On the unit-1-stable rank of rings of analytic functions

Publicacions Ma emá iques, Vol 36 (1992), 439-447 . ON THE UNIT-1-STABLE RANK OF RINGS OF ANALYTIC FUNCTIONS Abs ac JOAN JOSEP CARMONA, JULIÁ CUFÍ ANDPERE MENAL In his pape we p o e a gene al esul o he ing H(U) o he analy ic unc ions on an open se U in he complex plane which implies ha H(U) has no uni -1-s able ank and ha has some o he in e es ing consequences . We p o e also ha in H(U) he e is no o ally educible elemen s di e en om he ze o unc ion . In oduc ion Le A be a commu a i e ing wi h uni y . A pai o elemen s (al, a2) C- A 2 is said o be unimodula i he e exis s (b l , b2) E A 2 such ha a l b, + a2b2 = 1 . We will deno e by U2(A) he se o all unimodula pai s and by Uj(A) =A -1 he se o in e sible elemen s o A . One says ha he unimodula pai (al, a2) is educible i i is possible o ind x E A such ha al + xa2 E A -1 . The ing A is said o ha e s able ank 1i each unimodula pai in A is educible in A . This is a special case o he concep o s able ank n in oduced by Bass [1] . This no ion has been use ul in ea ing some p oblems in K- heo y . Mo eo e Vase sh ein [12] has calcula ed he s able ank o ings o con inuous unc ions and ings o di e en iable unc ions in Rn and ela ed i o he opological dimension o he domain space . Conce ning o ings o holomo phic unc ions P . Jones, D . Ma shall and T . Wol [3] p o ed ha he disc algeb a has s able ank l . P e- iously L . A . Rubel [7] had obse ed ha he same is ue o he ing H(U) o holomo phic unc ions'on he open se U CC . Di e en p o es o hese esul s can be ound in he pape o G . Co ach and F . Suá ez The i s wo au ho s dedica e his pape o he memo y o Pe e Menal, who in o- duced hem o he subjec o his pape du ing he las pe iod o his li e . Pa ially suppo ed by DGICYT PB89-0311 and DGICYT PB89-0296 . 44 0  J . J . CARMONA, J . CUFÍ, P . MENAL [2] , whe e he case o ings o holomo phic unc ions o se e al complex a iables is also conside ed . The mo e di icul p oblem o decide i he algeb a H'(D) o bounded analy ic unc ions in he uni disc D, has s able ank 1 has ecen ly been answe ed posi i ely by S . T eil [10] . In [4] P . Menal and J . Moncasi in oduced he concep o uni -1-s able ank . A unimodula pai (al, a2) EU2 (A) is said o be o ally educible i he e exis s an elemen uE A -1 such ha al +ua2 E A -1 . The ing A is said o ha e uni -1-s able ank i each unimodula pai in A is o ally educible . In [7] L . A . Rubel p o ed ha H(C) has no he uni -1-s able ank p ope y . The ques ion o decide i he disc algeb a has uni -l-s able ank a ose and was s udied by R . Mo ini and R . Rupp and by ou sel es . Mo ini and Rupp,communica ed o us he nega i e answe o his ques ion and using some ideas o hei p oo we ob ain a mo e gene al esul ha has some o he in e es ing consequences . This is he con en o he i s pa o he p esen pape . The esul o Mo ini and Rupp appea s in [5] whe e he ques ion o cha ac e ize he o ally educible elemen s o he disc algeb a is also conside ed . They ind a su icien condi ion o an elemen o be o ally educible in his algeb a . Gi en a ing A one says ha an elemen a E A is o ally educible i o each b E A such ha he pai (a, b) E U2 (A) hen (a, b) is a o ally educible pai . In he second pa o his pape we conside he o ally