How to solve an operator equation

Publicacions Nla e ná iques, Vol 36 (1992), 793-760 . HOW TO SOLVE AN OPERATOR EQUATION A bs ac MAI2TIN MATHILU This a icle summa izes a se ies o lec u es cleli e ed a he Ma h- e na . ics Depa nen o he Uni e si y o Leipzig, Ce inany in Ap il 1991, which we e e o e iew ecl niques o sol ing ope a o eclua ions en C*-algeb as connec ed wi h me hocis de eloped in a Spanish-Ge nan esea ch p ojec en "S uc u e ancí Applica ions o C*-Algeb as oí' Quo ien s" (SACA) . One o he esea che s in his p ojec was P o esso Pe e Menal un il his unexpec ed dea h his Ap il . To his me no y his pape shall be dedica ed . 1 . In oduc ion Sol ing equa ions belongs o l e undamen al asks o ina hema ics . Many p oblems in he sciences load e equa ions in ol ing numbe s, map- pings ; nd o he guan i ies . In aca, i equendy occu s ha e en ually a dues ion can be ph ased as an "equa ion", al hough, a i s , i appea ed ú be o a a he di e en na u e . To ind a solu ion o an equa ion gene - ally implies bo h he exis ence as well as he unidueness p oblem . The e is no uni e sal p ocedu e o sol ing ; bu he de ices in en ed seem , o be as mani old as he possible ques ions, asid only allow a a he ough classi ica ion such as nu ne ical, app oxima i e, algeb aic me hods e c . Flowe e , i is always an i npo an s op o de e mine he co mon ea- u es in sol ing a ce ain class o exa ples o he ai e o de eloping a machine y which enables o handle a speci ied collec ion o equa ions a one iene . In he p esen pape ; we will be conce ned wi h equa ions wi hin a non-commu a i e in ini e dimensional se ing . To be mo e speci ic, hey will be o he o m 'Chis pape is pa o a esea ch p ojec suppo ed by lic DAAD . 744  M . MATHIEU whe e, o each `pa ame e ' a, T«,~1,  , , en is a linea ope a o on a C*- algeb a A (wi h ce ain addi ional p ope ies) and we a e looking o elemen s xj E A sol ing he equa ion (1) (o be e , his sys em o equa ions) . We will i s ly collec some examples o ques ions which can be ph ased in an equa ion such as (1), hen desc ibe a gene al ool o ackle hem, and inally indica e solu ions which yield answe s o he ques ions lis ed . As a common ea u e, he ques ions in Sec ion 2 lead o equa ions in a C*-algeb a ; ha is, we a e looking o ce ain elemen s in a C*-algeb a sol ing he equa ion, while he condi ions ypically a e o mula ed in e ms o ope a o s de ined on he C*-algeb a . Needless o say ha he e a e many mo e ins an es which can be se led by he p oposed me hods . 2 . Examples We ha e selec ed ou examples om he ollowing ou classes o op- e a o s on C*-algebas : de i a ions, comple ely posi i e ope a o s, cen- alizing mappings, and gene a o s o dynamical semig oups . 2 .1 . De i a ions . Le A be a C*-algeb a and S a de i a ion on A, Le . a linea mapping om A in o i sel sa is ying Leibniz' ule S(xy) = x6(y) + b(x)y o all x, y E A . Each de i a ion S is au oma ically bounded whence i is meaning ul and wo hwhile o know unde which ci cums ances 8 is a compac ope a o , wi h espec o he no m o a weake opology . He e, we ask when S is weakly compac , ha is, when does S map he uni ball o A in o a subse whose closu e is compac wi h espec o he weak opology on A . (This is mo e closely ela ed o he poin o iew aken in his pape han he no m compac case, which, howe e , can be ea ed simila ly .) Specialize o he case A= B(H), he algeb a o all bounded linea ope a o s on some Hilbe space H . Since B(H) is he second dual o K(H), he closed ideal o all compac ope a o s on H, and S is con inuous wi h espec o he Q(B(H),K(H)*)- opology, S coincides wi h (S j )**, he second adjoin o he es ic ion 81 o S o K(H) . I is well known ha 6 1 is weakly compac i and only i (81)** maps K(H)** in o K(H) [15, VI .4 .2] . Mo eo e , by Gan mache 's heo em [15, VI .4 .8], 61 is weakly compac i and only i (ó1)** is weakly compac . Pu ing all his oge he yields ha 8 = (S1)** is weakly compac i and only i SB(H) C K(H) . In he gene al case we ha e o eplace K(H) by he ideal K(A) o all compac elemen s in A ; and, using app op ia e ep esen a ions, we ob ain he ollowing, c . [23, Theo em 2 .7] . HOW TO SOLVE AN OPERATOR