Partially ordered Grothendieck groups

Pub . ?1a . UAB Vol . 29 Ns 2 .3 No . 1984 PARTIALLYORDERED GROTHENDIECK GROUPS K . R . Goodea l 1 Mo i a ion o he s udy o pa ially o de ed abelian g oups has come om many di e en pa s o ma hema ics, o ma hema ical sys ems wi h compa ibleo de and addi i e (o linea ) s lk'c u es a e qui e common . This is pa icula ly e iden in unc ional analysis, whe e spaces o a ious kinds o eal- alued unc ions p o ide impe us o in es iga ing pa ially o de ed eal ec o spaces . In he pas decade, he obse a ion ha a G o hendieck g oup (such as K D o a ing o algeb a) o en possesses a na u al pa ially o de edabelian g oup s uc u e has led o new di ec ions o in es iga ion, whose goals ha e been o de elop s uc u e heo ies o ce ain ypes o pa ially o de ed abelian g oups o he poin whe e e ec i e applica ion o a ious G o hendieck g oups is possible . Such ecen de elopmen s in he a ea o pa ially o de edabelian g oups a e he subjec o his no e . We p esen a ske ch o he cons uc ion o G o hendieck g oupsas abelian g oups equipped wi h p e-o de ings ha a e o en pa ial o de ings, oge he wi h b ie ske ches o se e al si ua ions in which he heo y o pa ially o de edabelian g oups can be applied, ia he G o hendieck g oups KO , o he s udy o ce ain ings and C*-algeb as . This discussion is somewha cu so y,in he in e es o a oiding echnicali ies, and o easons o space . Fo 1) This exposi o ydiscussion con ains an expanded e sion o se e al lec u es gi en a he Cen e de Rece ca Ma emá ica, Ins i u d'Es udis Ca alans, Ba celona, du ing No embe and Decembe 1984, and he au ho wishes o hank he CRM o a anging his e y pleasan isi . An ea lie e sion o his ma e ialappea ed in [13] in connec ion wi h a lec u e gi en a he In e na ional Symposium on Algeb a and I s Applica ions in Delhi in Decembe 1981 . he sane easons, we do no discuss he ecen applica ions o pa ially o de ed abelian g oups o opological Ma ko chains [27, 3, 20, 21] o o posi i epolynomials and compac g oup ác ions [22, 23, 25, 26] . A G o hendieck g oup is an in a ian a ached o a collec ion o objec s, such as he ec o bundles on a gi en opological space, he ini ely gene a ed p ojec i e modules o e a gi en ing, o he p ojec ion ope a o s associa ed wi h a gi enC*-algeb a . The cons uc ion only equi es a collec ion o objec s equipped wi h a no ion o isomo phism and wi h some means o combiningany wo objec s om he collec ion in o a hi d objec . While i is adi ional o de elop G o hendieck g oups using sho exac sequences as he means o combining objec s, in many impo an applica ions his educes o di ec sums (o di ec p oduc s) . Since he cons uc ion p ocess is simple using di ec sums, we shall es ic ou discussion o ha case . Thus o ou basic da a we ake a se P o objec s in some ca ego y, such ha P has a ze o objec and e e y ini e se o objec s in P has a di ec sum (cop oduc ) in P . The easies algeb aic sys em o build using his da a is an abelian semig oup, whose elemen s a e he isomo phism classes o objec s in P, and whose ope a ion is addi ion induced om he di ec sumope a ion in P . Howe e , he semig oup ob ained in his manne need no ha e cancella ion, and so canno always be embedded in an abelian g oup . This p oblem can be ci cum en ed ei he by educing he semig oup modulo a sui able-cong uence ela ion, o , mo e con enien ly, by using an equi alence ela ion on P sligh ly coa se han isomo phism, as ollows . 7 8 Objec s A,B E P a e said o be s ably isomo phic (in P) i 'and only i he e is an objec  C E P such ha MC = BOC .  S able isomo phism is an equi alence ela ion on P, and we w i e [A] o he s able isomo phismclass o an objec A E P . Le G o (P) deno e he collec ion o all s able isomo phism classes in P . The di ec sum ope a ion in P induces an addi ion ope a ion in G o (P) , whe e [A]+[B] = [A®B]  o all A,B E P, and using his ope a ion, G o (P) +  becomes an abelian semig oup wi h cancella ion .  By o mally adjoining addi i e in e ses o G o (P) , we ob ain an abelian g oup G o (P), he G o hendieck g oup o P .  (Reminde : i sho exac sequences a e a ailable in P, he G o hendieck g oup cons uc ed om P using sho exac sequences may well be di e en om he g oupcons uc ed he e .) All elemen s o G o (P) ha e he o m [A]-[B] o A,B E P, and elemen s [A]-[B] and [C]-[D] in G o (P) a e equal i and only i A®D is s ablyisomo phic o B®C . Fo example, i P is he collec ion o all eal (complex) ec o bundles on some compac Hausdo space X, hen G o (P) is he eal (complex) K-g oup K 0 (X) . Fo ano he example, le R be a ing wi h 1,  and le P be he collec ion o all ini ely gene a ed p ojec i e igh R-modules (i .e ., all di ec summandso ee igh R-modules o ini e ank) . In his case, G o (P) is he algeb aic K-g oup  K 0 (R) .  Al e na i ely,  K 0 (R)  may be cons uc ed by aking P o be he ca ego y o all ec angula ma ices o e R . Then he objec s in P a e all idempo en squa e ma ices o e R, and he di ec sum o idempo en ma ices e and is he block ma ix 0 ) . In case R is a C*-algeb a, he se o idempo en ma ices o e R may be educed o he se o sel -adjoin idempo en ma ices (see [11, Chap e 19]) . In o de no o lose ack o he o iginal semig oup G o (P) + inside G o (P), we make i he "posi i e cone" o . a n o de . ela ion . Namely, o x,y E G o (P), we de ine x <_- y i and only i y-x lies in G o (P) . This ela ion is a p e-o de (i .e ., a e lexi e, ansi i e ela ion) which is in a ian unde ansla ion ( ha is, x 5- y implies x+z = y+z) . The combined s uc u e (G o (P),+,5) is called a p e- o de ed abelian g oup . In case he ela ion < is a pa ial o de (i .e ., an an i-symme ic p e-o de ), G o (P) is a p a ially o de ed abelian ou . Va ious ela i ely mild assump ions on P will o ce G o (P) o A common assump ion is ha all he objec s in P a e "di ec ly ini e", i .e ., objec s A,B EP can sa is y A®B - A only i B is a ze o objec . Fo example,i P is he collec ion o all ini ely gene a ed p ojec i e igh modules o e a uni al ing R, he objec s in P a e di ec ly ini e i and only i all squa e ma ices o e R ha sa is y xy = 1 also sa is y yx = 1 . pa icula , his holds i R is commu a i e, o i R is a di ec ed union o ini e-dimensional algeb as, o i R is noe he ian on I also holds i R is a uni - egula ing, meaning be pa ially o de ed . ei he side . ha o any x E R he e exis s a uni (in e ible elemen ) u E R such ha xux = x, o in ha case he objec s in P may be cancelled om di ec sums [18, Theo em 2 ; 9, Theo em 4 .5] . In he case ha Pis he collec ion o all ini ely gene a ed p ojec i e igh modules o e a uni al ing R, he module R plays a special ole in P, o e e y objec in P is isomo phic o a di ec summand o a ini e di ec sum o copies o R . As a consequence, [R] plays a special ole in K 0 (R) : gi en any x EKG (R) he e exis s a posi i e in ege n such ha x 1 n[R] . 8 0 In By i ue o his p ope y,  [R] is called an o de - uni o K 0 (R) . We may make K 0 in o a unc o om he ca ego y o ings wi h uni (and uni al ing homomo phisms) in o a ca ego y whose objec s a e all pai s (G,u) whe e G is a p e-o de ed abelian g oup and u is an o de -uni in G . The app op ia e mo phisms in his la e ca ego y a e no malized posi i ehomomo phisms : (G,u) -+ (H, ), ha is, g oup homomo phisms : G  H such ha (G + ) c H+ and (u) = .  We call his ca ego y he ca ego yo p e- o de ed abelian g oups wi h o de - uni . Any uni al ing homomo phism  : R -> S induces a unc o (-)ORS om he ca ego y o ini ely gene a ed p ojec i e igh R-modules o he ca ego y o ini ely gene a ed p ojec i e igh S-modules . Since his unc o p ese es di ec sums, i in u n induces a posi i e homomo phism  K 0 (~)  : K 0 (R) -> K 0 (S),  such ha K0(~)([A]-[B]) = [A®RS]-[B®RS] o all ini ely gene a ed p ojec i e igh R-modules A and B . As K0 (¢)([R])  = [S],  we see ha  K 0 (¢)  is a no malized posi i e homomo phism om (K0(R),[R]) o (K0(S),[S]) . Since he app op ia e unc o ialp ope ies a e clea , we ob ain a unc o om he ca ego y o ings wi h uni o he ca ego y o p e-o de ed abelian g oups wi h o de -uni , gi en by he assignmen s R¡-> (K0(R),[R])  and K0 W The cons uc ion o  K 0 (R)  o a ing  R  wi hou  1  is no based ob ained om he "uni i ica ion" o R, on he abelian g oup 7LxR, (m, )(n,s) = (mn,ms+n + s) . wi h he se {0}xR, on some class o£ non-uni al p ojec i e modules bu ins ead is namely he ing R 1 based wi h mul iplica ion gi en by he ule The o iginal ing R may be iden i ied which is an ideal o R 1  1 ,  and hen R /R - 7l . Then  K 0 (R)  is de ined o be he ke nelo  K0 o he na u al map R 1 1  7L,  ha is, he subg oupo  K0 (R 1 )  consis ing o hose elemen s  [A]-[B]  in  K0 (R 1 )  o which  A/AR  and  B/BR  a e (s ably) isomo phic abelian g oups, and  K 0 (R)  is equipped wi h he p e-o de ed abelian g oup s uc u einhe i ed om  K 0 (R 1) .  In gene al,  K 0 (R)  need no ha e an o de -uni .  To ake he place o an o de -uni , we may use he subse  D(R)  o  K 0 (R)  consis ing o hose elemen s  x E K 0 (R)  o which  0 <-- x <-_ [R 1 ] .  In he con ex o he ca ego yo no -necessa ily-uni al ings, K0 may be iewed ei he as a unc o in o he ca ego y o p e-o de ed abelian g . oups and posi i e homomo phisms, o as a unc o in o a ca ego ywhose objec s a e all pai s (G,D) whe e G is a p e-o de ed abelian g oup and D is a sui able subse o G + . G o hendieck g oups ha ing been cons uc ed, wo basic me a- ques ions a ise : Wha so o in o ma ion abou he da a Pis s o edin G o (P), and how may his in o ma ion be e ie ed? Fo ins ance, since G o (P) is an in a ian o P, he e can be si ua ions in which i can be p o ed ha da a P and P' a e no equi alen by showing ha G o (P) and G o (P') a e no isomo phic . Also, by i s cons uc ion G o (P) e lec s he a angemen o di ec sum decomposi ions in P, and we may ask how much o he di ec sum decomposi ion s uc u e o P, and wha o he s uc u alin o ma ion abou P, may be eco e ed om G o (P) . By way o illus a ion, we shall discuss a numbe o si ua ions in which hese ques ions ha e been success ully answe ed . Due o he au ho 's bias, hese examples a e KO 's o ce ain ings and C*-algeb as, bu he pa e ns o hese examples a e o be expec ed in G o hendieckg oups o o he ma hema icalsys ems . 8 2 a IRRATIONALROTATIONALGEBRAS . Le Tbe he uni ci cle in he plane, and le C(T) deno e he algeb a o all con inuous complex- alued unc ions on T . We may iew he unc ions in C(T) as bounded linea ope a o s on he Hílbe space  L 2 (T)  (whe e a unc ion om  C(T)  ac s on unc ions om  L 2 (T)  by mul iplica ion) . Gi en a posi i e eal numbe  a,  ie  p a : T -+ T  be coun e clockwise o a ion h ough he angle i a . The ule poc*( ) = pa hen de ines a bounded linea ope a o pc * on L 2 (T), and we le Aa deno e he no m-closed sel -adjoin subalgeb a o bounded linea ope a o s on L 2 (T) gene a ed by C(T) and p a* . This algeb a is known as " he ans o ma ion g oupC*-algeb a o he o a ion pa " . I is ai lyeasy o dis inguishamong he algeb as  Aa  o a ional a, and i is also easy o dis inguish he a ional cases om he i a ional cases . Howe e , dis inguishing among he i a ional cases is a sub le p oblem, which was only sol ed when Pimsne , Voiculescu, and Rie el calcula ed K0 o hese algeb as . Namely, o any i a ional numbe a E (0,1), he e is an isomo phism (in he ca ego y o p e-o de ed abelian g oups wi h o de -uni ) om (K 0 (A CC),[A a ])  on o  ( 71 . +a 7L ,1) ,  whe e he subg oup  71+aF  o  IR is gi en he usualo de ing [28, Co olla y 2 .6 ; 29, Co olla y 1 ; 30, Theo em 1] . Thus i A a - A6 o some i a ional numbe s a,s E (0,1),  he e mus be an isomo phism   o  ( 7L+aZ , 1)  on o  ( lZ+OF- , 1) .  (Since he opologies in  71+a1Z  and  7+gJ- may be de ined in e ms o he o de ing, mus also be a homeomo phism .) A e app oxima ing elemen s o  71+a1Z  and  lZ+BZ by a ionalnumbe s, clea ing denomina o s, and using he ela ion (1) = 1, i is easily seen ha  7L+a/Z =  1+1 1 .  