Structure of Commutative Normed Rings Citation Denby-Wilkes, John Edward (1950) Structure of Commutative Normed Rings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/tr8p-j588. https://resolver.caltech.edu/CaltechTHESIS:06252025-185103530 Abstract In a complex commutative normed ring the unit sphere at the origin has a vertex at the unit element. If the ring is finite dimensional, the radical translated to the unit element intersects this sphere only at the unit element. A finite dimensional ring containing an element of nilpotency degree equal to the dimension or the radical is a direct sum of a ring with a scalar product and a ring with a convolution product. Using this decomposition the conjugate space is made into normed ring, and a duality theory is obtained. General properties are given or completely continuous and weakly completely continuous elements or various types of rings. In a star ring, if uniform convergence with respect to the maximal ideals implies weak convergence, then the square or a weakly completely continuous operator is completely continuous. Some of the consequences of this result are: (a) no infinite dimensional ring of this type is reflexive as a Banach space, (b) all weakly completely continuous elements or infinite dimensional indecomposable rings of this type lie in the radical, (c) a new proof or Dunford's theorem that the square of a weakly completely continuous operator from L into L is completely continuous is given. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Aeronautics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Karlin, Samuel (advisor) Bohnenblust, Henri Frederic (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 January 1950 Record Number: CaltechTHESIS:06252025-185103530 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06252025-185103530 DOI: 10.7907/tr8p-j588 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17486 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 27 Jun 2025 22:33 Last Modified: 27 Jun 2025 23:17 Thesis Files PDF - Final Version See Usage Policy. 12MB Repository Staff Only: item control page

STRUC'l' UFi.3 OF COMilU 'i'll.TIV i! NOR1JI~D RH iGS Thesis by John 3dward Denby-Wilkes In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1950 The author wishes to express his profound gn.titude to Professors Karlin and Bohnenblust f or the kind encouragement and generous assistance t hey have given during the preparation of this thes i s. ii .fiBSTRACT In a complex commutative normed ring the unit sphere at the origin has a Tertex at the unit element. If the ring is finite dimen- sional, the radical translated to the unit element intersects this . sphere only at, the unit element. A finite dimensional ring containing an element of nilpotency degree equal to the dimension or the radical is a direct sum· of e. ring with a scalar product and a ring with a convolution product. Using this decomposition the conjugate space ie. made into e. normed ring, and a duality theory is obtained. General properties are given tor completely continuous and weakly completely continuous elements or various types of rings. In a star ring, it unifon convergence with respect to the maximal ideals implies weak con·urgenee, then the square or a weakly completely _continuous open.tor is completely continuous. consequenooa of thia result are: Some of the (a) . no 'infinite dimensional ring :or this type - is reflexive as a Banach spa,ce, (b} all weakly completely co,itinuoua elemen:ta or infinite dimensional indecomposable rings of thi.e type lie in the radical, (c) a new proor or Dunford 's _theorem that the square ot a weakly completely continuous operator from L into L is completely continuous is given. Hi This thesis cor1tains an investigation of three general problems in the theory of et,mplex commutative normed rings. denotad by the letter Such riugs will be A. The first prob~em dee.ls with the structure of the unit sphere of A. In a Banach space it ie well known that the unit spherl'J is a closed, convex, centrally symmetric body. !n the case of a nol"tlled ring an additional restriction is placed on tha r.io'f'm, and hence on the unit ephere, by the inequality • \\ xy II fl x II II y II • · It is shown in ~ ·. Part II th.at this implies that the unit sphere centered at the origin has a vertex at the unit element the beginning of that Part.) u. (The term vertex is explained at This result is then intetpretad in terms of the derivative of the norm as used by Ascol!. (1;53) t The second problem considered is that of defining a not"lDed ring . . . * conjugate to the ·Banach space of A. 1'he etl"tlcture in the space A poaeibility of accomplishing this was strongly suggested by sev0ral facts. For example, thore exist numerous similarities between results in normed ring theory _and in the theory of normed linear lattices'[ e.g. eee (2)]. It E is such a lattice there is a natural way of defining a normed linear lattice in (2;3) As anoth8T' example consider the set of pairs of complex numbers (x,y). This can be made into a Banach space, say A, by defining II (x~y) II • Nax( I xi, I y I)• Ite con.jugate f 'I'he first number within .the parentheses refers to a numbered refer•nce work at the end of this paper, ·and the second number refers to .the page number of that work. iv V space A*'!l { (u,v)} II (u,v) II : sup If ux1-vy H is also a Banach spac e if (l(x ,_y)ll=! : I u I + Iv! A and and is taken a.s norm in A* respectively as A*• (x,y)(x•,y•) s (xx',yy') (u,v)(u•,v•): {uu•.uv'+u'v) , .become normed rings. Now ii' products ore dafi ued iu (scal a r product) (convolutior1 product), A and A* This shows a duality between scalar and convolu- tion products. Furthermore A, of dimension ideals, whereas 'A* , also of dimension n : 2, has n :: 2, has k = 2 rnaximal n-k+-1 maxiwal ideals. '. This problem is discussed in Part III, where a complete duality theory foi- n- dimensional normed rings gives a partial answer. Thil"dly, an analysis of certain classical Banach space operators defined over normed rings is given in Pa.rt IV. In particular completely continuous and weakly completely continuous operators of the type T~ ; Ti1. y : xy are studied. The elements x which generate Tx are called completely continuous and weakly completely continuous, respectively. Partial results on completely continuous elements have been obtained independently by M. Ft'eundlich (3); however, results presented here are . believed to be more complete. A general theorem on weakly completely continuous transformation• is proved, and completely continuous elements ~re discussed as especial caea. For purposes of reference some definitions ana theorems from the theory of normed rings are given in Pa.rt I. .I ·. Introduction 1 II Structure of the Unit Sphere 5 III Structure of Finite Dimensional Normed Rings 12 IV Operators 22 Appendix 33 Reterencee • 3G vi PART I INTR0DUC'TI0N 1.1 Definition. A complex commutative normed ring A is a sot x, y, . .