One-sided division absolute valued algebras

Publicacions Ma e ná iques, Vol 36 (1992), 925-954 . Abs ac ONE-SIDED DIVISION ABSOLUTE VALUED ALGEBRAS ANGEL RODRÍGUEZ PALACIOS We de elop a s uc u e heo y o le di ision absolu e alued algeb as which shows, among o he hings, ha he no m o such an algeb a comes om an inne p oduc . Mo eo e , we p e e he exis ence o le di ision comple e absolu e alued algeb as wi h Ie uni o a bi a y in ini e hilbe ian dimension and wi h he addi ional p ope y ha hey na e no nonze o p ope closed le ideals . Ou cons uc ion in ol cs esul s om he ep esen a ion heo y o he so called "Canonical An icommu a ion Rela ions" in Quan um Mechanics . We also show ha homomo phisms om comple e no med algeb as in o a bi a y absolu e alued algeb as a e con ac i e, hence au oma ically con inuous . In memo iam Es e a ículo ha sido esc i o en homenaje a Pe e Menal i B u al . Acaso las ma emá icas no sean el medio más ap opiado pa a exp esa el dolo po la mue e de es e g an amigo así co no la ad ni ación que su abajo me me ecía . Las ma emá icas no obs an e p opicia on una bella amis ad y ue on c edo y lenguaje común en e noso os . 0 . In oduc ion A s ill unsol ed old ques ion is ha o he nonassocia i e ex ension o he Gel and-Maza heo em, namely i any di ision no med (nonassocia- i e) algeb a mus be ini o-dimensional (which would imply dimension 1 in he co nplex case, and 1, 2, 4, 8 in he eal one, by a heo em o R . Bo and J . Milno [6]) . This p oblem was explici ly poned by F . B . W igh [27] in 1953, who in he same pape ga e a pa ial a i ma i e answe p o ing ha di ision absolu e alued algeb as a e ini o-dimensional . Ano he olklo e pa ial posi i e esul abou his ques ion is ha one- sided di ision comple e no med co nplex algeb as a e iso no phic o he 926  A . RODRÍGUEZ PALACIOS complex ield (sea [14]), he case o ( wo-sided) di ision noncomple e no med complex algeb as as well as ha o di ision (o en comple e) no med eal algeb as emaining open . ln he con e se di ec ion, J . A . Cuenca [101 has ecen ly gi en examples o in ini a-dimensional one-sided di ision absolu e alued algeb as o a he ield o eal numbe s . Since he ac ha one-sided di ision absolu e alued complex algeb as a e isomo phic o he complex ield can also be conside ad as olklo e (sea P oposi ion 2 in his papa ) ; i sééms o be easonable o look o a s uc u e heo y o a bi a y one-sided di ision absolu e alued eal algeb as, and in ac o p o ide such a s uc u e heo y is he main pu pose o his pape . To unde s and he pliilosophy o ou wo k, i is sui able o ake in o accoun ha , easily, le di ision absolu e alued algeb as a e "iso opic" o le di ision absolu e alued algeb as wi h a le uni (sea P oposi ion 4) . So he . a en ion mus be cen e ed on hese las algeb as ; and hen he se (say P) o all le mul iplica ion ope a o s on such an algeb a is a subspace o bounded linea ope a o s on he no mad space (say X) o he algeb a con aining he iden i y ope a o and sa is ying (*)  11 T(x) 11=11 T 1111 x 11 o all T in P and x in X . lNe isola e his in o ma ion in he i s sec ion o he pape , and we p o e in Theo em 1 ha , i P is a subspace o bounded linea ope a o s on any eal no med space X con aining he iden i y ope a o (say I) and sa is ying (*), hen he ope a o no m on P de i es om an inne p oduc and ; o T in P o hogonal o I, he equali y T 2 =- IIT11 2 I holds . l ollows ha e e y elemen in P is an in e ible ope a o on X so, as a i s consequence, absolu e alued algeb as wi h le uni a e au oma ically le di ision algeb as . The con en o he abo e e e ed Theo em 1 is s a ed in he pa- pe in an equi alen ly- e o mula ed o m in ol ing ce ain p ehilbe ian quad a ic di ision-Jo dan algeb as wi h in olu ion which, cu iously, ha e been shown in [20] o be he only "smoo h no mad" nonassocia i e com- mu a i e algeb as . F om his e sion o Theo em 1 we de i e in Sec ion 2 he main esul in he pápe (sea Theo em 2) asse ing ha "uni al *- ep esen a ions" o hese Jo dan algeb as on hei own p e-Hilbe spaces na u ally gi e ise o (au oma ically le di ision) absolu e alued alge- b as wi h le uni , and ha by his cons uc i a me hod all absolu e alued algeb as wi h le uni a ise . This las asse ion in Theo em 2 ONE-SIDED DIVISION ABSOLUTE VALUED ALGEBRAS  927 means ha he no m o an absolu e alued algeb a A wi h le uni e comes om an i me p oduc ( . 1 .) sa is ying (ab 1 c) = -(b 1 ac) and a(ab) = - 11 a 112 b o all a, b, c in A wi h a o hogonal o e . In he hi d sec ion we use Theo eni 2, oge he wi h he basic ac s abou he so-callad "canonical an icommu a ion ela ions" o quan um mechanics (which he eade nay ind in he i s pagos o [8]), o p o e in Theo em 3 he exis en e o comple e absolu e alued algeb as wi h le uni o a bi a y in ini e hilbe ian dimension, and ha such al- .geb as can be cliosen wi h he addi ional p ope y ha hey ha e no nonze o p ope closed le ideals (no e ha e e y le di ision algeb a has no non ze o p ope igh ideals) . Since small pe u ba ions o he p oduc o a le di ision comple e absolu e alued algeb a gi e ise o new le di ision algeb as (P oposi ion 7), he exis en e o a wide col- lec ion o in ini o-dimensional le di ision