The almost isomorphism relation for simple regular rings

Publicacions Ma emá iques, Vol 36 (1992), 369-388 . A bs ac THE ALMOST ISOMORPHISM RELATION FOR SIMPLE REGULAR RINGS PERE ARA AND K . R . GOODEARL Dedica a la memó ia del nos e amic, Pe e Menal i B u al A longs anding open p oblem in he heo y o on Neumann eg- ula ings is he ques ion o whe he e e y di ec ly ini e simple egula ing mus be uni - egula . Recen wo k on his p oblem has been done by P . Menal, K .C . O'Mea a, and he au ho s . To cla i y some aspec s o hese new de elopmen s, we in oduce and s udy he no ion o almos isomo phism be ween ini ely gene a ed p ojec i e modules o e a simple egula ing . 0 . In oduc ion . In he ew pas yea s, he e ha e been some ad ances in he unde - s anding o di ec ly ini e simple egula ings . In 1988, Menal and he second au ho [GM, Theo em 5 .2] showed ha i R is a di ec ly ini e egula algeb a o e an uncoun able ield, and i R con ains no uncoun - able di ec sums o nonze o igh ideals, hen i is uni - egula . As a consequence o his, any s ably ini e simple egula algeb a R o e an uncoun able ield is uni - egula [GM, Co olla y 5 .4] . Mo e ecen ly, O'Mea a p o ed ha a di ec ly ini e simple egula ing sa is ying weak compa abili y is uni - egula [O, Theo em 1] . An al e na i e p oo o O'Mea a's Theo em was de eloped by he second au ho in p i a ely ci cula ed no es [G5] . We ake he oppo uni y o p esen his p oo he e . Ou s anda d e e en e o he heo y o egula ings is [G1], and o he heo y o pa ially o de ed abelian g oups is [G4] . The eade can e e o he e books o any unde ined e ms . The esea ch o he i s au ho was pa ially suppo ed by DGICYT g an P89-0296, and ha o he second au ho by an NSF g an 37 0  P . ARA, K . R . GOODEARL Le R be an associa i e ing wi h 1 . Deno e by P ( esp . Po) he class o ini ely gene a ed p ojec i e igh R-modules ( esp . he class o nonze o ini ely gene a ed p ojec i e igh R-modules) . I R is a egula ing hen L(RR) will deno e he la ice o p incipal igh ideals o R . Fo A, B E P, we will w i e A<B ( esp . A ~ B) i A is isomo phic o a submodule ( esp . p ope submodule) o B . Fo a posi i e in ege k and module A, we le kA deno e he di ec sum o k copies o A . A ing R is said o be di ec ly ini e i xy = 1 implies yx = 1, o x, y E R . We say ha R is s ably ini e i M  , (R) is di ec ly ini e o all n > 1 . I is no known whe he di ec ly ini e egula ings a e s ably ini e [G1, Open P oblem l] . The ques ion is open e en in he case o simple egula ings . A ing R is said o be uni - egula i o any x E R he e exis s a uni u E R such ha x = xux . E e y uni - egula ing is s ably ini e, bu he e exis s ably ini e egula ings which a e no uni - egula [G1, P oposi ion 5 .2 and Example 5 .10] . Howe e , he e a e some in e es ing classes o egula ings o which i is known ha di ec ini eness implies uni - egula i y . Fo example, his holds o egula ings sa is ying gen- e al compa abili y [G1, Theo em 8 .12], o igh k~o-con inuous egula ings [G2, Theo em 1 .4], and o k~o-comple e egula ings [Bu, Co ol- la y 1 .6] . An ou s anding ques ion in he heo y