• • having the following properties a >f elements (1) A is a complex Banach space. (2) For each in x, ye A, the 2roduqt xy is uniquely defined, lies A, e.nd has tha properties: xy = yx x(yz) : (xy)a x(y+z) : x( f y) xy+Xi = f-KY• v.ihere ,-.. is a complex number. - (3) A UDi t el emant u exists in . A such that (4) The !!2!!! II in A has the propertieei II ux: x for all II xy II~ Uxll IIYII I\Ull:le 1.2 Reniarks. The concept of complex comuutative normed ring was first introduced in 1932 by A. D. Michal and R. s. Martin, who also considered the ease where the scalar field is the real numbers. (4;69) . Henceforth complex commutative normed rings ae defined above will ba referred to simply as rings. Sate that are rings only in the al gebraic sense will be explicitly called algebraic rings. •The zero element of A ie denoted by ~- -to distinguish it from the scalar o. 1 2 If, as a -Banach space, A is n-dimensional, the ring is culled an n~dimeneional ring and is denoted by An• A is a !ill ring if for every element x*e A such that x(M) : x'A'(M} x~A thara is an element for all. maximal ideals . r., ( see section 1.4 below). 1.3 Topologies!!! A. Two topolo gies will be used .in stron15 (or !!2£!) t ·opoloa and the ~ topoloey. a. neighborhood ot an element that II x-x 0 II < c, x 0 is the set A, the In the first topology N(x 0 ;f.,) of ull where €, is an arbitrary positive number. x such A sequence { x: 11 } is conYergent, or more precisely at rongly convergent, if lim II xn • xinll n,m•oo II xy · 11 ~ topology ot xy ·1.1, exists. In this terminology condition ( 4) of Definition II x II II y II , expresses the continuity in the strong with respect to (x, y). Unless specific mention is made to _the contrary, it is the strong topology which will be used in In the weak topology a neighborhood X: 0 ia the set of all x t 1 • • • • , f m.. are where A• and e N(x 0 J f, , ••• , fYt'I. ; such that • \ f . (x-x 0 ) I<. . 'I, c, f.. ) A. of i • 1, .•• , m, m arbitrary coU1plex linear functionals ovet" is M arbitrary positive number. Since all functionals which will occur are complex linear functionals, they will be referred to ai111ply as functionals. A aequen_ce { x I\} is weakly convergent if exists tor every functional 1.4 Maxi-1 Ideale. (1 ) I ,I {~ }; {2) x,ytI and A, f over A subset· I lim f ( x.,.) 'r, ➔ oO A. of A is called an ideal if and x',y'fA imply xx'+yy•er. A maximal ideal is an ideal not contained in any other ideal. It has been ahoWD (5;8) that every ideal is contained in a maximal idenl. Ae e. consequence, the existence of the inverse :g- 1 of x is equivalent to 3 the st~ternent that x belongs to no maximal ide:w.l. ideul, the quotient space If M is a maxiroo.l A/M is isomorphic to the fiald of co~lex numbers. · This esta.bliehes a many-to-one mapping; of' A onto the complex numbersa to each. x A : x( i,1) depending · only on M. However, if A, ~(~} is a.saocinted a complex number Fol'" a fi:xed x it , is fixed and x( M) is a multiplicative functional over M • varies over the set of maxi rll-ll ideals of describes the spectrum of ro·i: whtch ( ri u.xf . hyperplanes of 1 A. x [ i.e., the set of all complex A's does not exist]• . The relation between functionals and A is as follows: the kernel of zero•s of f(x}] ie a hypaT'])lane of hyperplane K through ~ A such that · f(K) : O and K of f(x) [i.e. the set A passing through ~, and every uniquely determines a functional f(u) :: l. f(x) over In particular there is a one-to- one correspondence between multiplicative functionals over A and maximal id ool s. 1.5 Radical • . The radical maxlmal ideal•• R ·10 defined as the i -n tersection of all R is an ideal. It has been sho~ ( 5; 10) that the set of all generalized nilpotent elements or A, namely elements l I euch that 11m \Ix,.. II~ • ·o. Indeed, for all x E. A, lim M R contains all the nilpotent ele~nte. l~(i • Noma. ·A noi;m II 11• . is called 'admissible in A i 'f' 'A ia a .normed .ring .with reepeet to this norm, that is if (1) II h' makes II xy. II • (3) llull':1. ~ A a Banach space, II x 11• Uy u' , and • ( 2) An example of a non-admissible norm is tha pseudo--norm \Ix II I = ~P. \ ~(M) I, since for example II x 11, • O does not imply x = ~. norms I! U• · and x II x "'II n.' • sup Ix( M) I• l\-"<;1 \'\-C)Q In particular, R is Two II II", defined over A, are called isomorphi~ if 4 positive constants a Ux ll' ~ a, b II x I(" ~ b II x II'• respect to t wo norms If a normed vector space is co mplete wit h II • II II' and II II" • The Gonjugate space A~ of A is the set of all function als over sup I f{ x) I makes x, then The natural rn and II f ,u • ll l( U=l II II " and if II x ll " ~ b II x U1 II for all A. exist such that for all x EA, a re isomorphic. A* a Bana ch spa ce. 1.7 Isomonahism. Two rings are called: (1) isomorphic if they are a.lgebraic~,lly isomorphic; (2) equivalent if they are isomorphic under a norm-preservin g i somorphi am; (3) homeomorphic if they nre homeomorphic topological spaces. 1.8 Decomposability. A is decomposable if it can be written as a direct sum of two of its ideals. PART II S'l'R UC'l'UH3 or Tl-[; UNIT SPH ::RJ 2.1 Definition. A functional u. ~ 91. ~ m.u.1 sgber1 .a 2.2 if Definition. f(x) : 0 2.3 f{x) = 1 and if II f II f( u) :: l • 'f : [f} is called a !!Utl family of functionals f E 'f implies for all x ·• ···.g. (6;42) ilitam:ele. . Any set of n linearly independen~ functionals over the complex n-dimensional Buclideon space 2.4 . origin. ie called a ~upoorting el ene Definition. S Let CY\ is a total family. S denote the unit sphere centered at the ie said. to have u vertex at the point u if there exists a total family of functionals which are supporting planes of 2.s g{x) : sup Definition. Let g(x} u. be the real-valued function defined by ..!.. log II a '->t II , A complex. >.