non-absolu e- alued comple e no mad algeb as is asu ad . Le Lis emphasize also he exis en e o in ini e- dimensional comple e no med algeb as on which he ope a o s o le and igh mul iplica ion by any nonze o elemen a e su jec i e (a consequence o P oposi ion 8) . The concluding sec ion o he pape (Sec ion 4) is de o ed o p o e he au oma ic con inui y o homomo phisms om comple e no mad algeb as in o absolu o alued algeb as . Mo e pe ecisely, we show ha such ho- momo phisms a e con ac i e . TI - lis is pe haps he i s non i ial esul in ol ing a bi a y absolu e alued algeb as . Tho igh he axiom 11 ab 11=11 a 11 11 b 11 seems o be e y na u al, he s udy o absolu o alued algeb as has ecei ed he a en ion o a ela i ely slnall m .imbe o au ho s . Thus we a e only awa e o he pape s (ci ad c onologically) by A . A . Albe [1] (1947) and [2], F . B . W igh [27], K . U banik and F . B . W igh [26], K . U banik [24] and [25], M . L . El-Mallan and A . Micali [19], and M . L . El-Mallah [15], [16], [17] and [18] (1990) . We emphasize he esul in [26] (1960) asse ing ha R, C, o--0, and ® a e he only absolu o alued algeb as wi h uni (a ac ha , as i will be explained in Rema k 4(i), can be easily de i ad om Theo em 2) . Also le us men ion i s easy consequence, p e iously p o ed in [27] and al eady men ioned a lle beginning o his in oduc ion, ha di ision absolu e alued algeb as a e ini o-dimensional ; as well as he ele an esul in [19] showing ha absolu o alued algeb as sa is ying he iden i y a(ba) = (ab)a also mus be ini o-dimensional . Examples o in ini o-dimensional absolu e alued algeb as we e known in he classical li e a u a (sea [26], [24] ; [3], and Rema k 3(i)), bu pone o he algeb as in hese Examples a e one-sided di ision algeb as . 1 hopo ou papen, 92 8  A . RODRÍGUEZ PALACIOS as well as he almos simul aneous óne by J . A . Cuenca [10], could con ibu e as a e ulsi e o a subsequen lou ishing de elopmen o he heo y o absolu e alued algeb as . 1 . Subspaces o linea ope a o s whose nonze o elemen s a e mul iples o isome ies In his sec ion we will deal wi h subspaces P o bounded linea ope - a o s on a eal no med space X con aining he iden i y ope a o on X and wi h he p ope y ha 11 T(x) 11 = 11 T 1111 x 11 o all x in X and all T in P . The esul s ob ained he e will become he main ools o he p oo in he nex sec ion o he s uc u e heo y o one-sided di ision absolu e alued algeb as . In ac , we will p o e ha he ope a o no m on subspaces o ope a o s as conside ed abo e de i es om an inne p oduc . I will also u n ou ha hese subspaces a e ac ually quad a ic Jo dan algeb as o ope a o s whose nonze o elemen s a e in e ible, so ha hey co espond wi h some abs ac ma hema i- cal models p e iously . -conside ed in he answe o o he p oblems and ha a e closely ela ed wi h he "canonical an icommu a ion ela ions" o quan um mechanics . In ac , he possibili y o ep esen ing he canon- ¡cal an icommu a ion ela ions by means o bounded linea ope a o s on Hilbe spaces will allow us o p o e in Sec ion 3 he exis ence o in ini e- dimensional one-sided di ision absolu e alued algeb as wi h addi ional p ope ies . We begin ou a gumen wi h he ollowing (pe haps well-known) lemma . Fo a no med space X, BL(X) will deno e he associa i e no med algeb a o all bounded linea ope a o s on X . Lemma 1 . Le X be a Banach space, and T : X -> X be a linea isome y which is no on o . Then he open uni ball i , BL(X) wi h cen e T and adius 1 con ains only ope a o s wi h closed ange and which a e one- o-one bu no on o . P oo : Recall ha a linea ope a o T on X is said o be bounded below (by m > 0) i 11 T(x) J¡ > m 11 x 11 o all x in X . Since, o T in BL(X), T is one- o-one wi h closed ange i and only i T is bounded below, he lemina will ollow om he mo e gene al esul , we will p o e he e, ha , i T is bounded below by in and is no on o, hen he open ball in BL(X) wi h cen e T and adius m con ains only bounded below ONE-SIDED DIVISION ABSOLUTE VALUED ALCE13RAS  92 9 ope a o s which a e no op o . Easily, o such a T and any S in BL(X ) wi h II T - S 11 < m ; S is bounded below by i- . II T -S II . I a'c ually II T -S II < áand S is su jec i e, we ha e ha S is in e ible in BL(X) wi h II S -I II<- 1  < 2 , so 11 T- S II ~< ,"z) <II S" -I II -I ~ ni- 11 T -S ~~  i  2 and so T is in e ible in BL(X) [4, Theo em 2 .11], which is a con a- dic ion . In his way we ha e p o ed ha he open ball in BL(X) wi h cen e T and adius z con ains only bounded below ope a o s wich a e no op ó . l ollows ha he se S2 := {R E BL(X) : R, bounded below and no o o} is open . Now he open ball in BL(X) wi h cen e T and adius m is a connec ed opological space which is he disjoin union o i s in e sec ions wi h SZ and wi h he se o in e ible elemen s o BL(X) . Since hese in e sec ions a e open and he i s one is nonemp y, i ollows ha he e e ed ball is con ained in S2, as equi ed . A his i ase we ecall some concep s sui able o a easonably clea s a e nen o ou main esul in his sec ion . .A Jo dan .algeb a is a com- mu a i e algeb a (say J) sa is ying he "Jo dan iden i y", na nely x2-(y .2) = (x2 .y) .X o all x, .y in J . Tl e nos easy examples o .Jo dan algebas a e he so- called "Jo dan subalgeb as o associa i e algebas'', namely subspaces o an associa i e algeb a which ase closed unde he "Jo dan p oduc " a .b := 2 (ab + ba), whe e he associa i e p oduc has been deno ed as usually by jux aposi- ion . In case he associa i e algeb a is he one o all linea ope a o s on a ec o space X, such .Jo dan subalgeb as a e called "Jo dan algebas o ope a o s on X" . An algeb a is said o be quad a ic i i leas a uni 1 and, o e e y x in he algeb a ; he e exis A and l .c in he base ield such ha x 2 + x + l .cl = 0 . A no ned algeb a, is a eal o co nplex algeb a wllose ec o space is a 'no ned space wi h espec ó a nomas II . 