is whe he a di ec ly ini e simple egula ing is uni - egula [G1, Open P oblem 3] . We say ha a class o modules C sa is ies he cancella ion p ope y (wi h espec o he isomo phism ela ion) i A ® C=B®C implies A=B o A, B, C E C . A egula ing R is uni - egula i and only i P sa is ies he cancella ion p ope y, see [G1, Theo em 4 .5] . The main esul o Sec ion 1 s a es ha a di ec ly ini e simple egula ing is uni - egula i and only i P sa is ies he cancella ion p ope y wi h espec o he almos isomo phism ela ion (de ined in Sec ion 1) . We say ha R is s ic ly unpe o a ed whene e nA ~ nB implies A ~ B o A, B E P and n _> 1 . R is unpe o a ed i nA ; :S nB implies A<B o A,BEPandn>1 . Assume ha R is a di ec ly ini e simple egula ing . I is an open ques ion whe he R is (s ic ly) unpe o a ed . S ic ly unpe o a ed di- ec ly ini e simple egula ings ha e a numbe o in e es ing p ope ies . In pa icula hey a e uni - egula and, in he non-a inian case, hey a e close o being ings o ma ices o any size (see Sec ion 2) . Some echnical esul s needed o ob ain he la e s a emen a e included in an Appendix . Le R be a s ably ini e simple egula ing . Then (Ko(R), [R]) is a pa ially o de ed abelian g oup wi h o de -uni , see [G1, P oposi ion SIMPLE REGULAR RINGS  37 1 15 .3] . Le <D : Ko(R)  > A (S(Ko(R), [R])) be he na u al map, see [G4, Chap e 7] . Fo any compac con ex se S, deno e he s ic o - de ing on A (S) by «, ha is, « g i and only i (x) < g(x) o all x E S . By [B3, Theo em 3 .1 .4] and [G4, Theo em 4 .12], R is s ic ly unpe o a ed i and only i D ([A]) « <D ([D]) implies A ~ D o A, D E P . We can conside he ollowing weake condi ion : Fo any D E L(RR), he e exis s K >_ 1 such ha , o A E L(RR), i K,D([A]) « 4>([D]) hen A -< D . We will see in Sec ion 3 ha R sa is ies his condi ion i and only i R sa is ies he ollowing p ope y : The doubling condi ion (DD) : Fo any D E L(RR), he e exis s K >_ 1 such ha , o A E L(R R ), i A <_ D and K(D([A]) « 1>([D]) hen 2A~ D . Simila ly, he ollowing wo condi ions a e equi alen o a di ec ly ini e simple egula ing R : Weak compa abili y (O'Mea a) : Fo any D E L(RR) he e exis s n >_ 1 such ha , o A E L(RR), i nA < R hen A ; :5 D . (dd) : Fo any D E L(RR) he e exis s n _> 1 such ha , o A E L(RR), i A<DandnA ; :5 R hen2A<D . Obse e, in pa icula , ha he doubling condi ion implies weak com- pa abili y in any s ably ini e simple egula ing . We close he pape by s udying he e ec o imposing compa abili y wi h espec o he (app oxima ely) almos isomo phism ela ion on a s ably ini e simple egula ing . 1 . S able ange o simple egula ings . In his Sec ion we s udy he s able ange o simple egula ings, ob- aining a es ic ion on he beha iou o he s able ange on he amily o ini ely gene a ed p ojec i e modules . I is easy o show by using ou esul s ha i he e exis s a simple egula ing o s able ange 2, hen he e a e co ne ings o R wi h a bi a y ini e s able ange n> 1 . So, he si ua ion o simple egula ings di e s e y much om he si ua ion o a bi a y egula ings, see [MM ; GMM] . We will apply he esul s on s able ange o gi e he new p oo o O'Mea a's Theo em [O, Theo em l] . Recall ha a ing R sa is ies he n-s able ange condi ion ( o a gi en posi i e in ege n) i whene e a l , . . . . a n+ l E R wi h a l R+ +an+iR = R, he e exis elemen s bl, . . . , b n E R such ha (al + an+ibi)R + - . . + (a n + an+ibn)R = R . 