=1co I11\ be culled the S at The set G ,, { xi g{x) • 1 ) will G - sphere. 2.G 'l'heorem. g(x) has the f'ollowing p-roparties1 (1) g(.Q) • o, •• g(u) : l (2) g( 0( x): • (3) g(x+y) ~ g(x)+g(y) ( 4) g(x) ~ (5) x I~ · implies (ti) II x II~· eg(x} (7) If l°'I g(:x.} II x ll x eG g(x) > O [i.e., g(x):: 1), there exists a complex number absolute value 1 such that the line aagment from x lies in (Flatness eroperty) G. 5 c.. u to E. of 6 E!2.2f. of ill g(~) : By direct substit u~,i on g(u) = As;b l~I log II eAu.11 : -1 , 0 • sup >+1 111 1 o. Also, A+b J~-i lo g j e >. I • 1 • Proof of ( 2 j If • i 0, g(O( x):: sup - 1-log II 0""'11.II• l0<I su p _!__ l og l\eiiO(X!I 0( >-t o I 110<1 Mo I >-I If X: o, /cx\ g(x) • g{O( x) • O by {l). ~ 2! ill g(x+y) • sup A"+O ~ ~i~ f;~ :r 1 Ir.I fil ¼t ~ sup - >.-J.o log II/'" e"'::tll log II e >-x I! !I e 'Ay II (log II e>.xll + lo g II e>. 1 I\) 1 log II e>.x II t' sup - 1 1 1 ),,. ",t.o 1 "I log II e>-.y !I :: g{x) +g(y) ~ 2! ill II a"l( II = 00 u + [ imp liea 'r\.u l 1 -ltil II x II • II x H. "• o Ii\ I ticularthat g(x)<oo, for everyx. ' g(x) ,~ sup - ii'ro m this inequa lity it fo llows i n p; .; r,- Too, by(3) and(4) 3x)is continuous. Proporty ( 5) follows from the lemma: ~• property If for every complex II e f..l<. II ~ a, _ then c of absolute value 1, x ha s the ll x II ~ a. Proof of the ___,... lemma ,_.._........., By contour integration around the ci r-cla I6 l : 1 • Ill: l Proof 2! ill From the definition of numbers /\ i 0, g(x), g (x ) ~ _1_ log I >i. I it follows thut for all complex \I e ).)(I/. Hence for a ll A, \)-..lgtx ) U e c ).x II ~ a , u!·:d t her efore by tho prec e::.l iric lom'!la \11lg("-) T,:. kin g x / ~, a r,d DS'.J Ut;1it1g !\AXIi: \).I Bx H ~ tci . • this inequality gives which implies • g(x) > o. II x 11 ~ :J , end hence x : .;;, Cc contradict.ion.. Therefore Properties ( 4) ~ind ( 5) imply th~t g(x) is finite fot' all :-t. --~ Proof of 1 6' x: ~ If g(x) Since tha ine1uality is trivial. 1 > O, substituting A: g(x) Assume therefore in the inequality I )ij llx \\ ~ e )).tg(x) , usacl in proving {.5) , gives r (lX)-. \I x or n II ~ eg(x}. X f.!:22! £!ill Since g(x) : l, given ~ > :1 A there exists e. numbGr 0 such that II e ,.. If ~ e I >io l (1 - 6 ) • If e. AO : \ ' \, \ , th an ne Ac, ( + l u) II ~ e li\o i ( l - s ) + \/IO I = a \). .. ! ( a. - 0) ' so 0 )( X g{x+c.u)~2-6, or f; g(x+ E.u_) ~12 i.e., to each S>oth:,,re c, I e I : l, such that g (x ; c::u ) ~ 1- ~ ,. 'I'ok e now '-' o. sequence S--+ o, and the corrospondinr €.'I'\.. By ccmpa ctnoas of the corresponds an I'\ \ f.. I : 1 , a subsequence E., • can be selec:ted that is conStr\C e g'(X) iz 1'0 1\[,r.,JJ;\:,1 Yergent to s ay. Thus /\. ae, + lUJ' ~ 1. But, bec0. use G i s convex 2 :;;,' circle (x c, [by property (3) above]. 2.7 g(x+/u)~]. As defined e.bove the fur1ction Remark. nonn in A, and norm in A,,_, the f;lpaca of funct.ional s f Definition. f 2.s g ( x ~ cu) :: 1. Hence G ie its unit sphere. The g-!2.2!,! of Hence ov t:i T' g{x) g(x) is a ;:;,,nr;.ch induces Li H:iturd A. el~ is defined by I\ f IIg : g(x.hl sup ff( x) I • 2.9 f{u) • Theorem. l} is 0. Tho set of' funct1on!::i.ls tot ,:l family. Hence the sphere S has a v,w--tex nt u. 8 !!2..2.!• Let be such that x• f(x') : 0 for all J+ und f E g(x') 111 l (this last requirement _can always be met by nol"!ndizing . If . x• I 'Jt 0, i.e.,-. if of Thaoram 2.6, an g{u -t e.x') • 2. is ·not a total f amily, then by property (7) ot absolute ~alue E, x _: . t.x', then If g(x) : g( ex') : g(x•) : l • x. 1 exists such that g(u1-x) • 2; and by prop erty (2), Consider t he two-dimensional complex space spanr.ed by u and By a functional h defined over that apace such that exten$iOn f1 theorem of Banach (6;55}, th~re exists a to the full--space f 1 (u+x): 2. II h I~ : 1 By the Hahn,,, Bana:ch Theorem ( 6; 55 l h( u + x) : g( u + x) : 2. H~nce, x•) • A. Therefore, II f 1 llg fi(u)+f1 (x): 2 a nd lfi(u) I~ b end has an : 1 , and /lr~llg· g(u) •l • \ f 1 (x) I ~ II f 1 lk · g(x) :: l . i.e., 1, and at the same tim• · fl(x),, 1, a contradiction. Hence rl.. E: Consequently property ( 4)] and has has a vertex at 2.10 f!:22!• u is a total family. Therefore the But • S is contained in O . [Theorem · 2.'1, in common with G. It follows that S also u. I\ u • x .II : II u + x II~ II u II i mplies . x : Corollary. If u. 1 and f'i_ (x) : 1 , 'f x, and also • x •, must equal ~, so G • sphere baa a vertex at fl. (u): f is an element of the total family .Q. J+ . of 11 • f'(x) I : l f(u • x) \ ~ l \1 i" f(x) I:: f f(u+x) I~ 1. Together, these . rellltione imply f( x) : o. Hence x s ~. Theorem _2.9 mny be formulated e.nalytically by means of the --------~- derivative of the norm: 2.11 x0 The ri ght e..nd left ;derivatives of t h~ "t1orm at Definition. in the direction x a.re defined respectiv ely as g If for u fixed x, <.p+ (x 0 , x) = is said to be defined !:U, t.p_ (x 0 x), the derivati v e of the norm , in ill! direction x. 0 x; it is de::,o·t;ed by <f {x~, x). The derivative is defined .£i x 0 if it is defin ed at x 0 for every x. 2.12 - Proof. Whefl \lu• hxll -1 or, II u .. hx II + II u + hx II yields cp+ (u, - x) + <f- (u, x) ~ O h ➔ O,;-, Cf'+ (u, - x) ~ Cf>+(u, x). • c.p_ {u, x) ~ so, ~ II u -t- hx II - 1 h ? O. + h Hence, letting 2.13 Cf+(u, x). h ;;- o, the triangle ineoue.lity • II u - hx -t- u + hx II 2 : cp_ {u, x) ~ Theorem [t..scoli (1;53)]. But, clearly, - cp+ (u, - x ) : c.p_ (u, X '); <F+ (u, x). Theorem [ tiazur (7;75}]. If a. is a real number such that <.p_ (u, xl) ~ a ~ <.p..,. {u, xl.), where xl d8l'lotee a fixed element of' the space, there exists a functional F( x 4) a a, and F( x) ~ !:!22f• t F with the properties f!'(u) s l , II x II • Consider the linear subspa ce being real numbers. A~ : { x = .su +txl. } of A, s und by f(x) :: s t-t a is clearly diatributiv e '"nTid continuous and '- ttctl'ic c:J i s a functional. Since It has, for tx 1 : x- su; if llu+~xll Letting x&Al., the property h f(x) ~ Ai Cf>+ (u, x). ie positive ml<l sma ll (l+sh)'O), i hon -1. 6 + l+:h h ➔ O+, this gives q\(u, tx.) ~ t a 'l'he roa-1-valuad function defined over [nu +lY:h-txlll - 1]. 