11 sa is ying 11 xy x 11 11 y II o al] x, y in he algeb a . A uni al no ned algeb a is a, no ned algeb a wi h uni 1 sa is ying 11 1 11= l . Finally a s noo h no ned . algeb a is a, uni al no med algeb a whose uni is a smoo h poin o i s closed uni ball . All asse ions in he ollowing p oposi ion, ha a e no o s aigh o wa d e i ica ion, a e consequences o [20, Theo em 27] (see also [22, Sec ion 2]) . 93 0  A . RODRÍGUEZ PALACIOS P oposi ion 1 . I H is a eal p e-Hilbe space and we conside on R1 ® H he p oduc gi en by (A1 + 17) .01 + ~) := (AM - (77 I l))1 + (a1 +,U 7 7) o all A, M in R and 1 1, ~ in H, hen R1(D H becomes a quad a ic Jo dan algeb a o e R which, endowed wi h he p e-Hilbe no m 11 ~ 1 + 1 jj := (~ 2 + 11 77 112)1/2 is a smoo h no med algeb a . Mo eo e , e e y eal smoo h no med com- mu a i e algeb a a ises in his way . We will e e o he no med algeb as desc ibed in he abo e p oposi ion as he "smoo h no med Jo dan algeb as" . Rema ks 1 . i) An elemen x in a Jo dan algeb a J .wi h uni is said o be in e ible i he e exis s y in J wi h x .y = 1 and x .y 2 = y (see [13, De ini ion 1 .11 .51) . A di ision-Jo dan algeb a is a Jo dan algeb a wi h uni whose nonze o elemen s a e in e ible . I is easy o see ha all smoo h no med Jo dan algeb as a e di ision-Jo dan algeb as, so hey a e eal no med di ision-Jo dan algeb as unde any algeb a no m (which mus be g ea e han he canonical p e-Hilbe no m because his las no m equals he spec al adius) . I was p o ed in [14] ha no mo e eal no med di ision-Jo dan algeb as exis . ii) I is well-known ha i A is an associa i e algeb a wi h uni 1 and J is a Jo dan subalgeb a o A wi h 1 E J, hen an elemen x in J is in e ible in J in he abo e sense i and only i x is in e ible in A in he usual sense and i s associa i e in e se lies in J [13, p . 51] . In he pa icula case o J being (isomo phic o) one o he smoo h no med Jo dan algeb as, his ac is almos ob ious . Fo , i o x = A1 + 77 in J = R1 ® H we dono e by x* he elemen in J gi en by x* := A1 - 77, deno ing by yux aposi ion he associa i e p oduc o A we ha e x*x = XX* = a 2 1 - 77 2 = >, 2 1 - 77 .91 = (A2+ 117711 2 )1=11 x 11 2 1, so e e y nonze o x in J is in e ible in A in he associa i e sense, wi h associa i e in e se equal o 11 x 11 -2 x* (which o cou se lies in J) . iii) The no med space o a smoo h no med .Jo dan algeb a is ce ainly a nonze o eal p e-Hilbe space, and, gi en any nonze o eal p e-Hilbe space K, up o isome ic isomo phisms he e is a unique smoo hno med Jo dan algeb a whose p e-Hilbe space is K . This is so because, chosing any no m-one elemen u in K and deno ing by H he o hogonal comple- men o Ru in K, H is he only p e-Hilbe space sa is ying K = IRu® 12 H, ONE-SIDIE,D DIVISION ÁBSOLUTE VÁLUED ALGEBRAS  93 1 so he e is a unique commu a i e p oduc en he no med space o K con- e ing i in a smoo h no med Jo dan algeb a whose uni is u . Mo eo e , he choice o he no n-one ele nen u in K is i ele an hanks o he `' o a ion p ope y"' o p e-I-Iilbe spaces, namely any no n-one elemen can be ca ied in o ano he by means o a sui able su jec i e linea iso n- e y . Indeed, his is clea o dimension 1 o 2, and, in he emaining case, i u and 'a 'e no n-one elemen s in K and i we deno e by L he linea hull o {u, } as well as by cp a linea iso ne y om L , on o L wi h ~o(u) = ; hen he mappi ig 0 : l + l 1 -> cp(l) + 1 1 om K = Le) L' in o K is a su jec i e linea iso ne y sa is ying O(u) _ . As a consequence, gi en a ca dinal iumbé k~, he e exis s a unique Smóo Ii comple e'no med Jo dan'algeb a wi h hilbe ian dimension equal e 11 . Now we s a e and p o e he main esul in his sec ion . Theo em 1 . Le , X be a nonze o eal no med space, and P be a sub- space o bounded lineal- ope a o s on X con aining he iden i y ope a o and sa is ying II T(x) 11=11 T II II .c II o al,l T in P and all x in X . Thén P is a Jo dan algeb a o ope a o s on X 'which, endowed wi h lié ope a o no77n, is isome ically isomo phic o some o he smoo h no7 ned Jo dan algeb as : P oo .