F om his a quick compu a ion leads o he conclusion ha ei he S= a o S= 1-a, whence ei he p s = pa -1 o ps = pa F-algeb a ha is a union o a coun ableascending sequence o ma icial subalgeb as (equi alen ly, any F-algeb a ha is isomo phic o a di ec limi o a coun able sequence o ma icial F-algeb as and F-algeb a homomo phisms) . I is easily checked ha ma icial algeb as a e uni - egula . Hence, any uni alul ama icialF-algeb a R  is uni - egula , and so  K0 (R)  is a pa ially o de ed abelian g oup . Using F-algeb auni i ica ions, i ollows ha K 0 o any ul ama icialF-algeb a is pa ially o de ed . These pa ially o de ed abelian g oupsmay be used o classi y ul ama icial F-algeb as, ollowing a me hod o Ellio [7] . I R and S a e uni al ul ama icial  F-algeb as, hen R c S  i and only i he e exis s an isomo phism o (K0(R),[R])  on o  (K0(S),[S]) in he ca ego yo p e-o de ed abelian g oups wi h o de -uni [7, Theo em 4 .3 ; 9, Theo em 15 .26] . I R and S a e non-uni al ul ama icial F-algeb as, hen R = S  i and only i he e exis s an o de ed g oup isomo phism o  K 0 (R)  on o  K0 (S)  mapping  D(R) on o D(S) [7, Theo em 4 .3] . The pa ially o de edabelian g oups which can appea as KU o ul ama icial F-algeb as a e jus hosewhich a e isomo phic (as o de ed g oups) o di ec limi s o coun ablesequences o ini e p oduc s o copieso 71  [7', Theo ems 5 .1, 5 .5] . Howe e , i is 8 4 O ULTRAMATRICIAL ALGEBRAS . Fix a ield F . A ma icial F-algeb a is any F-algeb a ha is isomo phic o a ini e di ec p oduc o ull ma ix algeb as o e F .  (In case F is algeb aically closed, he ma icial F-algeb as a e exac ly he ini e-dimensional semisimple F-algeb as .) An ul ama icial F-algeb a is any usually impossible o checkdi ec ly whe he a gi en pa ially o de ed abelian g oup is isomo phic o such a di ec limi . Some ob ious p ope ies o he e di ec limi s a e ha hey a e coun able, hey a e di ec ed (upwa d and downwa d), and hey a e unpe o a ed (any x sa is y - _ng nx ?0 o some nE (N also sa is ies x ? 0) . A mo e undamen alp ope y, also easily checked, is he Riesz in e pola ion p ope "" : gi en any xl'x2'yl'y2 such ha xi --- y j o all i,j, he e e ; :is s z such ha x i <-_ z <_- y j o all i,j . The di ec limi so (sequences o ) ini e p oduc s o copies o 1 we e cha ac e ized by E os, Handelman, and Shen as exac ly hose (coun able) pa ially o de edabelian g oups which a e di ec ed and unpe o a ed and which sa is y he Riesz in e pola ion p ope y [5, Theo em 2 .2 ; 11, Co olla y 21 .8] . Pa ially o de ed abelian g oups wi h he la e h eep ope ies a e now called dimensiong oups . Thus, gi en a pa ially o de ed G and an o de -uni u E G, he e exis s an (G,u) on o (K0(R),[R]) o some uni al F-algeb a R i and only i G is a coun able Fo he non-uni al case, eplace- he o de -uni u such ha e e y elemen o G abelian g oup isomo phism o ul ama icial dimension g oup . by an upwa d di ec ed subse D c G and such ha any elemen o G+ which lies below an elemen o D mus lie in D . Then he e exis s an o de ed g oup isomo phism o G on o K 0 o some ul ama icial F-algeb a R, wi h D mapping on o D(R), i and only i G is a coun able dimension g oup .  (These esul s a e ob ained by combining Ellio 's esul s [7, Theo ems 5 .1, 5 .5] wi h hose o E os, Handelman, and Shen [5, Theo em 2 .2] . Fo example, he subg oup  {a/2 n 1 a E 71 is a sum o elemen s om D, and n E M }  o Fo example, he au ho and Handelman showed ha any nonze o pa ially o de ed abelian g oup wi h an o de -uni has a leas one s a e [14, Co olla y 3 .3 ; 9, Co olla y 18 .2] . Since K 0 o any nonze ouni - egula ing is nonze o andpa ially o de ed, i ollows ha any nonze o uni - egula ing has a leas one pseudo- ank unc ion [14, Co olla y 3 .5 ; 9, Co olla y 18 .5] . The au ho and Handelman also de eloped a ious c i e ia o a nonze o pa ially o de edabelian g oup G wi h an o de -uni u o ha e a unique s a e . Fo ins ance, his happens i and only i he eexis in ege s s .