'f + (u, x) • s + Cf+ (u, tx). in a ll c a ses; for if t;> O, But a~ 4>+ (u, x1 ) givos c.p + (u, i:.x l ) ; rind if tu~ - t a or tx ) ~ 1 'f- (u, Cf>+ (u, x) ~ 6 + ta : exists e. functiom,1 for x E: A, and But Banach has shown ( 6; 27} t hat there F(x) defined ov ,,r a ll of F{x(.= f{x) for t = O), { s : 1, f(x). F(u) ~ x & A1 • In particular, when l, und when x s x 1 (s • 2.14 Theorem. a vertex a.t x : 0( u, at f{u + hx} ~ ~ <p (u, x) h Therefore f E <p_ (u, x). c.p+ (u, x) = Suppose r::. . ll u+hx II. h < 0, and when S hava 'I'hen for h > O, Hance, '&ihan 11 u + hx 11 - l f'(x) • h tend to zero in these expressions gives ~ f ( X) • f{x) : <.p (u, x). But ({) + ( U, X) : Then f(x} • O< or f(x) ~ <.p_ ( U, X) : cp (u, x) for all r € J+ . Let f(x - J • But this implies x .. O< u : '1, or 0( O< q>+ (u, x) cp (U, X) • be the common u) = 0, for o.11 x = 0/ u, whi ch completes the proof of the necessity. (Sufficiency)• of the form If tp+ {u, x) for every ~- (u, x) / x:: o< u, Theorem 2.12 implies x not tp_ (u, x) < cp4-(u, x). I 0< u, there ie a number a l such that tp_ (u, xl.) <. al. < <.p+-(u, x 1 ). Then 'l"heorem 2.13 guarantees the So, if F h, h (p_ ( U, X) value }: be defined only in the di rection II f llgg{u ;- hx) ~ II u + hx II - l Letting and 1 u. f E "f , and an arbitrary real number ~ :'( x where Of. is any complex number. f!:2.2,! - (Necessity). f(x) x : u A necessary and sufficient condition that ia that u S ts 1 ), 0, £,• (x) ~ ii x I! II F' II : 1 , and hence that The conclusions of this theorem imply ie a plane of support of A, such that X 4 existence of a functional F{x) of norm 1 such that F(u) :s l and 11 of these functionals is a total f amily ma y be proved a S follo ws. x0 F(x 0 ) = 0 is such that of the form But if X 0 X ,/ 0 = 0( u, 0( U9 a real for all 0 proves that F0 (x 0 ) : in l F}, then thi a would imply ::!. S cunnot be f ( u} :: J.- / ( C>: u) 0( ' a0 x0 0( I 0 q,_ (u, Xe,) < ao < (f)+.(u, xo) for which ~• If can bo found such that and hence ul so a functional a 0 I 0, again a contradiction. S has a vertex at = i) ., u. So i;;, € { F} x 0 I ~, whi ch 3.1 ring Afl. Theorem. The radical R of the n-climansional normed is the set of al l nilpotent elements of Proof. is an ideal and R condition. Hence (8;ti4) R is nilpotent, i,e. there exists a p for which Rt:> : for every Conversely, if x • lim l"l.-.00 II x n /I rt : o, so • hence to 3.2 satisfies the d escend inf chain AY\. positive integer x 6 R. Ar.. x l ~ }. In particular, is nilpotent x "F': ~ Max I x( F) I M belongs to every mo.ximal ided R, Theorem. f!:22!• If Gelfand (5;8) If n >1, then An. has at least one maximal ideal, had no maximal ideals, then by a Theorem of An An would be isomorphic to the space of complex numbers. But any two isomorphic vector spaces have the same dimension, Hence n • lt a contradiction. 3.3 Definition. A set of maximal ideals 1'I 1 , ••• , :le. is said to be linearly independent if the multiplicative functionals M: 1 (x), •••, l\ (x) are linearly independent. are linearly independent maxi:ml If ideals of the ring A (which need not be finite dimensional) then there exists an element 1 = 2, 3, ••• - Proof. of y such that Ll l..(y) • O, e. Consider Since each A (i.e., a linear subspace of' dimension 12 is a hyperpl nne n - l.), any hyp ,"> rplane 13 containing can be expreased as a linear combination of L1 indepondant, the hyper'.)lane &I and therefore cannot contHin 1 1• have the point Hence points 3.5 is not ex.prossible in that form, in common. .Q Theorem. But intersects as they exist in y LI which A, any finite set of maximal ideals is linear- In ly independent. Proof• In an arbitrary set of k► 1 maximal ideals let M_ , •••, Mt.' be a maximal subset of' linearly independent 1 suppose t < k. Then if H t+t denotes one of the -but not in this subset, there exist complex numbers M•s aero such that ,,, = L C ( i. .,:t. Me +I ceding Lemma there is an element (xy) : • C<, ;-.;.{x) M. (y) ' .. I \.+ I V ( diction. O{. !\\ . (xy) ~=1 "\ " 't. 1. x, (y) , 0, and, for an arbit n, ry '" t+- I X This statos that [ , ) : C( not in oc. 1 M1 (x) rn 1 (y)/o. :r M,, "\. not all O; then by the pre-- X e : Mt +- I \, for which M, (y) / 0 and y Hence, for any ... ' e . Therefore ~I' Say -i.::t and in the set 0(. ~ I.i 's MI (y} ! Ht+i ( • rL t+- I y) and ~' ' ) ..i I \X • M1 aro the same mt-1.ximnl ideal, a contra- Thus e : k and the Theorera follows. Corollan. The maximal ideals of A, n are Uno~,rly independent. 3.'7 ~• If A11 has k maximal ideals, then dim ':l • n - k~ 14 Since the dim ti;• IC \. '.•i . 's .-:re l inot:~rly independet1t ~nd '- n - 1, dim R: dim :r 3.8 Remark. u -t-R 5 u +R and M1.. l - (k - 1) ' I impliaa the existence of II Ut-X II ~ 1. implies X E:R =n • k obtained by translation of If int~rior points of then s.=l. II u+x 11<1 Since (5;4), it follows that the subspace l/1 - /.,,,.k..~ R to u u carmot cont a in A• A,._ in common as shown by the following Theorem. 3.9 Theorem. f!:22!• In A.,,,, if x6R, x/~, then II u+x H>l. Let i,=p(x)be the smalleet positive integar such that xP • ~ for each xe:R Uu + x Ii ~ 1, I a na l ~ llu+x,llr.~ (see Theorem 3.1). Then, if Pt -:: p (X) , ll(u+x,>'111 or, n : l -! \'his states that A is finite dimensional, say have only the point X ... +--- u II . x 1 6 R, xii .;i, 15 As n ~ oo, the quantity in brackets tends to v.nd p n! tends to infinity. Hence, the rig:ht-hr nd side of the inequality tends to infinity, a contradiction. 11 Therefore u + x 1 II > 1. 3.10 Theorem. If An. k ~ 1 has maximal ideals, then tains exactly • k _ non-trivial idempoten:ts. contains the trivial idempotents f!:22!• and ,€l An. con- (Of course every ring u.) By Corollary 3.6 and Lemma 3.4 elements Y1 , ••• , Yk may be found for which Y. ( M.;) ,. l. . ={ ·The complex numbers 0 when J O(i. ;r O<t. O i I 1' . ) i • i • when may be assumed all distinct (distinct values can always be obtained by multiplying the y. 's k ,. non-zero constants). 'l'he element y: [ y~ by a pp r op riata then has the property t.=-1 y( Mi.): cc,, i.e., its spectrum consists of the k distinct points It follows from e. Theorem of Dunford and Hille (10;105) that contains exactly k idempotents j.,. = 8. ., . j. ' J:1 V 3.11 t.'