educible elemen s o he ing H(U) . In his case he si ua ion is comple ely di e en om he dise algeb a because we show ha he ze o unc ion is he only . o ally educible elemen o H(U) . We a e able o ob ain hese kind o algeb aic p ope ies o ings o analy ic unc ions by using deep heo ems o he unc ion heo y o one complex a iable . Uni -1-s able ank Rom now on we deal wi h he ing H(U) . I a E H(U) {0}, hen we deno e by Z a , he disc e closed se in U o he ze os a . Each ze o is conside ed wi h he co esponding mul iplici y . So when we w i e Z a = Zb we mean ha a and b ha e he same ze os wi h he same mul iplici y . We ecall ha a pai (a, b) wi h a, b E H (U) is unimodula i only i Z a l Zb = 0 . We can p o e he ollowing gene al esul . UNIT-1-STABLE RANK  44 1 Theo em 1 . Assume ha (an), (bn), (en), (d n ,) men s o H(U) sa is ying i) anbn + cndn = 1 o all n >_ 1 and bn, dn a e in e ible elemen s o H(U) . ii) The sequences (a n ), (c n ) a e uni o mly con e gen on compac subse s o U o a, c E H(U) {0} espec i ely . Then ei he Z a n Z c = 0o Z a = Z, a e sequences o ele- P oo : Fi s we p o e ha (a nbn ) is ano mal sequence in D Zaa . Fix a poin zo E D Zae . Then he e exis s a closed disc A C U Zac wi h cen e zo and a numbe S > 0 such ha lan(z)cn(z)1 >_ 5, o z E A, n > , la ge enough . Conside he sequence gi en by (a nb n ) i n >_ . We will p o e ha his sequence is no mal in he classical sense in A . Since bn is in e ible and an has no ze os in A we see ha anbn ne e akes he alue 0, n > . I an(z)bn(z) = 1 o some z E A, i ollows om i) ha cndn(z) = 0 . Bu dn is in e ible and cn(z) :y~ 0 i n >_ and his is a con adic ion . By Mon el's Theo em [8, p . 350] , (a,,,bn)n>  is a no mal sequence in 0 and so is (anbn)n>1 . Since zo was an a bi a y poin o he open se U Zae, i ollows om [8, p . 51] ha (anbn)n>1 is no mal in U Zac . Assume Z a n Z, z,~ 0and le us ix a poin a E Z a n Z, Take a closed disc O1 C U wi h cen e a and such ha (O1 {a}) n Zac = 0 . The esul will ollow i we p o e ha any ~3 E Zac is a common ze o o a and c wi h he same mul iplici y in a han in c . Fo such a (~ le 02 be a closed disc wi h cen e ,P and such ha (02 {,3}) n Zac = 0 . Le K = 01 U~2 . By ii) (cn) con e ges uni o mly o c in K and (1/c,) con e ges uni o mly on 8K . Since d, . = (1 - an b n )/c n we know ha a pa ial sequence o (d,,) is ei he uni o mly con e gen o uni o mly di e gen on 8K . In he i s case his pa ial sequence (dn) is uni o mly bounded in 801 and, by he maximum modulus p inciple, also in O 1 . Then a pa ial sequence o (dn(a)) is con e gen and he co espond- ing pa ial o ((cndn)(a)) ends o 0 . Since b,, = (1 - c,,,dn)/an and on 8K we ge ha same pa ial sequence o (bn) is uni o mly a a bounded on K . The e o e (an(a)bn(a)) ends o 0 and his oge he wi h he ac ha (cn(a)dn(a)) ends o 0 con adic s i) . The e o e (dn) is uni o mly di e gen on 8K . Since d n is in e ible, by he minimum modulus p inciple we ge ha (dn) is uni o mly di e gen on K . Since dñ 1 = anbndn 1 + c n we ge ha (-a n bn dn 1 ) con e ges uni o mly o c in K . Also (an) ends o a uni o mly on K and since Za, n 80 2 = 0, by Hu wi z's Theo em [8, p . 158] he unc ion a has he same ze os han c in 02 . 