- In a i s s ep we edúce he p oo e he pa icula case o X being a Banach space . To his end,' conside he comple ion X o X a id, o T -in BL(X ), le T deno e he unique ele nen in BL(X ) which ex ends T . Ti en he isome ic ho nomo phism T -> T om he associa i e no med algeb a BL(X) in o he Banach algeb a BL(X) maps P en e a subspace (say P) o BL(X) which clea ly inhe i s he p ope ies o P : P is a subspace o bounded linea ope a o s en X con aining he iden i y ope a o on X and sa is ies 11 S(y) 11=11 S 1111 y 11 o all S in P and all y in X . In his way, i he heo em is ue in he comple e case ; he in o ma ion i gi es abou 1' is easily ans e ed o P . In a second s ep we assume X o be a Banach space and we show ha hen e e y nonze o ele nen in P is in e ible in BL(X) . This ac being clea i he dimension o P is one, assume dim(P) > 2, so ha P {0} 93 2  A . RODRÍGUEZ PALACIOS wi h he opology o he ope a o no m is a connec ed opological space in which he subse {T E ^{0} : T is in e ible in BL(X)} is ce ainly open (see [4, Theo em 2 .11]) and nonemp y . Now his second s ep is concluded by e i ying ha he complemen a y subse Q := {T E P {0} : T is no in e ible in BL(X)} is also open in P {0} . Bu , i T is in 9, 11 T 11-1 T is a linea isome y om X in o X which is no on o and so, by Lemma 1, he open ball in P {0} wi h cen e T and adius 11 T 11 is con ained in 9, hence 9 is open in P {0}, as equi ed . Since in wha ollows complex me hods will be applied, i is sui able o summa ize ou si ua ion in he ollowing way . P is a eal subspace o a uni al complex (associa i e) Banach algeb a (in ou case, he no med complexi ica ion o BL(X) [4, P oposi ion 13 .3]) ha will be deno ed by A, he uni o A líes in P, and e e y nonze o elemen T in P is in e ible in A wi h 11 T -1 11=11 T 11-1 . No e ha his las p ope y implies (*)  I z 1=11 T 11 o all T in P and all z in he spec um o T ela i e o A . The hi d s ep in ou p oo consis s in showing ha , o T in P R1, he eal linea hull o {l, T} wi h he es ic ion o he no m o A is a copy o he euclidean space R2 . Chose z in he spec um o T ela i e o A and w i e A := Re(z) and S :=11 T -Al 11 -1 (T -Al), so ha S lies in P, 11 S 11 = 1, and he spec um o S con ains a numbe o he o m áE o some E in R . Then, o a bi a y a, 0 in R, a + ¡E/3 lies in he spec um o al +,PS so, by (*), we ha e 11 al + QS 11 2 =1 a + ¡Ep 1 2 = a 2 + 6202 = =a 2 + 1 aE 12 02 = a 2 + 11 S 112 X32 = a 2 + 02, so ha (a, /3 ) ---> al +,65 is a linea isome y om he euclidean space Ii8 2 on o LinR{1,T} . Ou concluding s ep o he p oo o he heo em will show as desi ed ha P is a eal Jo dan subalgeb a o A isome ically isomo phic o one o he smoo hno med Jo dan algeb as . The consequence ha he op- e a o no m on P de i es om an inne p oduc can be easily ob ained om he abo e s ep applied o he subspaces o BL(X) o he o m {S - 'T : T E P} wi h S any nonze o ixed elemen in P ; bu in ac he ONE-SIDED DIVISION ABSOLUTE VALUED ALGEBRAS  93 3 p ehilbe ian na u e o P will be eencoun e ed in wha ollows join ly wi h he emaining pa o he in o ma ion . We will use some concep s and esul s om he heo y o nume ical anges . Thus, ecall ha an elemen a in A is said o be he mi ian i (a) lies in R o all in he dual Banach space o A wi h 11 11= (1) = 1 . Clea ly, he se o all he mi ian elemen s in A is a eal subspace o A so, i we deno e by H he se o hose S in P such ha iS is an he mi ian elemen in A, H is a subspace o P, and we claim P= R1 ®ez H . Indeed, ob iously Vil l H = 0 and, by he hi d s ep o he p oo , e e y T in P is o he o m Al +S o sui able A in R and S in P wi h 11 al + OS II2= a 2 + 11 S 112 02 o all a, 0 in R, an equali y ha implies ~~ A1 + S II2= A2+ 11 S 112 and  lim  11 1 +,QS I1 - 1 = 0 PER {0},P-o ,Q so iS is he mi ian in A [4, Theo em 10 .10], and so S lies in H, concluding he p oo o he claim . A Theo em by B . Bollobás (see [5, Theo em 26 .7]) asse s ha , i a is an in e ible he mi ian elemen o a uni al complex Banach algeb a wi h (1 a 11=11 a -I 11= 1, hen a 2 = 1 . This applies in pa icula o elemen s o he o m T,-,',, S wi h S in H {0} o ob ain S 2 = - 11S11 21 o all S in H, hence he es ic ion o he no m o A o H comes om an inne p oduc ( . 1 .) . In passing om quad a ic mappings o associa ed symme ic bilinea mappings, we ind S .T = -(S 1 T)1 o all S, T in H . Finally, o e e y Al + S and pl +T in P = R1 ®I2 H, we ha e (A1 + S) .(M1 +T) = (Ap - (S 1 T))1 + AT + pS, so ce ainly P is a eal Jo dan subalgeb a o A which is a ma e ializa ion o a smoo hno med Jo dan algeb a . Co olla y 1 . Le X be a nonze o eal no med space, P be a subspace o BL(X) sa is ying 11 T (x)  11 = 11 T  x ~~ o all T in P and all x in X, and assume ha some elemen in P is in e ible (wi h in e so possibly ou side o P) . Then he ope a o no m 940  A . RODRIGUEZ PALACIOS (iii) =~> (i) This is i ial . An almos di ec consequence o he implica ion (i) => (iii) in he abo e p oposi ion is he ollowing Co olla y 3 . The co nple ion o a le di ision absolu e alued algeb a is a le di ision absolu e alued algeb a . In iew o P oposi ion 4, o ha e a sa is ac o y heo y o le di ision absolu e alued algeb as i is enough o s udy (au oma ically le di i- sion) absolu e alued algeb as wi h le uni . To his end i is use ul o in oduce some addi ional e minology . Gi en a Jo dan algeb a J and a ec o space X, a ep esen a ion o J on X will mean an homomo phism (say 0) om J on o a Jo dan algeb a o ope a o s on X . I J has a uni 1 and ~b(1) equals he iden i y ope a o on X, he ep esen a ion 0 will be called uni al . I X is a p e-Hilbe space, * is an algeb a in olu ion on J, and he ep esen a ion 0 sa is ies (O(x)( l) 1 ~) = ( 7 1 1 zb(x * )(I)) o all x in J and all l, ~ in X, hen we will say ha 0 is a *- ep esen a ion . When J and X a e no med, he ep esen a ion 0 will be called isome ic ( esp . : con ac i e) i , o all x in J, he linea ope a o O(x) on X is bounded wi h 11 O(x) 11=11 x II ( esp . : ~~ O(x) 11<11 x 11) . F om now on e e y smoo hno med Jo dan algeb a J = 81 ® H will be conside ed as algeb a wi h in olu ion * de ined by (A1 + 97)* := Al - 77 . This in olu ion can be in insically cha ac e ized as he only algeb a in olu ion * in J such ha , o e e y x in J, x + x* and x .x* lie in Hl . Lemma 2 . Le J be a smoo h no med Jo dan algeb a, K a nonze o p e-Hilbe space, and 0 be a uni al ep esen a ion o J on K . Then he ollowing asse ions a e equi alen : i)  is a *- ep esen a ion . ii) ~~ V)(x)(k) 11=11 x  k 11 o all x in J and k in K . iii)  is isome ic . i ) z~ is con ac i e . P oo . (i) = :> (ii) Being J a simple Jo dan algeb a and 0 a uni al ep esen a ion, he ango o 0 is a Jo dan subalgeb a o he associa i e algeb a L(K), o all linea ope a o s on he ec o space o K, isomo phic ONE-SIDED DIVISION ABSOLUTE VALUED ALGEBRAS  941 o J and con aining he uni o L(K) (namely, he iden i y ope a o on J ; which will be deno ed by I), hence by Rema k 1(ii) we ha e 0(x * W (x) = II x 11 , I o all x in J . The e o e, om he assu np ion (i), we ob ain o a bi a y kinK 11 VG(x)(k) 11 2 = (~ b (x)(k) 10 (x)(k)) = _ (k I O(x*)O(x)(k)) = (k 111 x 11 2 k) =11 x 11 2 11 k 11 2 (ii)  (iii) => (i ) These implica ions a e clea . (i )  (i) W i ing J = R1® H, i is enough o show ha , i he uni al ep esen a ion 0 o J on K is con ac i e, hen o any in H and all k I , k2 in K, we ha e (0( 7)(kj) I k2) = -(k 1,0(,«k2», o equi alen ly (0( l)(k) 1 k) = 0 o all no m-one elemen k in K . Bu , deno ing by a any posi i o numbe , we ha e (0( , l)(k) I k) _ ((I+c VG(7l))(k) I k) -1 < 11 1 +aO( , n) II -1 - a  ce = 11 0( 1 + cm) II -1 < 11 1 + 0 , 77 II -1 _ (1 + n 2 II 7,11 2 )1 / 2 -1 (whe e o he second inequali y we ha e used he assump ion ha 0 is con ac i e) . The e o e (o( 7 7)(k) I k) C lim  (1 + a 2 II l 1I2) I/2 - 1 = 0 a _o+ a and, changing / by - l, we ob ain (0( l)(k) 1 k) = 0 . Now we s a e and conclude he p oo o he main esul in his sec ion . Theo em 2 . I J is a smoo h nomned Jo dan alyeb a and 0 is a uni al *- ep esen a ion o J on he p o-Hilbe space o J, hen he no med space o J wi h p oduc O de ined by x O y := IP(x)(y) is an (au oma ically le di ision) absolu o alued alyeb a wi h le uni . Mo eo e , up o isome iic isomo phism, by means o his cons uc i o me hod all absolu o alued algeb as wi h, le uni a ise . P oo . The e i ica ion o he i s pa ag aph in he s a emen is e y easy . Since 0 is a uni al ep esen a ion, he uni 1 o .J as a, Jo dan 94 2  A . RODRÍGUEZ PALACIOS algeb a -becomes a le uni o he p oduc O, and, being also 0 a *- ep esen a ion, he ac ha J wi h p oduc O is absolu e aluad is a di ec consequence o he implica ion (i) =~> (ii) in Lemma 2 . Conce ning he p oo o he second pa ag aph in i e heo em, le A be an a bi a y absolu e alued algeb a wi h le uni , so ha by Theo em 1 he se J :={L a :aEA} is a smoo h no med Jo dan algeb a (o bounded linea ope a o s on he no med space o A) . Since A is an absolu e alued algeb a, he mapping u : a -> L a is a linea isome y o n he no med space o A on o he one o J and, as a consequence ; he mapping 0 : F -> UFU - is a uni al isome ic (hence *-, by he i iplica ion (iii) ==> (i) in Lemma 2) ep esen a ion o J on he p e-Hilbe space o J . The p oo will be concluded by showing ha A is isome ically isomo phic o he absolu e alued algeb a ob ained om he pai ( .1, 0) by he cons uc i a me hod in he i s pa ag aph . Bu he abo e conside ad su jec i e linea isome y u : A -> J is also an isomo phism om A on o (J, O), because o a and b in A we ha e u(a) O u(b) = V)(u(a))(u(b)) = uL a u -I (u(b)) = u(L a (b)) = u(ab) . Rema ks 4 . i) Mos o he in o ma ion gi en by he abo e heo em can be s a ed wi hou in ol ing Jo dan algeb as and hei ep esen a- ions on ec o spaces, as ollows .  The no m o any absolu e alued algeb a A wi h le uni e de i es o n an inne p oduc ( .  .) ; and, o a, b, c in A wi h a o hogonal o e, we ha e (ab 1 c) = -(b 1 ac) and  a(ab) = - 11 a 112 b . ii) The Albe -U banik-W igh heo em on absolu e alued algeb as wi h uni can be easily de i ed om he abo e ema k . Fo , i A is such an algeb a and 1 deno es i s uni elemen , aking b = 1 in he las equali y we ob ain a 2 = - 11 a 112 1 o all a, in A o hogo ial o 1, hence A is a quad a ic algeb a . Mo eo e he same equali y now yields o La = L a z o a, in A o hogonal o 1, and by symme y we ha e also R 2 = Rae ; hence A is al e na i a . Now A is a di ision quad a ic al e na i e algeb a, so i is isomo phic o R, C, H, o ® by he ex ended F obenius heo em (sea o example [12, Theo em 2 .26]) . iii) The examples in Rema k 3(i) show obs ensibly ha isomo phisms be ween absolu e alued algeb as can ail o be isome ic o e en con in- uous, a pa hology ha , as we will show in Sec ion 4, only can occu in absence o comple eness . Howe e , in he pa icula case o le -di ision absolu e alued algeb as i is no di icul o de i e om P oposi ion 