37 2  P . ARA, K . R . GOODEARL I n is he leas posi i e in ege such ha R sa is ies he n-s able ange condi ion, hen R is said o ha e s able ange n, and we w i e s (R) = n . I is well-known ha a egula ing has s able ange one i and only i i is uni - egula [G1, P oposi ion 4 .12] . The eade is e e ed o [V] o he basic p ope ies o he s able ange and o [W ;MM ;M] o he connec ions be ween cancella ion p ope ies o modules and he s able ange o he endomo phism ings . Lemma 1 .1 . Le R be a non-a inian simple egula ing . Then o each P E Po and o all k > 1, he e exis s Q E Po such ha kQ ;j, : P . P oo . Clea ly we can assume ha P= eR o some nonze o idempo- en e E R . Since R is no a inian, eR = e1R ® e2R o some nonze o idempo en s el, e2 . Since R is simple, e1R ; :5 n(e2R) o some n and so e 1 R=A l ® . . . ® A n wi h A Z:5 e 2 R by [G1, Co olla y 2 .9] . Then eR = Al ® (A2 ® . . . ®A n ® e2R) and clea ly 2A1 ;Z eR . Now, he esul ollows by induc ion . The ollowing esul is pa e ned a e an a gumen o Rie el [R] . Theo em 1 .2 . Le R be a simple egula ing such ha s (eRe) < k o some k > 1 and all idempo en s e E R . Then R is uni - egula . P oo . I R is a inian, he esul is well-known . So we can es ic ou sel es o he non-a inian case . Assume ha Pl ® nR= P2 ® nR o some ini ely gene a ed p ojec i e modules P l and P2 . I Pl = P2 = 0, we a e done, so we can assume ha Pl :7~ 0 . By Lemma 1 .1, Pl - kQ® U o some Q E Po, and clea ly we can assume ha Q =eR o some idempo en e E R . Now, we ha e kQ®U®nR=P 2 ®nR . Since R is simple we ha e R® V = sQ o some s >_ 1 and since kQ U®nR®nV=P2®nRE)nVweha ekQ®U®nsQ=P2®nsQ . By [W, Theo em 1 .2], kQ ®U=P 2 and so P l - P2 . By [G1, Theo em 4 .5], i ollows ha R is uni - egula . We need he essen ially-known ac ha he ini eness o he s able ange is Mo i a-in a ian . We include a p oo o his esul , which is a s aigh o wa d applica ion o he echniques in [W] . Lemma 1 .3 . Le P and Q be ini ely gene a ed p ojec i e modules o e a ing R . Assume ha he e exis s k >_ 1 such ha Q®U= kP and he e exis s i > 1 such ha P®T -- iQ, ha is, P and Q gene a e SIMPLE REGULAR RINGS  37 3 he same ca ego ies o modules . Then s (EndR(P)) < s (EndR(Q)) < oo . P oo .. We will ollow he p oo in [W, Theo em 1 .11] . Assume ha s (EndR(Q)) = < oo . Se m = ( + i - 1)k . Le N=Pl®K=Pi® . . .®P ;  ®L . whe e Pl =P j ' =P o all j . Adding T o his ela ion we ob ain M :=N®T=Qim .. .E)Qj®K=Qi® . . .eQí+T-i®V®L whe e Q p -- Qe - Q o all p, q . Applying [W, Theo em 1 .11], we ge a submodule B o M such ha M=B®K=B®C®L whe e C C Qi ® . . . ® Qi + - 1 ®V . Now we ha e N= (BnN)®K=[(B®C)nN]®L [(B®C)nK]®L=7 ((BE) C)nK)®L . i and only i and also (B (D C) n N = (B n N) e) [(B ® C) n K] . Le 7 be he p ojec ion o N on o Pi ® . . . m Pn, along L . Then I ollows ha N = (B nN) ® 7 ((B ® C) nK) ® L . By [W, Theo em 1 .6], s (EndR(P)) < m < oo . Rema k 1 .4 . I s (EndR(Q)) = hen by he abo e p oo we ob ain he ollowing bound : s (EndR(P)) < ( + i - 1) k . In pa icula , i e is an idempo en o R and R ; :5 n(eR) hen s (R) < s (eRe) + n - 1 . This is exac ly he same bound ob ained by Blackada o C*-algeb as, see [B1, Lemma A6], [B2, p .33] . Ou ollowing esul is an immedia e consequence o Theo em 1 .2 and Lemma 1 .3 . Theo em 1 .5 . Le R be a simple egula ing . Then one o he ol- lowing possibili ies occu s : (1) R is uni - egula . (2) s (EndR(P)) = oo o e e y P E Po . (3) s (EndR(P)) is ini e o e e y P E P and he se {s (EndR(P)) P E P} is no bounded . 37 4  P . ARA, K . R . GOODEARL 11 Rema k 1 .6 . Le R be any simple ing which is no s ably ini e . By simplici y, we hen ha e 2nR ; :S nR o some n > 1 . By [W, Theo em 1 .2], s (EndR(nR)) = oo . By Lemma 1 .3, his implies ha s (EndR(P)) = oo o e e y nonze o ini ely gene a ed p ojec i e R- module . Theo em 1 .7 . Le R be a simple egula ing . Assume ha whene e B, C l , C 2 E L(R R ) wi h R®B ; :-~, R®C Z o i = 1, 2, hen B ; :S C l ® C2 . Then R is uni - egula . P oo . Since i is easily seen ha he hypo hesis is inhe i ed by he co ne ings o R, i su ices by Theo em 1 .2 o show ha s (R) < 2 . Le a, b, c E R such ha aR + bR+ cR = R . The e is an idempo en e E R such ha cR = eR ® [cR n (aR + bR)] . No e ha e = c o some E R, and R = (aR + bR) ® eR . NowRR = Cl ® Dl = C2 ® D2 whe e Cl = .annR(a) and C2 = .annR(b) . We obse e ha le mul iplica ion by a induces an isomo - phism o Dl on o aR, and simila ly D2 - bR . By using his we see ha R® eR ;Z R®C Z o i = 1, 2 . By ou hypo hesis, we deduce ha eR ; :S Cl ® C2, so eR =E l ® E2 wi h each E i ; :S C z . De ine xE R such ha xDl = 0 and xR = xCl = El ; no e ha x = ex . Since aCl = 0 and aDl = aR, we ge (a + x)R = aR + El . Likewise, he e exis s y E eR such ha (b + y)R = bR + E2 . Then we ha e (a+c x)R+(b+c y)R = (a+x)R+(b+y)R = aR+bR+E l +E 2 = aR + bR + eR =R . This shows ha s (R) G 2 . a We now in oduce a key concep o his pape , namely he almos isomo phism ela ion . De ini ions . Le R be a egula ing and le A, B E P . We say ha A is almos subisomo phic o B, w i en A :5,a B, i o all nonze o C E L(RR) we ha e A < B®C . We say ha A is almos isomo phic o B, w i en A- a B, i A ;Za B and B ; :~a A . We say ha A is app oxima ely almos subisomo phic o B, w i en A ;~,aa B, i o all nonze o C E L(RR) he e exis s n> 1 such ha nA ;~-z n(B (D C) . We say ha A is app oxima ely almos isomo phic o B, w i en A - aa B, i A ; :Saa B and B ;~Saa A . The abo e no ions a e specially use ul whe i R is a simple egula ing which is no a inian . Since a inian simple egula ings a e i ial o ou heo y, we will equen ly assume ha ou simple egula ings a e no a inian . Lemma 1 .8 . Le R be a non-a inian simple egula ing and le SIMPLE REGULAR RINGS  37 5 A, B E P . Then : (a) A ;i-C a B ( esp . A ; :Saa B) i and only i A ; :S a B®C ( esp . A ; :Saa B ® C) o all nonze o C E L(RR) . (b) The ela ion ;!Sa ( esp . ; : - Z~aa) is ansi i e . P oo : (a) Assume ha A Z_ S a B®C o all nonze o C E L(RR) . Fix a nonze o D E L(RR) . Then since R is no a inian D = Dl ® D 2 o nonze o D I , D2 E L(RR) . So A ; :s (B ® Dl) ® D 2 =B® D . Thus A ;5_ 5 a B . The o he implica ion is i ial . The p oo o he ela ion ;!,z aa is analogous . (b) is p o ed in a simila way . Theo em 1 .9 . Le R be a di ec ly ini e simple egula ing .  