- '\, - :i dim L .1i. • u , j. /e, u. \, First fundamental structure theorem. where z, every representation k-1 X: L i=l Suppos€l tho radicd z such that Then in t~rms of this Re with the properties k i "' contuins an element r j 1 , ••• , Jk An 0(.1, z ,. xE.An has o. uni qus Hi potents of Theo-rem 3.10 1 and t he If y •s 0(-L is another element of Ah. e re complex numbers. with a r epr0s entat ion k-1 the sum y = L x + y is given by y( !/ J + JL !.c I k-l L[ x+y = + x{ ~ii.) + yC.:d] ji. i :. I and the product xy by k-1 x.y • L + x( :.\) y( ;s'. d j,:_ ,; CI which is of the mixed scalar and convolution type. representation of xy, k-t xy = [= i. Namely, in the n-k xy( U,:) j;. + [ Ii i. z i= 0 I ' the components are riven by and (convolution product). - Proof. where Ak , -I dimension 1 ■ 1, Consider the ring d ecompo sit ion k-1 : E9 /' n-k .. I { ) f;, x( Mi) j-t} n - kT1 since contains R, for if ... , An s Ak-1 A k- I is of dimension x e R, then x( Tu\) : 0 k - 1.) for k; and it also contains the aJ emetit ). complex. since the elements zo = \, z, za., 0 •• ' z n.- k The rin g wh ich is '.:'hor0fora, 17 independent, they constitute a b:-; sis in of A"' has a unique representation as L X : X 1 , $ X" and every elr~ment where n.-k k-1 x': .l\. ri.-k-t-l x(;w.) j., ,. i. L x" = and ,. : I O(i. z i In this decomposition • i:0 the sum and the product or two elements x • x • 1$ x", y .. y • e y" take the forms: X + y : -t- 'J I El, x" ;-- y" Xt k'-1 x' + y•,, [x(Mi) + y(t\)] j" [ ~=I where rt-k x" + y" : ( 0c {.. + [ (3i ) z,. l, =-0 and X'/ = X 'y e x"y" f k-r [ x•y• • x{ M,} ,. y{ M,) ,. j.,. i=t ( scalar -oroduct) where n-k x"y" • 11-k L Yi ;. ~,; l:: • 't, z • i:::O C¥eA-t•' (convolution product) 3.12 RemaY'k. The coefficients CXt , associated with each x functionals over (i.i;lll), say An. •> f i., (&,. ) : n - k 1'1.-k ~ L...., \ =O i -l:'i,J o, 1, •••, n - k by the representation above define are linearly independent for if' between O and i : and the f '\,•1 r.,. {x) 1'1.-k • °'-<.. • These functioncls f-i. rt, where i • is i. .. 0 i,,/d.' M.i. 'e / - L are constants, this would give ,I f.;_ fi (z 1. ) : o, n • k+ 1 e contradiction, since by 18 "' ! ii definition ' i, I ) (z : 1. It should also ba observed that ceding Theorem are normed rinps. in tho prek-l ?heir unit el 0rnei1ts are )""" L i:I a,nd jk norm in An• Indeed 9 o.ny norm admisai ble in A k-1 II x II: II x'@x" an admissible norm in and A l"\-k-+t II A~• : is nutomati ce.lly ilax ( and x• I I II ' A r, _ k,,. 1 II l\ ' rospocti vely, , In particular, one ~~y tak e the ~orms in to be . II X' II • : '.Jax . . Ix ( M. ) I '\. !~1.~k-1 11 x" An Conversely, if' and II" are admis~:d ble norms in Ak-1 then ' respectively, and the norm in thase rin gs i s the rw.me as the admissible in both II i .," I I! " : These will be referred to as tha natural !.1.£.! ~ of Ak-\ und and At\_ k ,.. 1 , will be called the n a t u r a l ~ ~ of This terminolof!Y is justified by the followin g Theorem. 3.13 Theorem. '"I'he natural fl ::.t norm of Ar,_ has the l1utnesa Property (7) of Theorem 2.ti. ~. If . 11 x' II' : I x(li. ) I : 1, and the maximum ~ then taking t: II x• + e u 11 = 2. I :: 1, then 6 :r O(" occuts say for property. II<ax I ' L .~1.~,._-1 ~.i 1, Namely to be Similarly, if n.-k = LI i.=o 0(, I O(_, I x('.: 1) gives r:ivas the desired II x" UIf n.-k 0-: ,_. ·' .... 1 1 ( y· .: ;; "' -l-- [ ·(:::: ! Thi3 Theorem follows im,uedi ately fr om these hm cases. 3.14 sion n Theorem. Any t wo l'lormec1 rin r: s with radic E: lS of dimension r A11 , f..n' bot h of di men- oatisfy i rip: th e com1it ion of' Theorem 3.11 are isomorphic. Proof• According to Thoorem 3.11 t v.10 general el ernnnt£ x E A,... 9 may be repras ~nted as k-1 L X: x(;~; _J ji. (=I k-1 L x•: 0(. I Z I i.. • '!. t::./ Ji. The :isomorphism arbitrhrily s etting J. \. and 1. z - A ~ i'\ now follo ws by defining I I'\. J{~..... j;_ '• So X«-+X' components a r e equal. a homeomorphism. when ev er corres pond in g 'fhis isomorphism is obviously It is an equivalence if tho norms in Ah.. 1:1.nd ;in.' a.re replaced by the naturul flat norms. 3.15 Se cond fundamental theoreme * = rr L of 1'heorem 3.11, and let A,-L space {as a. Bana ch spa ce) of sat:i. sfy the conditicm g, ••• ))- denot e the conjugate 9 Thon An• An, Let ~ j\, .. I > ce.11 be made il1to a normed ring by taking the functi on::,ls as a bnsis so that, for a given z, has a unique represent ation Ls n- i,_ i = L"" r - - - -•"\' f(j.) 1. ., ~. j I 1. + i.::.t L' ! ( Z ;, ) f . • i.:.: I 'rhe pro duct and norm of elements oi' r. - k i-) 1--t i =I ,. ~ "~ 2.0 k ll f U : Max { L i =- I a linearly independent sat; f'or if t i L fl . M• + ,. =I 1. thon l : f., (z1.) o, if '\. ~-=I ,ji.' · ;~ ........, . , M,(z'-) + f- I'\•· ~ ) ,. •> /',i f'L, 'I. .) ;\ 'I.. f ( z,. ) : and - ... , so that every is a basis for a representation n-k k f = To evaluate the >-"' '--' i:. I f, 's U, I!i · ! l. .... + >' L;--1 '\, • form n-k )--J' f-;_ Mi (je > + i 1, 1.,:J k r (je) = ),, f. =r L i=I Similarly, r- - ( . r.1 . Z e) l. l. The-ref ore, n-k k f = L i.= I f(.1.) tI.'l + L i.:: I f{ z 'I.) f. ~ • f e.' A n.* has .the properties of a product in a normed rin g ., Th ,:, unit cl J:210nt i<J ,1-k • ,{ • + ) ~, ~ F'inally , the ;:iorm is t,d~.!1issibl0 s:i.nct) i-t 1s tho f. .. ~ L:: I natural norm correspondinr: to the mixed ·oroduct o ~ 3.Hi Theorem. ~ • idempotent s, and - Proof. In A V\ A r'I. ha s the el emer1t s n - k -+· l fc.g in Theorem 3.15. :,,re max5.:nal ideals. The idempotancy of the definition of 1·\ , ••• , f f1.. _ k follows directly from the ~~xhibiting in each case the components of t.heir generic elements, tho n • k + 1 mt,xirrw.1 i dods are {(o, r(J:i)' •••t f(jk), f(z),r(z 2 ), f(z 3 ), {(f(j, ), f(ji}, f(f(j,), f {j2.)' ... , ... , Corollaa. is the dual of ~ A'fl. tt : A"-, so 0 •• , f(z "' -k))} f{ji.-), O,f(z:.),f(z'3 ), •• 0' f(z"-~))} f(jk), f(z), () 9 f(z~}, • ':'he normad ring 0.' f ( zn.-k) }t, defined in Thaorem 3.15 Ar~ taken with the nntur-0.l n~t norm, that is An. is re:tlexive as a normed ring. PART I V UP~RATORS 4.1 Definition. A set B is called {weakly) saqu 8ntially compact if every infinite sequence t "n.} in B contains a subse- quence converging {weakly) to u point in 4.2 Remark. The word operator wil l be used or1ly in t he sonse of linear operato~ (i.e., distributive and continuous). 