44 2  J . J . CARMONA, J . CUFÍ, P . MENAL The ollowing simple esul shows ha he hypo hesis abou he in- e ibili y o b,, and d n canno be weakned . P oposi ion 1 . Le a, c E H(U) .  Then he e exis sequences (an), (b n ), (cn), (d n ) such ha (a n ) and (cn) a e almos uni o mly con e gen o a and c espec i ely such ha a n b n +c nd n = 1 and b n is in e ible o all n > 1 . P oo .. Fo each > 0 le us conside he se AA= {z E Uja(z) + = 0} n Z, . Since >, 0Y implies AA n Ay= 0, he se o A such ha Aa :,A 0 is a mos coun able . The e o e we can choose a sequence ( , n) o posi i e numbe s ha con e ges o 0 such ha Aa n = 0 . Pu an(z) = a (z)+A n and c n = c, n >_ 1 . Clea y he sequences (a n ) and (c n ) con e ge uni o mly o a and c and Z Qn n Z, is emp y o all n _> 1 . Since H(U) has s able ank 1 his implies ha he e a e b n in e ible and d  such ha anbn + c nd n = 1 . P oposi ion 1 says in pa icula ha he opological s able ank o H(U), in he sense o Rei el [6], is 2 . F om he Theo em 1 we can deduce he esul o Mo ini and Rupp Co olla y 1 . Le be a nonze o elemen o H(U) . Then has some ze o in U i and only i he e is a posi i e in ege n such ha he un¡- modula pai ( , 1 - n 2 ) is no o ally educible in H(U) . P oo . Since (n ) +(1-n 2 ) = 1 i is clea ha ( , l-n 2 ) is o ally educible when has no ze os . Con e sely, assume he e exis un E H(U) -1 such ha n = u, + 1/n - 2 E H(U) -1 o all n >_ 1 . Then 1 = u n ñ 1 + (1/n - 2) ñ 1 . I ollows om Theo em 1 ha has no ze os . Co olla y 2 . Le A be a sub ing o H(U) . I E A is o ally educible in A, hen has no ze os in U o is iden ically 0 . Le co : A - B be a ing homomo phism . We say ha cW has s able ank 1 p o ided ha o any x, yE A wi h xA+yA= A he e exis s c E S wi h cp(x) +W(y)c E B -1 . I c can be chosen o be in e ible in B, hen we say' ha cp has uni -1-s able ank . Co olla y 3 . Le U be an open se o C and le A be a ing . I ep : A -~ H(U) is a ing homomo phism wi h uni -1-s able ank, hen W(A) C C . UNIT-1-STABLE RANK  443 P oo .. Co olla y 1 implies ha cp(a) E H(U) -1 when a E A, a 7¿ 0 . Le a :,1 : 0 and assume ha W(a) is no cons an . Then W(a)(U) con ains an algeb aic numbe and so he e is a nonze o polynomial P E Z[ ] such ha P(W(a)) has some ze o in U . Since P(W(a)) = ~p(P(a)) we conclude ha P(a) = 0 and so P(cp(a)) = 0 . This shows ha W(a) akes only ini ely many alues and so i mus be cons an which is a con adic ion . Co olla y 4 . Le EC C[z] be he se o all polynomials wi hou ze os in he closed uni disc . Then he ing R = C[z] has s able ank 1 bu no uni -1-s able ank . P oo .- Clea ly we can iew R as a sub ing o he disc algeb a A(D) . I (a, b) is a unimodula pai in R , he e exis s an elemen E A(D) such ha a + b is in e ible [2] [3] . Since can be app oxima ed uni o mly by polinomials we can assume ha i sel is a polynomial . Then a + b E R and R has s able ank 1 . I ollows om Co olla y 3 ha R has no uni -1-s able ank . Ano he applica ions o Theo em 1 a e some esul s ha gua an ee he exis en e o a ixed disc con ained in he image o he uni disc o each elemen o some classes o unc ions . In his line we ecall he classical esul s o Bloch [11, p . 262], Koebe [9, p . 197] and also he in e es ing one e e ed in [9, p . 502] . He e we conside he class o unc ions g whe e is a ixed unc ion and g is a holomo phic unc ion wi hou ixed poin s in an open se and also he class o all he uncions .g whe e E S and g is as be o e . We w i e S o he se o all E H(D) such ha is one o one and (0) = 0, '(0) = 1 . P oposi ion 2 .  