Then he ollowing condi ions a e equi alen : (a) Fo all A,B, C E P, A®B ;i5a A®C implies B ;5-a C . (b) Fo all B, C E L(R R ), R®B ; :5 R ®C implies B ;!- :5 a C . (c) R is uni - egula . P oo . (a) => (b) : This is clea . (b)  (c) : Apply Theo em 1 .7 . (c)  (a) : This is immedia e om he cancella ion p ope y o uni - egula ings . Lemma 1 .10 . Le A, B, C be ini ely gene a ed p ojec i e igh mod- ules o e a egula ing, such ha A®C = B®C . Le n E N . Then he e exis decomposi ions A = A' ® A" and B = B' ®B" and C = C' ® C" such ha A' = B' and A" ® C" = B"® C", and also n(A" ® B") ;z5 C . P oo . By [G3, Lemma 2 .2], he e a e decomposi ions A = Al, ® A12 and B= B,, ® B12 and C=C11 ® C12 such ha Al, = B,, and A12 ® C12 =B 12 ® C12, while also A12 = Cli . Applying his lemma epea edly, we ob ain decomposi ions Ai_1,2 = A¡, ®A i 2 and Bi_1,2 = Bil ®B i and C i _ 1 ,2 = Cil ® Ci2, o i = 2,3, . . ., such ha A ¡ , l-- - B ¡ , and A i 2 ® Ci2 = Bi ® Ci2, while also A i 2 - Cil . Now se A1 = Al, ®A21® . . . ® A, and A2 = A n 2, and de ine B,, B2, Cl, C2 simila ly . Then A= A1® A2 and B = B l ® B2 and C = Cl ® C2, wi h A 1 -B l and A2®C 2 =B2®C 2 . Since A2 =A n 2 <A,- 1,2 < ... < A 1 2, we also ha e nA2 ; :5 A 1 2 ® A22 ® . . . ®A n 2 = Ci1® C21® . . . ® Cnl = C l . Finally, we Apply he abo e p ocedu e o he isomo phism B2 ® C2 A2 ® C 2 . We ob ain decomposi ions B2 = B3 ® B4 and A2 = A3 ® A4 and C2 = C3 ® C4 such ha B3 = A3 and B4 ® C4 = A4 ® C4, while nB 4 ;Z C 3 . Se A' =A 1 ®A 3 and A" = A 4 , and de ine B', B", C', C" 37 6  P . ARA, K . R . GOODEARL simila ly . Then A= A' ® A" and B = B' ® B" and C= C® C", wi h A' - B' and A" ® C"=B" ® C", while also n(A" ® B") < nA2 ® nB4 ,<  Ci®C3=C' . E Now we can gi e a di e en p oo o [O, Theo em 1] . Theo em 1 .11 . (O'Mea a) Le R be a di ec ly ini e simple egula ing sa is ying weak compa abili y . Then R is uni - egula . P oo . . We show ha R sa is ies condi ion (b) in Theo em 1 .9 . Le B, C E L(R R ) wi h R ® B <R®C . Gi en 0 :7~ D E L(RR) he e exis s, by weak compa abili y, a posi i e in ege n such ha nT <R implies T<D o anyTEL(RR) . By Lemma 1 .10 he e exis s a decomposi ion B= B' ® B" such ha B' ; :5 C and nB" ;5- R . Consequen ly B" < D and B ; :~ C®D . I ollows ha B ~a C and hus R is uni - egula by Theo em 1 .9 . 