4 and Theo em 2 ha isomo phisms mus be isome ic . ONE-SIDED DIVISION ABSOLUI E VALUED ALCEBRAS  94 3 3 . Exis en e o one-sided di ision absolu e alued algeb as By Co olla y 3 he comple ion o a le di ision absolu e alued al- geb a is a le di ision absolu e alued algeb a, and by P oposi ion 4 and Theo em 2 e e y le di ision comple e absolu e alued algeb a is a Hilbe space . Then one can ask na u ally o hose ca dinal numbe s 3` o whicli he e exis le di ision comple e absolu e alued algeb as wi h hilbe ian dimension equal l` . Since in ini o dimension he answe is clea ly 1`2 = 1, 2, 4, o 8 (see P oposi ion 3), we will cen e ou a en ion in he in ini o-dimensional case, and in aca we will p o e he exis en e o (au oma ically le di ision) comple e absolu e alued algebbas wi h le uni o a bi a y in ini e hilbe ian dimension and wi h he addi ional p ope y ha hey ha e no nonze o p ope closed le ideals (no e ha e e y le di ision algeb a has no nonze o p ope igh ideals) . By in ok- ing Theo em 2, he e i ica ion o his ac is equi alen o p o e ha e e y in ini o-dimensional smoo h comple e no med Jo dan algeb a has an "i educible" uni al *- ep esen a ion en i s own Hilbe space . A his espec we ecall ha a sel -adjoin se S o bounded linea ope a o s on a eal o complex Hilbe space K is said e ac i educibly on K i he only closed S-in a ian subspaces o K a e 0and K . While o complex K his concep has been widely s udied, his is no he case o he eal con ex in which we a e mainly in e es ed, so we begin ou a gumen wi h he ollowing Lemma 3 . Le K be a complex Hilbe space, S a sel -adjoin sub- se o BL(K) ac ing i educibly on K, and Q be any nonze o p ope S-in a ian closed eal subspace o K . Then K = Q ®12 áQ . As a conse- quence, S ( ega ded as a sel -adjoin se o bounded linea ope a o s on he eal Hilbe space Q) ac s i educibly on Q . P oo . Fi s no e ha , being Q n iQ a complex p ope closed S- in a ian subspace o K and ac ing S i educibly on K, we mus ha e Q 1 áQ = 0 . Deno ing by n he ( eal linea ) o hogonal p ojec ion om K on o Q, by he S-in a iance o Q and he sel -adjoin ness o S, i commu es wi h e e y elemen in S . Now, ega ding complex numbe s as linea ope a o s on K, 7 - i7 i is a bounded co iplex-linea ope a o on K commu ing wi h he elemen s o S, hence, by he i educibili y o S on K, we ha e 7 - i7 i =a + i,3 o sui able a and,3 in R (see [7, P oposi ion 2 .3 .8]) . Mul iplying on he igh his equali y by 7 , we ha e i - i7 i7 = a7 + i,(j7 so, since Q 7~ 0 94 4  A . RODRÍCUEZ PALACIOS and Q n iQ =0, we ob ain a = 1 and -T i7 = j7 . By aking adjoin s in he las equali y i ollows ha ,0 = 0 and he e o e - - i7 i = 1 . Thus, since -¡i ¡ is clea ly he o hogonal p ojec ion om K on o iQ, we ha e ha iQ is he o hogonal complemen o Q in K, ha is K Q ® 2 iQ . Fo he consequence asse ed in he s a emen , no e ha any nonze o S-in a ian closed ( eal) subspace R, o Q is a eal subspace o K which also sa is ies he assump ions on Q, hence he abo e p o ed ac abou Q applies o R gi ing clea ly R= Q, and ce ainly S ac s i educibly on Q, as desi ed . The exis en e o "i educible" uni al *- ep esen a ions o smoo h com- ple e no med Jo dan algeb as on ( eal) Hilbe spaces will ollow om he abo e lemina and he nex p oposi ion, which con ains basic ac s abou he "canonical an icommu a ion ela ions", and is aken almos li e ally om [8, pp . 6-11] (see p ecisely [8, P oposi ion 5 .2 .2]) . P oposi ion 5 . Gi en a complex Hilbe space H, he e a e a nonze o complex Hilbe space K ( he so called Fe mi-Fock space o H) and a conjuga e linea mapping - a( ) ( he "annihila ion" ope a o ) om H in o BL(K) sa is ying he ollowing he e p ope ies : i) 2a( ) .a(g)* = ( ig)I and a( ) .a(g) = 0 o all , g in H ("canon- ¡cal an icommu a ion ela ions"), whe e " ." deno es Jo dan p od- uc and I deno es he iden i y ope a o on K . ii) K is ini e dimensional whene e H is so, while, i H is in ini e- dimensional, he hilbe ian dimension o K equals he one o H . iii)  The sel -adjoin se o ope a o s {a( ), a(g)*  : , g E H} ac s i educibly on K . Call a ep esen a ion o a Jo dan algeb a on a nonze o Hilbe space i educible i i s ange is a sel -adjoin se o bounded linea ope a o s ac ing i educibly on he gi en Hilbe space . P oposi ion 6 . E e y smoo h .co cple e no med Jo dan algeb a has an i educible uni al *- ep esen a ion on a ( eal) Hilbe space . Mo eo e o such an algeb a (say .J) he ollowing asse ions a e equi alen : i) J has an i educible uni al *- ep esen a ion on i s own Hilbe space . ii) J has a uni al *- ep esen a ion on i s own Hilbe space . iii) The hilbe ian dimension o J equals 1, 2, J , 8, o any in ini e ca dinal numbe . ONE-SIDED DIVISION ABSOLUTE VALUED ALCEBRAS  94 5 P oo . Fo any ca dinal nu be K, conside he complex Hilbe space o (complex) hilbe ian dimension ?