The e exis s a uni e sal cons an > 0 such ha he dise D(0, ) is con ained in he image o e e y unc ion g, whe e E S and g E H(D) has no ixed poin s in D . P oo .. Assume ou conclusion is alse . Then o each posi i e in ege n, he e exis zn wi h 1zn1 < ñ, n E S and 9n wi hou ixed poin s such ha n9n - zn E H(D) -1 . W i e 9n (z) = z - hn(z), whe e h n E H(D) -1 . We ob ain 1 = nun + (z n - z n) n I '  wl h  un, n E HA-1- Bu S is ano mal class [9, p . 200] , so he e exi s a pa ial sequence o ( n) uni o mly con e gen o some , E S . Applying Theo em 1 we conclude ha Z = Z, , a con adic ion . 444  J . J . CARMONA, J . CUFÍ, P . MENAL P oposi ion 3 . Le U be an open se wi h 0 E U and le be a unc ion ha is nei he in e ible no ze o . Then he e exiss a cons an = ( ) > 0 such ha he disc D(0, ) is con ained in he image o e e y unc ion g , whe e gE H(U) has no ixed poin s in U . P oo . I he s a emen is no ue, hen o each n he e would exis z,, 1, wi h Izn1 < ñ and gn wi hou ixed poin s such ha .gn - z n E H(U) -1 . P oceeding as be o e we ob ain 1 = un + (z - zn) ñ 1 ,  wi h  u n , n E H(U) -1 . Now we aply Theo em 1 and he conclusion ollows . The cons an ha appea s in P oposi ion 2 is less o iqual han 1, by Koebe's Theo em . Conside ing he unc ions (z) = z, g(z) = ez -1 one can see ha <_ é . I would be in e es ing o ind a cons uc i e p oo o P oposi ion 2 and he bes alue o . To ally educible elemen s We a e going now o conside he o ally educible elemen s o he ing o analy ic unc ions in an open se . Le A be a Banach algeb a . Fo his special case e e y elemen u E A -1 is o ally educible . In ac , o each E A, he pai (u, ) is unimodula and u + e E A -1 i e < 11 u 1 11 ' Fo he disc algeb a A(D) o med by he unc ions which a e con inuous on D and holomo phic in D, Mo ini and Rupp p o ed ha each ou e unc ion in A(D) is o ally educible [5] . The si ua ion is comple ely di e en o he ing H(U) as he ollowing heo em shows . Theo em 2 . Le U be an open se in C . A unc ion E H(U) is o ally educible in H(U) i and only i is he ze o unc ion in U . Fo he p oo we conside sepa a ely he wo di e en cases U= C and U 7~ C . P oo o he i s case : Le be o ally educible in H(C) . We know, using Co olla y 2, ha = eh wi h h an en i e unc ion . I he un¡- modula pai (e h , z) would be o ally educible hen he e would exis k, l E H(C) such ha ehe k +ze l = 1 . ÜNIT-1-STABLE RANK  44 5 The unc ion ze l ne e akes he alue 1 and akes he alue 0 only once a he o igen . By he g ea Pica d Theo em [8, p . 353] ze l mus be a polynomial and his con adic s he ac ha i ne e akes he alue 1 . To p o e he Theo em 2 o U z,~= C i s o all we ema k ha he exis en e o some E H(U), :,A 0 o ally educible implies ha each unc ion in H(U) -1 is also o ally educible and his exis en e is equi a- len o an in e pola ion p oblem as he ollowing lemma shows . Lemma . Le U be an open se o C . Then he ollowing a e equi alen . i) The e exis s a unc ion E H(U) -1 which is o ally educible in H(U) . ii) Fo each closed and disc e e se {zn}, coun ing e e y z n wi h some mul iplici y, he e exis s a unc ion h E H(U) whose ze os a e {z n }, wi h he co