> > 0 such ha gi en any x,y E G + wi h x+y = u, he e is some n E [N o which ei he n x 1 nsy o n y < nsx . As a consequence, a nonze o uni - egula ing R possesses a unique pseudo- ank unc ion i and only i he e exis in ege s s > > 0 such ha gi en any o hogonal idempo en s e, E R wi h e+ = 1, he e is some n E 11`1 o which ei he he di ec sum o n copies o eR embeds in he di ec sum o ns copieso R o he di ec sum o n copies o R embeds in he di ec sum o ns copies o eR [14, Theo em 4 .6 ; 9, Theo em 18 .6] . o PSEUDO - RANK FUNCTION SPACES AND TRACE SPACES . The collec ion P(R) o all pseudo- ank unc ions on a egula ing R (wi h 1) can be iewed as a subse o he eal ec o space  IR R  o all eal alued unc ions on  R .  I  1R R  is gi en he p oduc opology, i is a locally con ex Hausdo linea opological space, and  JP(R) is a compac con ex subse o RR [9, P oposi ion 16 .17] . In ac , IP(R)  is a a he special kind o compac con ex se known as a "Choque simplex" [9, Theo em 17 .5] . (Choque simplices a e in ini e- dimensional analogs o classical ini e-dimensional simplices, and may be cha ac e ized as exac ly hose compac con ex subse s o locally 92 con ex Hausdo linea opological spaces which a ise as in e se limi s o ini e-dimensional simplices .) In a simila ashion, he collec ion S(G,u)  o all s a es on a p e-o de ed abelian g oup (G,u) wi h o de -uni , called he s a e space o  (G,u),  is a compac con ex . subse o he p oduc space  [ñ G [9, P oposi ion 17 .11] . I G is a dimension g oup (mo e gene ally, i G sa is ies he Riesz in e pola ion p ope y), hen S(G,u) is a Choque simplex [17, Theo em 1 .2 .5 ; 5, P oposi ion 1 .7] . Fo he case o S o he egula ing R, he canonical bijec ion be ween he á a e space S(K0(R),[R])  and he pseudo- ank unc ion space  JP(R) is an a ine homeomo phism [9, P oposi ion 17 .12], i .e ., an isomo phism in he ca ego y o compac con exse s . Hence, o ealize a gi en Choque simplex K as  FP(R) o some egula ing R, i su ices o ealize K as S(K0(R),[R]) . In he me izable case, his was done by he au ho [8, Theo em 5 .1 ; 9, Theo ems 17 .19, 17 .23] . We ske ch an easie p oo o his esul , aking ad an age o ou abili y o ealize any coun able dimension g oup as K 0 o an ul ama icial algeb a . Thus le K be an a bi a y me izable Choque simplex, and le A (K) be he pa ially o de ed eal Banach space o all a ine (i .e ., con ex-combina ion-p ese ing)con inuous eal- alued unc ions on K . (The o de ing in A (K)  is he poin wise o de ing o unc ions, and he no m is he sup emum no m .) F om he me izabili y o K, i ollows ha A (K) is sepa able . Also, since K is a Choque simplex, A (K)  sa is ies he Riesz in e pola ion p ope y [4, Théo éme ; 32, Theo em 5] . Consequen ly, we may cons uc a coun able dense addi i e subg oup G o A (K)  such ha G con ains he cons an unc ion 1 and G has he Riesz in e pola ion p ope y . Then G is a dimension g oup and 1 is an o de -uni in G . Since G is dense in A (K), he es ic ion map S(A (K),1) - S(G,1) is an a ine homeomo phism . On he o he hand, a s anda d olklo e esul is ha he e alua ion map K -> S(A (K),1)  is an a inehomeomo phism, and hus K is a inely homeomo phic o S(G,1) . As he e exis s a uni al ul ama icial algeb a R o which (K0(R),[R]) - (G,1), we conclude ha  FF>(R) is a inely homeomo phic o K . By using he s ic o de ing on A (K)  (unde which < g only i (x) < g(x) o all x E K), we can ensu e ha he dimension g oup G is simple, whence Ris a simple algeb a . simplices as ace spaceso uni al C*-algeb as . Gi en a uni al complex C*-algeb a A, he se Asa o sel -adjoin elemen s o A becomes a pa ially o de ed eal ec o space wi h posi i e cone and o de -uni 1 [11, P oposi ion 6 .1] . A s a e on Ais any linea unc ional  A y C  which es ic s o a s a e on  (A sa' l) .  Since all s a eson  (A sa,1)  ex end uniquely o s a es on  A,  he collec ion o all s a eson  A  may be iden i ied wi h he s a e space o  (A sa ,1) .  A acial s a e on A is any s a e such ha (xx*) = (x*x) o all x E A, and a no malized ini e ace on A is he es ic ion o A  o any acial s a e . The ace space o A is he collec ion sa T(A) o all no malized ini e aces on A . We iden i y T(A) wi h he collec ion o all acial s a es on A, which is a compac con ex subse o  S(A sa ,1) .  