~, le K and a be espec i ely he complex Hilbe space and he conjuga e-linea mapping om H in o BL(K) gi en by P oposi ion 5, and de ine a ( eal-linea ) mapping s om H in o BL(K) by s( ) := i(a( ) + a( )*) . F om asse ion (i) in P oposi ion 5 we ob ain o all and g in H, so ha he se s( ) .s(g) = - Re( 1 g)I J :={AI+s( ) :AER, EH} is a eal Jo dan algeb a o ope a o s on K ha , algeb aically conside ad, is acopy o he smoo h comple e no med Jo dan algeb a o ( eal) hilbe - ian dimension 21` + 1 (ac ually one can sea ha , when endowed wi h he ope a o no m, his copy is o en an iso ne ic copy, bu his ac is i ele an o ou a gu nen ) . Mo eo e , lince clea ly S( )* = -s( ) o all in H, J appea s uni ally *- ep esen ad on he eal Hilbe space KR unde lying K . In a i s ins an e, aking in o accoun ha he smoo h comple e no med Jo dan algeb a o hilbe ian di nension 2?~ can be uni ally *-embedded in he one o hilbe ian dimension 2?~ +'1 and applying asse ion (ii) in P oposi ion 5, his a gu nen shows ha any ini o-dimensional smoo h no mad Jo dan algeb a can be uni ally *- ep esen ad on a . nonze o ini a-dimensional ( eal) Hilbe space and, o ob ain i educible uni al *- ep esen a ions, i is enough o pass o he es ic ion o he ope a o s in he ango o he exis ing ep esen a ion o a subspace which is inimal among he nonze o subspaces ha a e in a ian unde i e ango o he gi en ep esen a ion (such a n inimal subspace always exis s because o he ini e dimensionali y) . Re aking he ini ial a gu nen in i e in ini a-dimensional case, J is he smoo h comple e no mad Jo dan algeb a o a bi a y in ini a hilbe ian dimen- sion 1` (= 21~ + 1 in his case), and he iden i y ope a o on J is a uhi al *- ep esen a ion o J on he Hilbe space K R which, in iew o asse - ion (ii) in P oposi ion 5, has also hilbe ian dimension equal o ?~ . I his ep esen a ion is no i educible, by he de ini ion o J he e us exis a nonze o p ope elosed eal subspace Q o K in a ian unde i e sel -adjoin se o bounded complex-linea ope a o s S := {s( ) : E H} . 946  A . RODRÍGUEZ PALACIOS Bu , i ollows om he de ini ion o he mapping s and he conjuga e- linea i y o he mapping a ha a( )-_ s(Z )-is( ) anda( )*=- s(i )+is( ) 2  2 o all in H . These equali ies, oge he wi h asse ion (iii) in P opo- si ion 5, show ha S ac s i educibly on K (in he complex sense) . By Lemma 3, S ac s i educibly on Q (in he eal sense) ha is, he mapping T -> T/Q om J in o L(Q) is an i educible uni al *- ep esen a ion o J on he eal Hilbe space Q which, in iew o he equali y K = Q®iQ (sea again Lemma 3) and he in ini e-dimensionali y o K, is also o hilbe ian dimension equal o k~ . Thus ; in any case, he in ini e-dimensional smoo h comple e no med Jo dan algeb a J o hilbe ian dimension ?~ has an i e- ducible uni al *- ep esen a ion on a Hilbe space o hilbe ian dimension IZ . Since he ini a-dimensional case has been conside ad p e iously, his concludes he p oo o he i s pa ag aph in he p oposi ion, and e en p o es he implica ion (iii) => (i) in he in ini e-dimensional con ex . To inish he p oo o his implica ion no e ha , i o i = 1, 2, 4, 8 we deno e by A i he absolu a alued algeb a R, C,H, ® espec i ely, he smoo h no med Jo dan algeb a JZ o dimension i can be ecognized as he Jo dan algeb a o ope a o s on A i gi en by {L a : al E A Z }, and hen he iden i y mapping on Ji is an i educible uni al *- ep esen a ion on he Hilbe space o A i which o cou se has dimension i . Since he implica ion (i) =~> (ii) is clea , le us conclude he p oo o he p opo- si ion showing ha (ii) => (iii), namely, i a smoo h comple e no med Jo dan algeb a J has a uni al *- ep esen a ion on i s Hilbe space ; any ini e dimension di e en om i = 1, 2, 4, 8 mus be excluded o J . Bu his ollows om he i s pa ag aph in Theo em 2 oge he wi h he implica ion (iii) =~> (ii) in P oposi ion 3 . 91 Wi h P oposi ion 4 and Theo em 2, he abo e p oposi ion leads di- ec ly o he ollowing Theo em 3 . Le di ision comple e absolu a aluad algeb as o hilbe - ian dimension ?~ exis i and only i H equals 1, 2, 4, 8, o any in ini e ca dinal numbe . Mo eo e , o such a ca dinal l~ he e exis in ac (au- oma ically le di ision) comple e absolu a alued algeb as wi h le zeni o hilbe ian dimension k~ wi h he p ope y ha hey ha e no nonze7o p ope closed le ideals . Rema ks 5 . i) Since he comple ion o a smoo h no med Jo dan algeb a is a smoo h no med Jo dan algeb a, i ollows om he i s pa ag aph in P oposi ion 6 ha e e y smoo h no mad Jo dan algeb a ONE-SIDISD DIVISION ABSOLUTEVALUED ALGEBRAS  94 7 has a uni al *- ep esen a ion on a Hilbe space . Wi h he i nplica ion (i) => (ii) in Le nma 2, his shows ha e e y smoo h no med Jo dan algeb a can be iewed as a subspace P o bounded linea ope a o s on a sui able no med space (which ac ually can be chosen o be a Hilbe space) sa is ing he assump ions in Theo em 1 . Now ce ainly we a e su e ha Theo em 1 canno say mo e . ii) E e y in ini e-dimensional smoo h comple e no med Jo dan algeb a J has noni educible uni al *- ep esen a ions on i s Hilbe space . Fo ; i ~b is any uni al *- ep esen a ion o J on i s Hilbe space ; he napping V) ® 0 om J in oL(J ®l2 J), gi en by 'O ® ~( X) (y, z) = (0 (X) (Y), 0(x)(z)) o al] x, y, z in J, is a noni educible uni al *- ep esen a ion o J on he Hilbe space J (D ¿2 .I which has he sa ne hilbe ian dimension ha o J . Via Theo em 2, his ac e iec s on he exis en e o comple e absolu e alued algeb as wi h le uni (o a bi a y in ini e hilbe ian dimension) ha ing nonze o p ope closed le ideals . iii) I A is an in ini