2 . The almos isomo phism ela ion . Le R be a non-a inian s ably ini e simple egula ing . By [B3, Theo em 3 .1 .4], he ela ion ~aa is cancella i e, Le . A®B ~aa A ® C implies B <aa C . So he app oxima ely almos isomo phism classes o ini ely gene a ed p ojec i e modules o m a cancella i e abelian semi- g oup S (since i is easy o show ha di ec sum gi es, a well-de ined ope a ion) . Deno e by [A]a, he class o A in S . We de ine a pa ial o de on S by [A]a < [B]o i and only i A ~aa B . I is easy o show ha his ela ion is well-de ined and ansla ion-in a ian , so ha S becomes a pa ially o de ed abelian semig oup . Also, he ela ion <_ is cancella i e, Le . x + y < z + y implies x <_ z o x, y, z E S, again by [B3, Theo em 3 .1 .4] . Le K . '(R) be he abelian g oup ob ained by adjoining in e ses o mally o S . Because o he cancella i n p ope y o _<, he ela ion x - y < z - i x + <_ z + y o x, y, z, E S becomes a pa ial o de in Kó (R) . I is easy o show ha his pa ial o de is ansla ion-in a ian and so Kó (R) becomes a pa ially o de ed abelian g oup, in which we ix he o de -uni [R]a . P oposi ion 2 .1 . Le R be a non-a inian s ably ini e simple egula ing . Le ~P : KO(R) -j A (S(Ko(R), [R])) be he na u al map . Assúme ha oP(Ko(R)) is endowed wi h he pa ial o de < g i (x) < g(x) o all xES(Ko(R), [R]) . Then 4)(Ko(R)) = Ko (R) as pa ially o de ed abelian g oups wi h o de -uni . P oo .. We ha e a su jec i e posi i e homomo phism a : KO(R) -~ Ká (R) gi en by c¿ ([A] - [B])'= [A] a - [B]a . We will show ha Ke (a) _ SIMPLE REGULAR RINGS  377 Ke (-D) . I is clea ha -D([A] - [B]) = 0 whene e A =o,a B, bu his happens exac ly when [A] Q = [B] o , . Con e sely assume ha 4>([A] - [B]) = 0 . Le 0 :,~ C E L(RR) . Then 4)([A ® C] - [B]) » 0 and so, by [G4, Theo em 4 .12] he e exis s m >_ 1 such ha m([A ® C] - [B]) > 0 . By using [B3, Theo em 3 .1 .4] we ha e ha he e exis s n _> 1 such ha nmB z5 nm(A ® C) . Consequen ly B :Saa A . Analogously A <aa B and so [A] a - [B]a, = 0 . I ollows ha we ha e a g oup isomo phism -y : Ko (R) - <D(Ko(R)) gi en by y([A] a - [B]a) = <D([A] - [B]) . Since R is no a inian y is posi i e . Con e sely, i -b([A] - [B]) >_ 0 hen by he same a gumen as be o e we ob ain ha B ~S . A and consequen ly [A],, - [B]o, > 0 . Hence o h, we will iden i y Ko (R) wi h <D(Ko(R)) . The p oo o he ollowing lemma is s aigh o wa d . Lemma 2 .2 . Le R be a s ably ini e simple egula ing . (a) Le S = lim M,,,(R) . Then Ko(S) = Ko(R) ®Q . (b) Le S  , = lim M~  k (R), o a ixed m > 1 .  Then Ko(S m ) _ Ko(R) ® D,  whe e D  , = {a/m k 1 a E Z, k > l} . By [B3, Theo em 3 .1 .4], he ings S = lim M  (R) and all S  z = lim M  ,k (R) a e unpe o a ed uni - egula simple ings p o ided R is a s ably ini e simple egula ing . I ollows ha Ko(S) and all Ko(S m ) a e simple dimension g oups . The e o e we can use [G4, Theo em 14 .14] and Lemma 2 .2 o s udy he ques ion o when oD(K o (R)+) is dense in A (S(KO(R), [R]))+, leading o he ollowing lemma . Recall ha o any pa ially o de ed abelian g oup G and any subg oup H o Q, he enso p oduc G ® H is a pa ially o de ed abelian g oup wi h posi i e cone (G ® H)+ = {x ® y 1 xE G+, yE H+} . Lemma 2 .3 . Le H =Q ( esp . H = D  , .), endowed wi h he usual o de . Le (G, u) be a pa ially o de ed simple abelian g oup wi h o de - uni , and assume ha G ® H is a simple dimension g oup .  Le d> G ---> A (S(G, u)) be he na u al map .  