In ac ,  T(A)  is a Choque simplex [31, Theo em 3 .1 .18] Fo a ma ix algeb a Mn (C), he ace space T(M n (E)) is a single on, as is he s a e space o  (K 0 (Mn((E)),[Mn( C)]) .  Using he 94 Pa allel p ocedu es can be used o ealize me izable Choque AS a .= {x E A sa  1  spec um(x) c  R +} obse a ion ha he unc o s T(-) and S(K O con e ini e p oduc s o ini e cop oduc s and con e di ec limi s o in e se limi s, i ollows ha he ace space o any uni al complex AF C*-algeb a A is a inely homeomo phic o S(K0(A),[A}) [1, Co olla y 3 .2} . Gi en any me izable Choque simplex K, he e is a coun able simple dimension g oup (G,u) wi h o de -uni such ha S(G,u) is a inely homeomo phic o K, as indica ed abo e . Hence, by choosing a simple uni al complex AF C*-algeb a A o which (K0(A),[A}) - (G,u), we ob ain Blackada 's esul ha any me izable Choque simplex K is a inely homeomo phic o T(A) o some simple uni al complex AF C*-algeb a A [1, Theo em 3 .9} . o METRICALLY COMPLETE REGULAR RINGS . A no m-like unc ion N* may be de ined on any egula ing R (wi h 1) by se ing N*(x) equal o he sup emum o he alues N(x)  o N E {P(R) . I is easily checked ha he ule d(x,y) = N*(x-y) hen de ines a pseudo-me ic d on R [12, Lemma 1 .2] . In case d is a me ic and R is comple e wi h espec o d, we say ha R is N*- comple e . Fo ins ance, i he e exis s a posi i e in ege n such ha all nilpo en elemen s x E R sa is y x n =0, hen N*(y) k 1/n o all nonze o elemen s y E R, and so R is N*-comple e [12, Theo em 1 .3} . This occu s, o ins ance, i R can be embedded in a di ec p oduc o nxn ma ix ings o e di ision ings . The unc ion N* on R co esponds o a no m-like unc ion on K0 (R), de ined using s a es in place o pseudo- ank unc ions,as ollows . Gi en any p e-o de ed abelian g oup  (G,u) `wi h o de -uni ,`wemay de ine Ilxll = sup{Is(x)I : s E S(G,u)} o all x E G . Al e na i ely,  Ilxll may be compu ed as lixll = in {k/n l k,n E IN and -ku <_ nx <_ ku} [17, Lemma 1 .6 .1] . Then II-ll is a nónnega i e eal- alued unc ion on G such ha IImxII = ImI .llxll and llx+yll < llxll+lly!I o all m E 71 and all x,y E G [ibid] . I he pseudo-me ic d' on G de ined by d'(x,y) = lix-yll is ac ually a me ic, and i G is comple ewi h espec o d', we say ha (G,u) is no m -comp le e . Hecause o he canonical bijec ion be ween  FP(R) and S(K0(R),[R]), i ollows ha i¿[xR]II = N *(x) o all x c' R, and so N*-comple eness o Ris ela ed o no m-comple eness o . B : 0 (R) . Speci ically, in case he egula ing R is N*-comple e, he au ho p o ed ha (K0(R),[R]) is an a chimedeanno m-comple e dimension g oup [12, Theo em 2 .11] .  (Fo a pa ially o de ed abelian g oup G o be a chimedean means ha whene e x,y E G wi h nx S y o all posi i e in ege s n, hen x <_ 0 .) Consequen ly, some s uc u e heo y o N*-comple e egula ings may be ob ained om co esponding s uc u e heo y o a chimedeanno m-comple e dimension g oups . Fo example, he e exis s a comple e ep esen a ion o any a chimedeanno m-comple e dimension g oup (G,u) in e ms o a ine con inuous eal- alued unc ions on i s s a e space . The only es ic ions placed on he unc ionsappea ing in his ep esen a ion a e he alues allowed a disc e e ex emal s a es .  (A s a e s on (G,u) is disc e e i s(G)  is a disc e esubg oup o IR . A poin x in a con ex se K is ex emali x does no lie in he in e io o any line segmen wi hin K .) Se ing A = {q E A (S(G,u)) l q(s) E s(G)  o all disc e e ex emals a es s}, he au ho and Handelman p o ed ha he e alua ion map G - A (S(G,u)) gi es an isomo phism o (G,u) on o (A,1)  (as o de ed g oups wi h 96 o de -uni ) [15, Theo em 5 .1] . This a ine con inuous unc ion ep esen a ion o a chimedean no m-comple e dimension g oups in u n p o ides an a ine con inuous unc ion ep esen a ion o K 0 o he N*-comple e egula ing R . Unde he canonical a ine homeomo phism be ween S(K0(R),[R]) and ?