esponding mul iplici y, and such ha h ne e akes he alue 1 in U . iii) Each unc ion g E H(U) -1 is o ally educible in H(U) . P oo o he lemma : i) =~> ii) Le E H(U) -1 be o ally educible . Gi en he se {zn} ake k E H(U) wi h {zn} as i s ze o se . By i) he e a e a, b E H(U) -1 such ha a + bk = 1 . So a = 11 bk and h = bk sa is ies he equi emen s o ii) . ii) ==> iii) Le g E H(U) -1 and le l E H(U) be a bi a y . The pai (g, l) is unimodula . Le {z,,} be he ze o se o l . By ii) he e is some hE H(U) ha akes he alue 1 on {z n } and h(z) :7É 0 . Now he unc ions b = 11 hh E H(U) -l and a = s E H (U) -1 e i y ag + bl = 1 . We need he ollowing esul s [11, p . 215-204] . Theo em (Ahl o s) . Le E H(D) and assume ha lim sup  T ( )  = +oo . -1  log 11 1 Then akes any alue in ini ely o en in D wi h one possible excep ion . He e T( ) is he cha ac e is ic unc ion o Ne alinna [11, p . 196] . Theo em . Le E H(D) and le {zn} be i s ze o se . Assume ha -°° 1 ( 1- I zn I )1+P = + oo, o all p > 0 . Then lim sup,1 io = + oo . 44 6  J . J . CARMONA, J . CUFÍ, P . MENAL P oo o he second case : Assume i s ha U=D . Le (zñ) be a sequence such ha lim n ~~ zñ = 1 and _,' 1 (1 - jzñ¡)'+P = + oo o all p > 0 . Le {z  ,} be he sequence o he same poin s bu doubling hei mul iplici y . We show ha ii) o he lemma is no sa is ied o {zn} . Le h E H(D) be any unc ion ha anishes on {zn} . We ha e h = hó o some ho E H(D) anishing a {zñ} . By Ahl o s Theo em ho may omi only one alue and so i s squa e h akes any complex alue in ini ely o en . Fo a gene al open se U le A be any disc A C U such ha he e is a poin a E ao n aU . l su ices o ake a sequence {zn}, z n E A as be o e wi h lim e -,,, zñ = a . Fo his sequence, condi ion ii) o he lemma is no sa is ied in U, since i is no sa is ied on A . Re e ences 1 . H . BASS, K- heo y and s able algeb a, Publi . Ma h IHES 22 (1964), 5---60 . 2 .  G . CORACH AND F . DANIEL SUÁREZ, S able ank in holomo phic unc ions algeb as, Illinois J . o Ma h . 29 (1985), 627-639 . 3 .  P . W . DONES, D . MARSHALL AND T . WOLFF, S able ank o he disc algeb a, P oc . Ame . Ma h . Soc . 96 (1986), 603-604 . 4 .  P . MENAL AND J . MONCASI, K1 o Von Neumann egula ings, Jou nal o Pu e and Applied Algeb a 33 (1984), 295-312 . 5 .  R . MORTINI AND R . RUPp, To ally educible elemen s in ings o analy ic unc ions, Comm . i n Algeb a ( o appea ) . 6 .  M . A . RIEFFEL, Dimension and s able ank in he K- heo y o C*-algeb as, P oc . London Ma h . Soc . 46 (1983), 303--333 . 7 .  L . A . RUSEL, Linea composi ions o wo en i e unc ions, Ame . Ma h . Mon hly 85 (1978), 505-506 . 8 .  S . SAKS AND A . ZYGMUND, "Analy ic unc ions," Else ie Pub . Company, 1971 . 9 .  G . SANSONE AND J . GERRETSEN, "Lec u es on he heo y o unc- ions o a complex a iable," Wol e s Noo dho , 1969 . 10 . S . TREIL, The s able ank o he algeb a H°° equals 1, P ep in Lening ad Uni e si y (1991) . 11 . M . Tsu,Ii, "Po en ial heo y in mode e unc ion heo y," Chelsea Publishing Company, New Yo k, 1975 . UNIT-1-STABLE RANK  44 7 12 .  L . N . VASERSTEIN, S able ank o ings and dimensionali y o opo- logical spaces, Func ional Anal . Appl . 5 (1971), 102-110 . Depa amen de Ma emá iques Uni e si a Au ónoma de Ba celona 08193 Bella e a (Ba celona) SPAIN Rebu el 2de Ma C de 1992