^,, Then ^ . • i M f o r ™><*0 Theorem 1: Let vf, be the mapping defined above of into the set of Borel-measurable functions defined on v% with range In the set of complex k*I-tuples. Let * v be the set of a l l Borel-measurable functions JLv.*>=U,.wa, ,V^) defined on cr% such that i s f i n i t e * Then X.^  i s a Banach space under the given norm. Proof: It i s only necessary to show that V^. i s complete. By the same technique as used i n the following proof one can show that i f and only i f $ i s a v k - n u l l function; as usual, we agree to i d e n t i f y functions whose difference i s n u l l . Suppose "23 i s Cauchy i n X^. We show f i r s t that there i s an such that ^ c r * ^ i n v a-measure. Assume that t h i s i s true f o r j4.j e. Then t *<r0 ^Z}- ^> >^.,o, , j f o r i f We remark that V"\K i s measurable and is.nowhere o on <rK. Then For i f ^ y U « > i f ^  ^ i - S - ' S i ^ * * Thus {21 ^\^<\\ converges i n measure. By the induction as sumption \ ^ * \ f or converges i n measure. Therefore converges i n measure. Therefore ^ ^ O ^ Y N X converges i n measure so, since A. i s defined and measurable, V^<A converges i n measure. This completes the induction. We show now that ^ i s i n Let nowu^:^,. . ^ b e the image of an arbitrary function i n Evc-o & , then { 2 1 ^ ^ ^ ^ ^ ^ converges i n measure to ZL> cv-oc>4\oo • ( \ <-<^^\j<^X also i s Cauchy inVAsj^, thus i t must converge to 2!--C^A-O i n • Therefore C i k. vv ^> \\A ^ S\K W 9 f| T*A \ ,i \ and \ i s i n X ^  . Also 24 \ ^ ^jc*> c y * v V"* ***** * v ° v * c * N X ^' ^ and \»v^w---C i n X ^ , Corollary I t The mapping,; ^  ,, defined above may be extended to an isomorphic and isometric mapping of 1^. into X K s u c h that i f ^ V X J O ^ 1 * ) and v ^ u ^ t then ( i ) ^ V ^ V ^ * ' * Proof; By lemma 2 the mapping ^ i s an isomorphic and isometric map of a dense subset of X ^  into X ^ which s a t i s f i e s the condition (1). This may be extended to an isomorphic and i s o -metric map of X K into . Let l^ „"\ i°Vo> ^ V Y - ^W^ • Then H> <^ V>J) - ^  ; and f o r any v|*£Cs with <f,^*>we have V * 1 ^ " ) 3 W , ^ V ^ j , ^ ^ . SO , ^ £ - ^ V » ) l » 0 Corollary 2: I f X i s r e f l e x i v e <^  maps X,K onto Proof; I f > i s r e f l e x i v e , £ A < T ^ X i s reflexive a n d ^ 6 : ^ v ) ^ : (Jtie-^X). by the corol l a r y to lemma 1. Two d i s t i n c t elements of X * define d i s t i n c t continuous l i n e a r functionals on , therefore ^ maps X K onto X - K • To obtain a representation of ¥. i t s e l f we s e t ^ v * . ^ . .^y S t X , withtV^-^j . . . j ^^C-vcr^V** on g-^ and 4 k i^-. . . . - = O on <rK • I f we consider the elements of CX,as l i n e a r transformations of CX, or i t s r a d i c a l , "H, ,. t h e n T i s s t i l l a spectral operator and E t<r) i s i t s resolution of the ide n t i t y ( l e t V be the l i n e a r 25 space spanned by the l i n e a r functionals ^ I H V K * ^ ^ ) • CX,is s t i l l generated by the ^ . i ^ and T . I f "T s a t i s f i e s ( i i ) and CX, or*K t s a t i s f i e s ( i i i ) c o rollary 1 y i e l d s a representation of OC» or t l , .• For Ot,* I i s to be taken as * G ; for"U, ,Vl i s to be taken as x e . Since CM., i s isomorphic to the direct sum of "H, and the es s e n t i a l l y bounded functions we can obtain a representation of CX^in both cases ( i n the second, i f ^-*4 V»^, ^Ulj.y.a^.. ./..^  and -.T( set CVuU O J ). Moreover (or>VM*u) i s obtained i n a very natural manner from «^V^ and This i s just l i k e multiplying polynomials i n one indeterminate, X, modulo ( X m + 1 ) . I f the space t on which crtx acts i s separable then CX, s a t i s f i e s condition ( i i i ) . Let be a countable dense subset of X , be a countable fundamental subset of quired by ( i i i ) . We now sketch two examples. The f i r s t i s to show that the mapping defined i n corollary 1 i s not always onto;, the second example i s of an algebra 0l vwhose representation i s no longer given by function spaces with a "sup" norm,,, e. g. a space of continuous functions, or a space of essentially-bounded functions. 1) Let a be the set u i , . . . \ . Let X be the space of functions,, \ , defined on <r such that 26 i n the uniform norm* Let T be the continuous l i n e a r trans-formation defined by Then <r =. <r<JT) and we may take x*. to be the function and -yo* to be the measure Then Ve«. \ +J,^- J 4 „ , >*^o=o A function, ,. defined on <S i s the image of an element of X only i f But i f A i s a bounded function on <$" and h u o \= ><,,\>V^ O then i s contained i n the space 3C * of theorem 1, 2) Let 2. be the d i r e c t sum of V.,W* and so that i f x = o> xa , x, £v,\o,i~i. , \ asA^o,fv, \\x\v - \\x,u Let S be the operator on ^ d e f i n e d by multiplying the elements of ^ V ^ a n d E^VS by-^ <oO-> » Let ^ tX-^TLo,^ ; l e t M be the operator which sends onto cy^©Q H i s nilpotent and \\Nl\=\\y\ since and w Co<£> ^  \\ z \\ Ifl=$-^M , "v i s a spectral operator and the algebra, generated by"T and i t s resolution of the i d e n t i t y i s isomorphic to the d i r e c t sum of the essentially-bounded, with respect to Lebesgue measure, measurable functions on ^°>*\ andV^car) t <s a measurable subset o f l o , ^ . vanishes o f f & and i s no-where O on <$ • ViS<r) i s the image of the r a d i c a l * 27 BIBLIOGRAPHY 1 Bourbaki, N,, Elements de Mathematique, Livre IV,- Integ-r a t i o n , P a r i s , Hermann et C* e, 1952. 2 Chevalley, Claude, A New Kind of Relationship between Matrices, American Journal of Mathematics, v o l . 65, 1943. 3 Chevalley, Claude, On the Theory of Local Rings, Annals of Mathematics> v o l . 49, 1943. 4 Cohen, I . S., On the Structure and Ideal Theory of Complete Local Rings, Transactions of the A. M. S.. v o l , 59, 1946. 5 Dixmier, Jacques, Les Algebres d'Operateurs dans I'Espace H i l b e r t i e n . Paris,, G a u t h i e r - V i l l a r s , 1957, 6 Dunford, Nelson, Spectral Operators, P a c i f i c Journal of Mathematics, v o l , 4, 1954. 7 Loomis, L. H., An Introduction to Abstract Harmonic Analysis,. D.. van No strand, 1953, S Northcott, D, G., Ideal Theory, Cambridge University Press, 1953. 9 Wermer, John, Commuting Spectral Measures on Hi l b e r t Space,, P a c i f i c Journal of Mathematics, v o l . 4, 1954. 10 Zaanen, A. Linear Analysis. New York., Interscience Publishers Inc., 1953, 

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