4.3 Def'inition. The operator T ls {weakly ) compl et ely con- tinuous if it transforms bounded sets into (weakl y ) se ,1u : mtially compact sets. 4.4 Definition. The element xEA completely continuous if the opere:1.tor Tx is said t o be {weakl y ) definoc by 1: y : xy is (weakly) completely continuous. 4.5 Remarks. The standard abbreviations c. c ~md w. c. c. will be used to designate completely continuous and wookly completely continuous operators and elements. as defined above. In A 'r'I. every x c. c. e.nd w. c. c. ie trivially low• thie is not true of infinite dimensional rings. Used will ba indecomposable. ~ or A has the Peirce decomposition 4.6 .and if - Thoorem. x is Proof. idempotents. Ce If c., shown be- Some of tho rings It ia recnlled that such rings cannot contain idempotents other than then As u, for if A contains j s ~, u, A : (u - j) A EB jA. A is i nfinit e climenaional and inda to ".'lposable, than xe r o In vi,;w of the preceding Remark, A eo?1t n ir1s 110 non-trivi al Hance, by a 'r'h eor om of Lor ch (9; 41 6 ) the npecirum 22 . 6" (x) of x is cormec·ted.. arbitrary poini in CS" (x) Tirnrefor-e, if and is proved), connectedness implies th&.t a ssquence of points v (x) be found in 3ut as c. c., it follows from a Theorem of Riasz (11;90) thut t he values is /J.. for which ( r U • omitting tha origin, Hence Aon. ~ Ao . which converges to () (x) x) -1 doen not exist tire discrete, i a 0 .. , { ~ } is cliscrete•, a contn di etio:1. () (x) : reduces to the point (\ : o, and therefore z E. R. If 4.7 Theorem. (1) A is infinite dimensional and indecomposable, (2) f( /\) is a function which is 1rnalytic on ;:; rid innide tho circle I AI : r, (3) II x ll ~ ( 4) the olamant (1} there exists a unique r, f( x) is c. c., then {2) is a root of /\0 - Proof. Ao such that X - /\ 0 u E:. R r( Ji). The hypothesis implioa that re A> has ar: expansion CXJ L - L Therefora, si~ee valid fo1· 'r\.:: 0 00 f(x) - .. ('. is un element of ""'tl X A. By the preceding Thoorom n:o f(x) E R. Hence, for every maximc.1 ideul • L I'\. an[x( t )] But that is - L anxl"l( n ) ,s.:::: o 1\:0 : f ( x) f(x) { L~ ) :t o, oC 00 f( x( M) ) M, on = o. !x( LJ ) I ~ 11 x II~ r, £Hld f( A) has only rt finite number of' 2. 4 zeros in I A I ~ r. Thus 0- (x) that contradicts the fact that 0-{x) A ie indecomposable ( 9; 416) unless consists of only one point: CS-(x) :: x( M) • i\ 0 •It follows that is the only value of (x .. . Au} ( M) : O for all A. 0 for all ti. A for which that is for which E'urth er- x - /\ u E R. f ( /\ 0 ) .: o. more and is finite and hence isolated, and 4.8 Corella!:%• If A is inrinit-e dimonaional end indecompo sable, X n. ii e. e., then x E:R. Proot. Since >. :: 0 is the only root of r( A> : Ar. , . this - root is /\ 0 • H811ce., x • /\ 0 U • x ER. Remarks. • The converse of Theorem 4.6 is not true. 4.9 There exist indecomposable infinite dimensional rings whose r e.dic u.ls cont a in elements ' • that are . not series c. c. a :a For example, consider the ring of convergent power a I"\, x n. for which lan./<OO. the product of two · elements in this ring a re defined as ::>a 00 Do¢ L L • L anx n.:O n. + bl'\xn: 'I\ ::.o · where C 't\ a \,, b l\,•'I., • L (an.t- b,Jxn., and ab• II a I) i:;:u The only maxilllal ideal, hence R, a0 • o, that is a • L L • L I C n, xn I'\ ::co Th e norm is given by oO :---, a+ b • oO YI= Ci n. The sum and consists of all n, ;::.O a•s for ~hich .. an.:x 1 ". The element a • t\: I which is in R io not e. c. For, consider in A the bounded aet 00 C' , I"\ k- . The transformed set C)h.k X ., X • 1b (k) } where b Ck): [ l'lal • a l'I.\• I 00 I', ab -k'I ; is the sat of all elements 1 , a nd every X r,_:o k< r-~ r.~ C, - sequence of these elemants diverges bec r: use, t akin g II ab(k) - ab{!) II : :n+I :t-k•n ) ~l... ;,;, 2 - 2 --3- - + e, 2f-lt HoweveT, there are indecomposable infinite dimensional ringu whose radic nls are made up entirely of c. c. elements. For exmnple, the t ring of operators of the Volterra type Tf g: { f(t - s) g ( s) ds, 0 O ~ t ~ l, from the space of real continuous functions ..the unit interval and into itself'• .Michal and mconin. (12) ' 'l'he product 1 t by · r3 : o ~ t T.f :r Tf 11 Tf r, 11 : is defined o~t~ I I f{t)f. This ring has no '. unit element, but may be imbadded in the space of couples with complex. Defining the product by ( A, \ ,~T{ +A?~~+ T+. T+,t) I Al + II T II .(1,0). ( /\, Tt) ., ) ( \ , T+ )( \., T,c and the norm by 1, is t aken to be t 11r11,: sup ~ I and the norm of T, 0 thenonuof f(t)t ',! r,(t- s) f~(s) ds, over 'l'hi s ring has been discussed by Each continuous function .gives rise to an operator Tf. ' g( t) ' . II (i\, Tf) II= the space becomes a normed ring with the unit element Now The radical is the only maximal ideal, consider the bounded sequence of clements • I Arel, II gn// ( K for all n. element of R, the elements (O, form an equi-continuoue f 0mily. is closed, , one can find f( s) Then if {O, 'rf) is an ~{ rbit ra t"'/ An Tt + Tt Tgn) :: ( 0 , Tf ){ /',,.,,, 'lg'r,.) For, since the i nterval is uniformly continuous and hence given 6 > O such that e I f( e 1 ) - f ( s~) I < 3K O ~ s ..$ 1 E. > 0 whenever 26 taking I SI - s .t I < 8 • Therefore, c \ .. t:t < 8I = '.iin ( 3 II f l! K ~ ' 1, rt 0 ~ and wri ting , . + l f(t-s)~n.(s)dS I ·o < t , ~ 1, t,2. "" a nd for '" g~ gives for all n ... /In[ f(t,) - f(t,)] + ... •l l..i_ f(t~ - s)gn.(s)ds 0 I;,\ •.[ f(t,) _ f(t,) J - l t4 + [r(t 1 • s) - f(t, - s)] v,,_( s)de Ot + jr<t 1 - s) r.n(s)ds t. ~ \r(t,) - f(t.z.) j K + lr(t 1 - ( t I - t ;_) /I f I! K t c c 3K 3K ~-+ s) - f(t:i. - s)j K + 8, llrl/ K ~ E. • It follows by Ascoli 's Theorom that the swqu en ce uniformly convergent subsequence that Tf is 4.10 Theorem. gr'\. t I 0 f\._' contains a converges, so Hence c. c. The set C of all c. c. elements of i, is a closed ideal. f!:22.!• xEC Consider in a subsequence A the bounded sequence { zn. } exists such that I Similarly if { z n,} such that .. • limits gives lim x• EC 1 im II xzn na., h'I~ •♦ 00 J. - xzm,. ti (x +x')zn 2. fl • o. (x + x')zl"h 'rhen if /1 xzrc - xz 11, II lim "\/Y\ -+OQ there is a subsequence "Iv_ >IT\2,.➔ ()GI { z n.} . { zn.a.} I I of Combinir1g the two t. II II o. Therefore c_ (' X t ., ,., q c::; v o ii xyz 11 lim _, · m, ➔ 00 ·,) ' t.e., Also, if - is un arbitrr1 r-y element of y I!' X2: n, XJZ 111, I Thus x.y~C. C is an ideal. To show tha t consider a convergent sequenc e of' elements operators The convergent sequence. that is x defined over Xn C is clos r,id in xri_-+x C9 by A Hence the limit op erat or say. f orm a is Tx c. c. (G;9fi), c. c. is Lemma. 