Then he ollowing p ope ies a e equi alen : (a) D(G+) is dense in A (S(G,u))+ . (b) Fo each 0 ,-A x E G+, o each n >_ 1 ( esp . o each n = mk, wi h k > 0), and o each e > 0 he e exis y,, y2 E G+ such ha n<D(y2) « -D(x) K wD(yl) nob(y1) - d>(x) <G e , D(x) - n-D(y2) K 6 38 4  P . ARA, K . R . GOODEARL 1 .  Mo eo e , gi en ini ely gene a ed p ojec i e modules A, B, ei he nA~ nB o some n> 1, mB ~ mA, o some m > 1, o A -aa B P oo : Since M,, (R) has a unique ank unc ion he abo e p oo applies e ini ely gene a ed p ojec i e modules . P oposi ion 4 .3 . Le R be a di ec ly ini e simple egula ing sa is- ying a-compa abili y . Then R is uni - egula . P oo : T is clea ha R sa is ies 2-compa abili y . By [O, Co olla y 2], R is uni - egula . By a sligh modi ica ion o he p oo in [G1, P oposi ion 8 .2], we ob ain he ollowing esul . P oposi ion 4 .4 . Le R be a di ec ly ini e simple egula ing . R sa is ies he a-compa abili y condi ion, hen so does M,, (R) o all n > 1 . P oo : We can assume ha R is no a inian . We will p e e ha o ini ely gene a ed p ojec i e modules A, B, ei- he A <a B o B <a A . By induc ion, assume he esul is ue o p ope submodules o (n - 1)R, and le A, B wi h A, B < nR . W i e A=A l ®A2, B=B l ®B2 wi h Al, A2, B,, B2 -< (n-1)R . Now ei he Al ; :S a Bl o Bl ca A l , and ei he A2 :!Sa B2 o B2 ca A2 . We need only conside he case whe e Al ;~a, B l and B2 ca A 2 . Le 0 :y~ C E L(RR) be such ha B l ®C ~ (n - 1)R and A2 ® C ~ (n - 1)R . Then A l ; :5 B i ®C and B 2 ; :5 A 2 ® C, so B l ®C = B' ® Bi , A 2 m C = AZ ®A2 wi h B' -A l and A' 2 - B2 . So ei he B' ~a AZ o A" < aB',' . Assume ha Bi <a A2 . Then B l ® B 2 (D C= B' ® Bi® A2 ; :Ea B' (B A 2 l' e Al A l ® A2 ® C, and so Bl ® B2 Ca A l ® A2, since R is uni - egula by P oposi ion 4 .3 . P oposi ion 4 .5 . Le R be a simple egula ing wi h a unique ank unc ion N . Then he ollowing a e equi alen : (a) R sa is ies he a-compa abili y axiom . (b) R is s ic ly unpe o a ed . (c) Fo x, yE R, n(xR) -< n(yR) implies xR --< yR P oo : We can assume ha R is no a inian . (a) ==> (b) : Assume ha nA --< nB . I B Z-5 a A hen w i e nB = T l ®T2 wi h T l - nA and T 2 :7¿ 0 . Choose 0 :7~ T wi h nT -< T2 . Then B ; :5 A®T so nB ; :~ nA® nT -< T I ®T2 = nB, con adic ion . So A Ga B . Applying SIMPLE REGULAR RINGS  38 5 he same a gumen o A ® T, we see ha A® T ; :S o , B . Consequen ly, A®T  B®T and so, A -< B because R is uni - egula . (b)  (c) : Ob ious . (c)  (a) : Since R has a unique ank unc ion, we see om P oposi ion 4 .1 ha ei he xR ~aa yR o yR ;z : a , o , xR . I xR ; :Saa yR, hen o 0 :~ C E L(RR) we ha e n(xR) -< n(yR ® C), and so xR -< yR® C p o ided ha yR® C ;i5 R . We can always assume his excep in he case whe e yR =R . Bu i yR = R hen xR ;S a , R= yR ob iously . Consequen ly, R sa is ies a-compa abili y . Appendix . We p o e he ollowing esul : P oposi ion A1 . Le R be a s ic ly unpe o a ed non-a inian sim- ple uni - egula ing . Le -D : Ko(R) - j A (S(K O (R), [R])) be he na - u al map . Then D(Ko(R)+) is dense in A (S(Ko(R), [R]))+ . To p o e P oposi ion A1, i clea ly su ices o p o e a co esponding esul o pa ially o de ed abelian g oups (Theo em A3) . No e ha i G is a pa ially o de ed abelian g oup hen i s o sion subg