(R),a disc e eex emal s a e s on (K0(R),[R]) co esponds o an ex emalpseudo- ank unc ion P wi h a disc e e ange o alues . Speci ically, i s(K 0 (R)) = (1/m)j  o some m E H, hen P(R)  = {0,1/m,2/m, . .. . l},  and his occu s i and only i  R/ke (P)  is isomo phic o an mxm ma ix ing o e a di ision ing . Se BP = (1/m)3 : in his case, and o all o he ex emal pseudo- ank unc ions P se BP = F R . Then he e is a na u al isomo phism o (K0(R),[R]) on o (B,1), whe e B =  {q E A (IP (R))  q (P)  E Bp  o all ex emalpseudo- ank unc ions P} [12, Theo em 4 .11] . To gi e an easy applica ion o his a ine con inuous unc ion ep esen a ion o K 0 (R), assume, o some ixedposi i e in ege , ha all simple a inian ac o ings o R (i he e a e any) a e x ma ix ings (o e some o he ings, no necessa ily o e di ision ings) . Then i P is an ex emal pseudo- ank unc ion and R/ke (P) is isomo phic o an mxm ma ix ing o e a di ision ing, mus di ide  m,  whence  l/ E BP .  As a esul , he cons an unc ion  l/ belongs o he g oup B gi enabo e . F om he isomo phism o (K0(R),[R]) on o (B,1), i ollows ha [R] = [C] o some [C] E K 0 (R)  .  Since  R  is uni - egula [12, Theo em 2 .3], he module R is isomo phic o a di ec sum o copies o C, whence he ing R is isomo phic o a x ma ix ing (o e he endomo phism ing o C)  [12, Co olla y 4 .14] . In pa icula , i R has no simple a inian ac o ings, hen R is a x ma ix ing o all posi i e in ege s . he collec ion L(R R ) o p incipal igh ideals o ms a la ice, wi h ini e in e sec ions o ini e in ima and ini e sums o ini e sup ema [9, Theo ems 1 .1, 2 .3] . The ing R is said o be - con inuous i£ he la ice  L(R R )  is  'Ng -con inuous in he sense ha (a) e e y coun ablesubse o L(R R ) has an in imum and a sup emum in L(R R );  (b) whene e A E L(R R ) and B 1 <- B2 ~-l . . . in L(R R), hen  A n (V Bi )  =  (An B i ) ;  (c)  whene e  A E L(R R )  and B 1 ? B2 . 1  . . .  in  L(R R ) ,  hen  AV (A Bi )  =  MA V Bi ) .  (Since  L(R R ) is an i-isomo phic o he la ice o p incipal le idealso R [9, Theo em 2 .5], his de ini ionis le - igh symme ic .) Equi alen ly, R is ieo -con inuous i and only i gi en any coun ablygene a ed igh (le ) ideal I o R, he e exis s a p incipal igh (le ) ideal J =) I such ha e e y nonze o igh (le ) ideal con ained in J has nonze oin e sec ion wi h I [9, Co olla y 14 .4] . Handelman p o ed ha e e y !? C -con inuous egula ing is uni - egula [19, Theo em 3 .2 ; 9, Theo em 14 .24], and he au ho p o ed ha e e y iz o -con inuous egula ing is N*-comple e [12, Theo em 1 .8] . Hence, he s uc u e heo ies o a chimedean no m-comple e dimension g oups and N*-comple e egula ingsyield a s uc u e heo y o X o -con inuous egula ings . Howe e , a s uc u e heo y o -con inuous egula ings was i s de i ed om a s uc u e heo y o mono one a-comple e dimension g oups, as ollows .  (A pa ially o de ed se P is mono one a-comple ep o ided ha e e y ascending 9 8 o ALEPH-NOUGHT-CONTINUOUS REGULAR RINGS . In any egula ing R, (descending) sequence x 1 :1 x 2 5 . . . (x 1 ? x 2 ? . . .) in P which is bounded abo e (below) in P has a sup emum (in imum) in P .) Handelman, Higgs, and Law ence p o ed ha K 0 o any -con inuous egula ing Ris a mono one 6-comple e dimension g oup [24, P oposi ion 2 .1], and ha such g oups a e a chimedean [24, Theo em 1 .3] .  Since  K0 (R)  is a chimedean, hey ob ained ){ke (P)  I PE  [P (R) } = {0}, om which i ollows ha he in e sec ion o he maximal wo-sided ideals o R is ze o [24, Theo em 2 .3] . I M is any maximal wo- sided ideal o R, hen he exis ence o a s a e on (K0 (R/M),[R/M]) implies he exis ence o a pseudo- ank unc ion P on R/M, and ke (P) = {0} because R/M is a simple ing . As a consequence, R/M con ains no uncoun able di ec sums o nonze o p incipal igh o le ideals, and using his coun abili y condi ion, Handelman p o ed ha R/M is a igh and le sel -injec i e ing [19, Co olla y 3 .2] . Thus, since he in e sec ion o he maximal wo-sided ideals o R is ze o, R is a subdi ec p oduc o simple igh and le sel -injec i e ings . A s uc u e heo y o mono one C-comple e dimension g oups was de eloped by he au ho , Handelman, and Law ence [17] and applied o K0 (R) . Fo example, he a ine con inuous unc ion ep esen a ion o such g oups led o a comple e ep esen a ion o  K 0 (R)  in e ms o a ine con inuous unc ions on  [P(R)  [17, Theo em 11 .15 .1] . As a consequence, i all simple ac o ings o R a e x ma ix ings ( o some ixed posi i e in ege ), hen R is a x ma ix ing [17, Theo em II .15 .3] . o FINITE RICKART C*- ALGEBRAS .  