4.11 A9 The limit of a convergent sequence of w. c. c. w. c. c. operator. operators is a The proof of this Lemma is entirely similar t o th ut given by Banach ( 6; 96) in the corresponding ease of c. c. operators, and will be omitted. 4.12 Theorem. The sat of all ~i w. e. c. elements in A is a. closed ideal. f.!2.2.!• , r i zn., t since can be found such that is bounded, if { z", } { z,..,} t z,,_} be boundoo. If x E: ~ , a subsequence Let t converges weakly. x'E: W, a subs equence ~ f l x'zl'\a. 1 exists such that {x + x' )zn. xzn, yf:A, { YZn 1 } there exists a subsequence converges weakly. Thus the Tx,., TX w. c. c. ' 80 that is also bounded. :Ient:e for operators Tx. is w. c. c., xrtE W defined by 'l'hat and i.e., xE VI . f\. ~ 1 xe W ,, is stron rlY closed lim ll xr.- - XII • o, th 0n n. ➔ ;:)Q TX { z Y'\. } y(xzr.J • x(yz.._J.) for which W ie an ideal. For if H follows that Again since f Y'n:.. J follo•ns from Lemma 4.11. l z ~"- } of x + x • v; . converges weakly, i.e., is bounded, if converges weakly• Similarly y • Xn.Y converge to 4.13 Definition. A ring t~st satisfies the t wo condit ions ; (1) The ring is a star ring (2) Uniform convergence of the sequence xn.(;'-'.) vJit ?l Y"G 3pHct to the maximal ideals implies weak convcrgonce of' -will be denoted by -'It' A, V •• n. and its cor1ju gat e s pa ce as e Be.na ch spnca A with the ~ r· topolog,v; of Gelfand and Silov (13;3 ')) \Yill be ca lled oo'e it is denoted by A • The space of maximal id,m.ls of l. "" a compact Hausdorff space (13;31)J. C( oo"'C) space of complex valued functions over , ar,d C{ ----oY'C )* the C( o'(C, ) • conjugate space of 4.14 Theorem. i~'very element element of C( ooG )* • E!:2.2.f• 1,""t(:; will desi gYw.te the -* f E- A can be extended to an 3y a Theorem of Gelfand (13~34) every <__p (c.) EC{ o)"t;) / is a uniform limit of functions xrf :) . 4_.13 implies the weak co nvergence of f E- C( o'tG ). every f( cp ) Conditi::m Ci) of Definition x n.' -1~• a ., f( <f ) : Define lim l im n. ➔ oo f{x,). f 'f x) n "X~'·ts '-'-" . :for '.i'he function f l - oc, is cl oarly independent of the cho: ce of the s equeYJCG It will now be verified that l\.-.. l>Gl C( '{'ft.~ ) ~ 0 H c.p - lim • is a contirmous function ov s r <-p E C( ofG ) , and u se quence Consider a fixed i.e., f n. Cf0 II C( ooe) _ : o. Then for every t:1 ere exists n e. subsequence { x n, } such that - rn Tn h'I < - 1n !r(xn• I ) r( cpn > I < 1 I! X "'1 whenever n1 ~ NI It follows thot o q:>0 • um Ir< ro Tn > - r( w" T , >I • o, l'\ ➔ oO every -. ---- 'it" ' r E c( arc > x n..( :·r) converges uni f'ornily to " Hence, usi ng Cond ition ( 2 ) of ~)e finition 4.13, f E C( '()j'(; ) * o or , this property is unique. The follo winf~ Lem % of' ['. akutu ni 's {14;1012) is given ,;ithout pro of. 4.16 Lo'oma. Any functional i' JJ,- {Z) sent ed as a completely additive measure _., ,- :, the smallest Borel collection of sats of C{ nY.'> ) over crrn be r spre- with respect to containing the opm1 subsets m. 4.1 '7 - Proof. -~ -it' Lemma. A - - * are isomorphic. C( '.f'(G ) and Indeed, the mapping defined by the identity mrrpping of 'A • is by Corollary 4.15 a 1:1 onto mi :pping of Moreover, as But II x II ~ 1 II : sup ii~ II~ I I f • re x) l - II xsupII ~I If(x( t'.)) I • I x( E) l II~ 1 implies sup M II X to f(x( M)) :r f(x), it follows that r 11-lf A II -* A ; ~ II X ii~ l is included in the set where llx► .:f i. e., t he sat ' sup I x( E) \ ~ 1. Her:ee, .M. II f ''ccml • s~J.p I x{ M) I• \\ f !IC( mf .. Whence, it follows from a result of Banach (6;41) that - A 'flt is -,t- C( YrG isomorphie to 4.18 Corolla.a• weak convergence in 4.19 then Theorem. } • Weak convergence in ~ GU C( ~ A is equivalent to >*. If T is a w. c. c. mapping of A into itself, T 2. ie -* the functional f( T x). Using the Consider over A notation (x, r) (Tx, 'l',. f). Let convergent, Tx"----+ x() that 2. 2. to denote f (x) gives in that case (T l x n.} be a bounded sequence; then Tx 'ti say. 2 I"~ x, f) :; is weakly To complete the proof it must b0 shown T x I'\. converges uniformly with respect to all functionals f of norm II f II ~ 1. I f the contn:ry is assumed, then there 0xi sts o. r I"~ 's sequence of I! rh. 11 ~ 1 of norm > 0 has been preassigned arbitrarily. where co suppose nv = n, n • 1, 2, ••• (6; 100) where it is proved that for such that T w. c. c. is similar As 0 Tc. c. T* i a al so implies ?or convenience, w. c. c. [ see 7* c. c.; the p·roof J, a subsequence of { Tf~ } can 'be selected {which again ie supposed to be the full sequence { Tfn.}) which ia weakly convergent as elements, scy. Thus the proof of the ~'haorem will be complete if the followin ;:r L e :'l'ta t'urnishing the desired condition can be established. (1) r ,) r each ( 2} xn.( t: } n ( 3) ",:-;; 9 xn.<i'Jj> is uniformly bounded, i.e., and ovary the functionals (f n. , Xn) !:!:2.2.!• X 0 ( IA ), I xri.( M) l ~ K for ev ery M elements to then converges to f n € C( ooC { converge waakly as f? converges. By Lemma 4.16 and the weak convergence of that if f'-n. f- n ( ~) converges to r r.. it follows f n,, then is the measure function associated with Now,· j(rl'\, x,) - (f0 , xJ/ :$ I {fl\, x 0 ) - (f0 , x 0 ) / + /(rn., x 0 } - (±'0 , xn)/ :II,l+II2.l. Consider :r I first. By y; goroff•s Theorem (15; 18), ar1 when n ~ n0 ;~ exists for , such thnt 31 U ,, ( ?!c) - _<. 1 'ttt • c c- w•n,.,"' ...• e , Th e remark a bove sh ows t hat as n ~ 00 , to f0 so fy,. ( ,/ ) II, I~ E., weakly, Secondly consider I2 "'t, c d. (motes t h e complement of l_l : f-'f\. (:{ ) ~ 2 c for :l tends t o n ~ nQ. in f oC/) Hence, a s f r.. tends n ~ n1 • for I1 • xl\( M) - xo( M) J drn. m • S. [ xn.( b ) • x { I:;:)J dffl +0 e: • I a.I + .l.22. ) • J The weak convergence of is a constant. ~ f re. II fr. II ~ Y •, implies t hut 'oVher e l\ ' Clearly, theref ore, £• (Total Var. of f-ri.> ~ E. II ffl. // ~ €, K' . Also, I I,_, t I ~ 4-C e for n sufficiently l a rge. by combining the inequalities for 4;.21 Theorem. If A The Le m:"'.la now f ollows I Ii I , / I~,I / , and / I :i.~ / • is infinite dimensional, it c a r.not be reflexive. f.!:22.!• Assume A is reflexive. Then A is weakly complete (ltiJ 423 ). Hence the identity operator Theorem 4.19 (Tv..