oup T is a con ex subg oup, and so G/T is a pa ially o de ed abelian g oup wi h espec o he induced o de ing . A special case o a esul o Ellio [E, Theo em 4 .5] says ha i G is a s ic ly unpe o a ed in e pola ion g oup, hen GIT is an unpe o a ed in e pola ion g oup . Since he p oo o his case is much easie han he p oo o [E, Theo em 4 .5], we gi e he de ails . P oposi ion A2 . (Ellio ) Le G be a di ec ed s ic ly unpe o a ed in e pola ion g oup, and le T be i s o sion subg oup . Then G/T is a dimension g oup . P oo . I is clea ha Since G is di ec ed ; so is G/T . Fi s conside x E G and n E N such ha n(x + T) >_ 0 . I x E T, hen x+T = 0, and so we may assume ha x 1 T . Now nx+T = y+T o some y E G+, and y > 0 because x q T . Then k(nx - y) = 0 o some k E N, whence knx = ky > 0 . Since G is s ic ly unpe o a ed, x > 0, and hence x +T> 0 . Thus G/T is unpe o a ed . Now conside xl, x2, YI, y2 E G such ha xi +T < y j +T o all i, j . I x +T = ys + T o some , s, hen xi + T< x + T < y 3 -}- T o all i, j . Hence, we may assume ha xi + T< yj +T o all i, j . Consequen ly, he e a e nonze o elemen s wij E G+ such ha xi + wij + T= yj +T . The e is some k E N such ha k(xi + wij - yj) = 0 o all i, j, whence 386  P . ARA, K . R . GOODEARL kxi < ky j and so xi < y j o all i, j ; by s ic unpe o a ion . Thus he e exis s z E G such ha xi < z _< yj o all i, j, and hence xi+T < z+T <_ y j +T o all i, j . The e o e G/T is an in e pola ion g oup . Theo em A3 . Le (G, u) be a s ic ly unpe o a ed simple in e po- la ion g oup wi h o de -uni , such ha G+ con ains no a oms . Le -P : G , A (S) be he na u al map, whe e S= S(G, u) . Then 1)(G+) is dense in A (S)+ . P oo . Le T be he o sion subg oup o G, and no e ha G/T is simple and ha he elemen u' := u +T is an o de -uni in GIT . By P oposi ion A2, G/T is a dimension g oup . Suppose ha (GIT)+ con ains an a om, say x+T whe ex E G+ . Since x canno be an a om in G+, he e exis s y E G such ha 0 < y < x . Bu hen 0 + T< y + T < x +T (because T n G+ = {0}), con adic ing oú assump ion abou x+T . The e o e (GIT)+ con ains no a oms . Le 7 : G -> G/T be he quo ien map, and se S' = S(G/T, u') . The induced map 7 * : S' ~ S is an a ine homeomo phism, and hence he induced map 7 ** : A (S) -> A (S') is an isomo phism o o de ed Banach spaces . The e is a commu a i e diag am as ollows, whe e V is he na u al map . Acknowledge nen . G A (S) G/T A (S') Since (GIT)+ = 7 (G+), i su ices o p o e ha V ((G/T) + ) is dense in A (S')+ . Thus he e is no loss o gene ali y in assuming ha G is a simple dimension g oup, wi h no a oms in G+ . Since G+ has no a oms, G is no cyclic . The e o e, by [G4, Theo em 14 .14], 4)(G+) is dense in A (S)+ . a I is a pleasu e o hank J . Meneas¡ o his help ul commen s . No e added in p oo . E . Pa do has p o ed ha any non-a inian s ably ini e simple egula ing sa is ies condi ion (D) . SIMPLE REGULAR RINGS  38 7 Re e en es [B1] B . BLACKADAR,A s able cancella ion heo em o simple C*-algeb as, P oc . London Ma h . Soc . 47 (1983), 303-305 . [B2] B . 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