A Ricka C*- algeb a is a C*-algeb a A in which he igh annihila o o any elemen x ( ha is, he igh ideal {a E A 1 xa = 0}) equals he p incipal igh ideal gene a ed by some p ojec ion  p  ( ha is,  p = p* = p 2 ) .  This is a gene aliza ion o he concep o an AW*- algeb a , which is a C*-algeb a in which he igh annihila o o any subse is a p incipal igh ideal gene a ed by a p ojec ion . In pa icula , all on Neumann algeb as (W*-algeb as) a e Ricka C*-algeb as . A ini e C*-algeb a is a uni al C*-algeb a A such ha all elemen s x E A sa is ying xx* = 1 also sa is y x*x = 1 . The K- heo y o a ini e Ricka C*-algeb a A can be in es iga ed wi h he aid o an auxilia y  ?~ O -con inuous egula ing  R  which is alsó *- egula , i .e ., he e is an in olu ion * on R such ha e e y p incipal igh ideal o R is gene a ed by a p ojec ion . Handelman p o ed ha  A  is a *-sub ing o an  ?~ O -con inuous *- egula ing  R such ha he only p ojec ions in R a e hose in A [19, Theo em 2 .1] . The ing R is essen ially unique (up o a *- ing isomo phism which is he iden i yon A), and is called he egula ing o A . Handelman also p o ed ha he inclusion map A ~ R  induces an isomo phism o  (K0(A),[A])  on o  (K0(R),[R]) .  (A p oo o he case ha A has no one-dimensional ep esen a ions is gi en in [13, Theo em 5 .2] .) In pa icula ,  K0 (A)  is a mono one  Q-comple e dimension g oup, and he s uc u e heo y o such g oups yields a co esponding s uc u e heo y o A, in exac ly he same manne as o ?ZO -con inuous egula ings .  (Howe e , his s uc u e heo y was i s de i ed om he s uc u e heo y o  c¿ 0 -con inuous egula ings, ia he egula ing o A .) Fo example, A is a subdi ec p oduc o simple AW*-algeb as, and so A can be embedded in a ini e AW*-algeb a [24, Theo em 3 .1] . Fo ano he example, i he dimension o e e y ini e-dimensional i educible ep esen a ion .o A (i he e a e any) is di isible by a ixed posi i e in ege , hen A is a x ma ix 10 0 ing o e some o he ini e Ricka C*-algeb a [17, Theo em 111 .16 .8] . REFERENCES 1 . B . E . Blackada , "T aces on simpleAF C*-algeb as" J . Func . Anal . 38 (1980) 156-168 . 2 . O . B a eli, "Induc i elimi so ini e-dimensional C*-algeb as" T ans . Ame . Ma h . Soc . 17 1 (1972) 195-234 . 3 . J . Cun z and W . K iege , "Topological Ma ko chains wi h dicyclic dimension g oups" J . eine angew . Ma h . 320 (1980) 44-51 . 4 . D . A . Edwa ds, "Sépa a ion des onc ions éelles dé inies su un simplexe de Choque " C . R . Acad . Sci . Pa i s261 (1965) 2798-2800 . 5 . E . G . E os, D . E . Handelman, and C .-L . Shen, "Dimensiong oups and hei a ine ep esen a ions" Ame . J . Ma h . 102 (1980) 385-407 . 6 . E . G . E osand C .-L . Shen, "App oxima ely ini e C*-algeb as and con inued ac ions" Indiana Uni . Ma h . J . 29 (1980) 191-204 . 7 . G . A . Ellio , "On he classi ica ion o induc i e limi s o sequences o semisimple ini e-dimensional algeb as" J . Algeb a 38 (1976) 29-44 . 8 . K . R . Goodea l, "Algeb aic ep esen a ions o Choque simplexes" J . Pu e Applied Algeb a 11 (1977) 111-130 . 9 .  , _Von Neumann Regula Rings London (1979) Pi man . 10 .  , "A inian and noe he ian modules o e egula ings" Communic . i n Algeb a 8 (1980) 477-504 . 11 .  , No es _on Real _and Complex _C*- Algeb as Nan wich (Cheshi e) (1982) Shi a . 12 .  , "Me icallycomple e egula ings" T ans . Ame . Ma h . Soc . 27 2 (1982) 275-310 . 13 .  , "Pa ially o de ed G o hendieck g oups" in Algeb a _and _I s Applica ions (H . L . Manocha and J . B . S i as a a, Eds .),  pp .  71-90 New Yo k (1984) Dekke . 14 . K . R . Goodea l and D . E . Handelman, "Rank unc ions and K 0 o egula ings" J . Pu e Applied Algeb a 7 (1976) 195-216 .