• T~) it is a lso atld c. c. so by Th erefore the unit sphere is sequentially compa ct, and i t follows that A ie finite dimensiona l (6;84), a contradiction. 4. 22 Corollary. If every el emer1t of A is c. c., t h en A is finite dimen sional. Proof. If Cs A, then A ie reflexive. (3) The Coroll ur y now follows from Theorem 4. 21. 4.23 . If T is a Theorem [Dunford and Pettis (17;385) arid Phillip s ( 1 8;536 )] • w. e. c. operator mapping L into L, then T2. is c. c. 32 isomorphic to C( is But J1 ) , where w. c. c. is similar], 0nd and if Yf - A-ring. Co C it 00 Hence, by 'I'heo rem 4.1 9, is then XE: R. Theorem •l.19 implies that Corollary 4. '7, w. c. A is infinite dimensional and indecomposabl09 is w. c. c., Proof. is an is ~s u. comp1:,ct Hausdorff spa ce. and, since Theorem. X J1 [ soe (6;100); the proof il'i the 11 ) C( is 4.24 * a fundamental result of Kakutani (14;1021) !~ X 2. is Hence, by x E: R. 4.25 Theorem. If (1) A is infinite dimensional arld indecomposable (2) f( ~} is a function which is analytic on and inside the ci \·cle (3) ll x II ~ r ( 4) tho element (1) there exists a unique f(x) is w. c. c. , then ~• such that x • /\ 0 u E. R The result follows immedia-c.aly frorn Theorem 4.7 upon observing that and /\ 0 2. [ f{ /\ ) ] satisfies the conditions of that Theorem. 4.2G Corollapr. If A is infinite dimensional nnd indecomposable, n. x is w. e. c., then $everal problems related to topics in Part III nr e dis cussed in / t,his ;Appendix. 1. Proj ectio1;1 operators g"!fil:, Ao Definition. into A projection over of A A is an operator b .ki ng A A such that (1) P(x + y) = PX+ '?y (2) Pxy • PxPy (3) P'-x:, P(Px) • Px Jection P • An i d empotent Remark. defined by j Px: jx. trivial projection opera.tors: over P generates over t he pro- In this s ense ovary ring ad mits two Px = x und A is not necessa:rily of the type example shows. A Px a ~. Px • jx Gver-y projection as the following This example, it should be observed, involves ~ot more- - ly an algebraie ring, but a normed ring. Thus, aven in t~e presence of a norm the representation of P in the form Px ■ jx is not possi ble in general. :~xa:imle. (where Consider the ring s ' Y\ t ~ A3 of elements are complex) with product x a { S lJ r"j , C ) xx'• 5 •• ~vi· + s·~ ' ( ?;" •), and norm II X ll Max( 15 I + I tr I This Y'ir1g !·w s two non-trivi e.l i d empot ont s, ' i ?; I J• JI • (1, o, 0) and j 2. = ( 1, 1, 1}. Their products with tho r onarul ( ~ \ ( s, ~ , ~ ) are • ( a, o, ~ ). Now the operator element Xa jl X: ( s' ~ ' 0) u defined by 33 Ux • arid ( ~ .J,._ X s' o, 0) 34 is clea rly a projection. ( ~ '' x a rbi t rary ........ Indeed, t here i s no fixed element . , ~ However, for ~.) Ux • x'x such that ••. re quire f 5 ' :: f , whence f·= 1, vi·=--1s ' for all .f'. - For this would Xt• ~ ~' = o, I 1' + f'q = o, and c; ' = 0 which mear1s that x' is not constant. If P ia of the form Theorem. If f.!:2.2!· The case Px:: jx, obviously o.n ideal. For n Px:, jx then j: ~ the following result may be stated. PA is a closed ideal of is trivial, so assume PA j ,/· ~. e II jxn - y II ~ II _j II , is jxn. ➔ Y • In it consider a convergent sequence sufficiently large A. II jx,.. • jy II • so II J4 Xn - jy II • II J ( j x n • Y} II ~ II j I/ ill • i • Thus jxn. ~ jy : y, the ring . ideal. f ( PA A"3 P is of the form Px : jx, will not be a maximal ideal of A. For example, in { (o, o, ~ }} ie not a maximal above P:t A~ • j2. A3 , The maximal ideals in that ease are { ( o, Y) , \ ) } and S' Y{ ' O) } • 2. Homomorphisms. Theorem. If B is a normed ring with zero radical, and a homomorphism from h PA is eloaed. F.Nen when a projection Remark. in general i.e., h is B whose kernel is a closed sat, then A fil!12 is continuous. f!:22!• The kernel of The mapping H ot h is defined by K• 1xeA / h(x) • .Q}. A/K onto B that is induced by algebraic isomorphism. Since K, h which is an ideal of is an A, is closed, it follows trom a Theorem of Gelfand (5; 17) that H is a homeo- morphism. X is the coset Hence H is continuous. tJoreover, if 35 of A/K ' whi ch corresponds to an arbitrary h{x) : H(X ), and so X is an ideal of that A, II h(x) I/= /I H{ X) II ~ x'E A, then l\ H II · HXii• But , since uX ii a infEX II x' u ~ II n. It follows II f{x) Ii 6: II H II II x II, ,c x' i.e., h is bounded ·and hence continuous. 1. ASCOLI, G. Sugli spazi lineari metrizi a le loro vari ata lineari, Ann. di lint., vol. 10 (1932) PP• 33-81. 2. KR£IN, M. and s. Sur l 'eapaee des fonctions continues definies sur un bieompaet de Hausdorff at sea sousepacee semiordonnas, Trane. Moscow Math. Soc., vol. 55 (1943) PP• 1-35. 3. i<'R.JUND.LICH, M. Completely continuous elements of a normed ring, Duke Math, J., vol. 16 {1949) PP• 273-283~ 4. ;acHAL, A. D., and MARTIN, R. s. Some expansions in vector space, J. de·Math., vol. 13 (1934} PP• 69-91. 5. GZLFAND, I. Normierte Ringe, Rec. Math., vol. 9 (1941) PP• 3- 2_3. 6. BANACH, s. Theorie des operations lineairas, Warsawa, 1932. • 7. MAZUR, s. a. JACOBSON, t-1 . Uber konvexe Menge in linearen nor:nierten Rhuemen, _S tudia }!clth., vol. 4 (1933} PP• '70-84. The theory of rings, Amar. Ma.th. Soc. Surveys n, New York, 1943. • 9. LORCH, e. R. The theory of analytic functions in abelian vector rings; Trans. Amer. Math. Soc., vol. 54 (1943) PP• 414-425. 10. HILL£;, ~;;. Functional analysis and semi-groups, Amer. Math. Soc. Coll. Publ., vol. XXXI, New York, 1948. 11. RI.i;;S4 9 F. Uber linearen Funktionalgleichungen, Aeta. Math., vol. 41 _ (1918) PP• 71•98, 12. MICHAL, A. D., e.nd 3LC0lUN, V. Completely integrable differential equatione in abstract spaces, Acta. Math., vol. 68 (1937) PP• 71-107. 13. Gfil.FAJ~D, I., and ' ~ILOV, G. Uber verschiedene Methoden der • lli.nfuhrung der Topologie in die Menge der maximalen Ideals einee normierten Ringes, Rec. Math., vol. 9 (1941) PP• 25-39. 14. KAKUTA NI, s. Concrete representation of abstract Oc:)- spaces, Ann. of Math., vol. 42 (1941) PP• 994-1024. 15. SAKS, s. Theory of the integral, Warse.wa, 1937. 3(, 37 16. i'~...L"i'IS, B. J. A note on regul nr !3anach spaces, Bull. Arner. ;~th. Soc., vol. 44 {1938) PP• 420-428. 17. DUMi'ORD, N., and P.:a~rrs, B. J. Linear operations on summable functions• Trans. Amer. Math. Soc., vol. 47 (1940) pp. 323-392. 18. PHILLIPS, R. s. On linear transformations, Trans. Amer. Math. Soc~, vol • . 48 (1940) PP• 51&-541.