e-dimensional le -di ision absolu e alued algeb a, hen all ope a o s o igh mul iplica ion on A a e nonin e ible . This ollows om i nplica ion (i) ==> (iii) in P oposi ion 3 . The es o his sec ion will be de o ed o ob ain so e in e es ing consequences o he exis en e o in ini e-dimensional one-sided di ision absolu e alued algeb as . Tl e i s esul we will p o e in his di ec ion is ha small pe uba ions o he p oduc o a . le di ision comple e abso- lu o alued algeb a gi e ise o new le di ision algeb as, hus p o iding in iew o Theo em 3 a e y wide collec ion o in ini e-dimensional com- ple e no med le di ision algeb as . The a bi a i y o he pe u ba ion, oge he wi h he s uc u e heo y o le di ision absolu o alued alge- b as (P oposi ion 4 and Theo em 3), shows ha he algeb as ob ained by his p ocedu e canno be in gene al absolu o alued algeb as . As a ma e o ac , we will ealize ha all hese algeb as ail o be di ision algeb as . P oposi ion 7 . Le A be a le di ision comple e absolu o alued al- geb a, le El be any con inuous bilinea p oduc on he Banach space o A, and le d deno e he dis an e om l] o he p oduc o A . Thenwe ha e : i) I d < l ., he ec o space o A wi h he p oduc 111 is a le di ision algeb a . ii) I A is in ini e-dimensional and d < 1, hen all ope a o s o igh mul iplica ion, on A 7-ela i e o he p oduc 0 a e nonin e? , ible . 94 8  A . RODRÍGUEZ PALACIOS P oo .. Fo a in A, le us deno e as usual by L a and R a espec i ely he ope a o s o le and igh mul iplica ion by a ela i e o he ini ial p oduc , and by Lo and R .° he ones ela i e o he p oduc El . I d < 1, since A is a le di ision absolu e alued algeb a, o e e y a in A {0} we ha e ha L a is in e ible and ~~ La - L°  lic d 11 a jj<jj a 11=11 La 1  11 -1 so by comple eness o AL° is in e ible, and (i) is p o ed . Le us assume A in ini e-dimensional and d <_ 1 . Then, o a in A wi h 11 a 11= 1, R a is a linea isome y o m A in o A which is no on o (see Rema k 5 (iii)) and ~~Ra - Ro li~d11ajj< 1 , so Lemma 1, oge he wi h he ac ha he se o nonin e ible elemen s o a Banach algeb a is closed, gi es ha Rao is no in e ible, and (ii) ollows . Gi en an algeb a A and an elemen A in he base ield, he A-mu a ion o A, deno ed by A( A ), is de ined as he algeb a wi h he same ec o space ha o A and p oduc gi en by (a, b) -> Aab + (1 - A)ba . Co olla y 4 . Le A be a le di ision comple e absolu e alued algeb a, and A be a eal numbe . Then we ha e : i) I A > 2, hen A1`1 is a le di ision algeb a and, i in addi ion A is assumed o be in ini e-dimensional, hen all ope a o s o igh mul iplica ion on A(A) a e nonín e ible . ii) I A = 2 a ad A is in ini e-dimensional, hen all ope a o s o le (= igh ) mul iplica ion on A( - ) a e nonin e ible . iii) I A < 2, hen Ahl is a igh di ision algeb a and, i in addi ion A is assumed o be in ini e-dimensional, hen all ope a o s o le mul iplica ion on A(A) a e nonin e ible . P oo .. (i) and (ii) ollow di ec ly om P oposi ion 7 by aking as he p oduc 0 he one gi en by allb := ab + 1  ba, while (iii) ollows om (i) applied o 1 - A aking in o accoun ha he opposi e algeb a o A(' -- ) is A( A ) . ONE-SIDED DIVISION ABSOLUTE VALUED ALGEBRAS  94 9 Rema k 6 . A consequence o he abo e co olla y is he well-known ac ha all A-mu a ions wi h A 7~  o a ini e-dimensional absolu e á alued algeb a a e di ision algeb as . Ou concluding esul in his sec ion, oge he wi h Theo em 3, will show he exis en e o in ini e-dimensional comple e no med algeb as such ha he ope a o s o le and igh mul iplica ion by any no lze o elemen a e su jec i e . Un o una ely, in ou examples all hese ope a o s will ail o be one- o-one . Ou a gumen begins wi h he easy obse a ion in he ollowing le nma . By in olu ion on a eal ( esp . : complex) ec o space we mean a linea ( esp . : conjuga e-linea ) ope a o on he space wi h squa e he iden i y ope a o . Lemma 4 . Le A be a eal o complex comple e absolu e alued algeb a whose Banach space is a Iiilbe space, le - be an in olu ion on he ec o space o A, and de ine a new (bilinea ) p od-ac 0 on A by a0b := Rb (a) (whe e, as usual, R .b deno es he ope a o o igla mul iplica ion by b ela i e o he inicial, p oduc ) . Then, o e e y b in A {0}, che ope a o R° o igh mul iplica ion by b ela i e o he p oduc E is su jec i e . P oo . No e ha R° = Rb, so i is eno lgh o p o e ha R* is a su jec i e ope a o o any no m-one elemen b in A . Bu , being A an absolu e alued algeb a, o such a b, R .h is a linea isome y so, since he Banach space o A is a Hilbe space, we ha e R*R = I ( he iden i y ope a o on A), hence ce ainly R* is su jec i e . Ou ollowing obse a ion, which is also o easy e i ica ion, in ol es s anda d e minology o H*-algeb as . 1{ollowing [23] and [111, a le , se?ni-H*-algeb a will be a eal o complex Hilbe space A oge he wi h a con inuous bilinea p oduc on A (deno ed usually by jux aposi ion) and an in olu ion - on he ec o space o A sa is ying o all a, b, c in A . (ab 1 e)=(b1ác) Lemma 5 . Le A be a eal o7- complex le semi-H*-algeb a whose in olu ion - is isome ic,  _and de ine a new p oduc 0 on A by a0b - R* (a) .  Then - is an algeb a in olu ion on (A,[ :]) ( ha is, a0b = bOa o all a, b in A) .