EQUATION  74 5 P oposi ion 1 . A de i a ion S on a C*-algeb a A is 2ueakly compac i and only i b**A** C K(A) . Again, K(A) is b-in a ian and hus b induces a de i a ion b en he gene alized Calkin algeb a A/K(A) . Co olla y 2 . I b is weakly compac , hen b = 0 . Suppose b we e inne , Le . b = b ., whe e 6,(x) = xa - ax, and he elemen a belonged o K(A) . Then, b is weakly compac by [41, The- o em 3 .1] . On he o he hand, b** is always inne by Sakai's heo em . The e o e, he o iginal ques ion o weak compac ness o b leads o he ollowing ope a o equa ion . (1 .1)  Can & = 0 be sol ed in K(A)? Going one s op u he we can ask a simila ques ion o he p oduc 6162 o wo de i a ions 51, 62 on A (which ; in gene al, is no longe a de i a ion) : when is 5162 (weakly) compac ? This ques ion should be e- la ed o 11e Dun o d-Pe is p ope y o a commu a i e C*-algeb a which implies ha T1T2 is a compac ope a o whene e T1, T 2 a e weakly com- pac on A . By simila a gumen s as abo e, i can be o mula ed in e cos o ope a o equa ions as ollows . (1 .2)  Can & i b, 12 = 0 be sol ed in K(A)? Ques ions o his kind a e s udied in [25] and [27] . 2 .2 . Comple ely posi i e ope a o s . Recall ha a linea mapping T on a C*-algeb a A is said o be com- ple ely bounded i he no nls JIT,,11 o he canonical ex ensions T,, o T o he ma ix algeb as Al,,(A) o e A a e all bounded by some eal numbe , and T is comple ely posi i o i all T,, a e posi i o ope a o s on M  (A) . The p o o ypes o comple ely bounded ope a o s a e 11e eleme a a y op- e a o s gi en conc e ely as mappings o he o m 5 :x~-+ xb j wi h x E A, a1, . . . , a, l , b1, . . . ; b, L E 1Vl(A), whe e A11(A) deno es 11e mul iplie algeb a, o A . This is jus i ied by he ep esen a ion heo em o comple ely bounded ope a o s and he ac ha ce ain comple ely bounded ope a o s can be app oxima ed 746 . M . MAT1-HEU by elemen a y ope a o s, c . [12] . A na u al ques ion in his con ex is : wha does a comple ely posi i e elemen a y ope a o S look like? Al hough his is in ol ing ineguali ies, we immedia ely a e led o an ope a o equa ion . Deno e by M a ,b he ( ino-sided) mul iplica ion x >-> axb . I S = 1 1Vl a ~,b~ is posi i e, i is he mi ian-p ese ing om which 111 l b -' a AI a  b~ j-1 ollows- As a esul we a e o conside he ollowing ope a o equa ion . (1 .3)  Which elemen s .x j , y j EM(A) sol e Z :" 1 A j , yj = 0? This ques ion has eme ged o be no only an example, bu o unda- men al signi ican e o ou app oach, c . [28] . 2 .3 . Cen alizing mappings . Le R be a ing . An addi i e mapping F : R -> R is cen alizing i , o e e y .x E R, we ha e [x, F(x)] = xF(x) - F(x)x E Z(R), he cen e o R . In many cases ; he exis en e o ce ain cen alizing mappings yie1ds commu a i i y c i e ia o R . Fo example, i R is a p ime ing ; hen R, is commu a i e i he e is a non-ze o cen alizing de i a ion on R [38, Theo em 2], see also [30], o i he e is a non-iden ical cen alizing au omo phism on R [31, Theo em] . In he con ex o ope a o algebas, he e a e analogues o he e esul s as ollows . P oposi ion 3 . The e is no non-ze o cen 7-alizing de i a ion on a C*-algeb a . This seems o be a olklo e ex ension o Singe 's classical esul ha he e a e no non-ze o de i a ions on commu a i e C*-algebas . In ac , i b is a cen alizing de i a ion on a C*-algeb a A, i easily ollows ha bA C_ Z(A) . Hence, he es ic ion 6 1 o S o Z(A) anishes so ha 62 = 0 . The iden i y 2 b(x)y5(x) = 6 2 (Zyx) - xb 2 (yx) - b 2 (xy)x + xb 2 (y)x  (x, y EA) he e o e yields Aló(x),a(x) = 0 o all x E A, whence b = 0 . The case o au omo phisms equi es some mo e wo k and was i s s udied by Mie s . o equi alen ly, o equi alen ly, HOW TO SOLVE AN OPERATOR EQUATION  74 7 P oposi ion 4 . [32, Theo em 5] Le ce be a cen alizi ~g *- au omo phism on a on Neumann algeb a A . The e is a cen al p ojec- ion e E A such ha a(e) = e, aJA, = idA, and A(1-e) is commu a i e . Whe he his esul emains ue o a bi a y (no necessa ily *- p ese ing) au omo phisms was answe ed only ecen ly by B esa , who also ob ained a gene al s uc u e heo em o cen alizing mappings en on Neumann algeb as as ollows . P oposi ion 5 . [8, Theo em 2 .1] Le F be a cen alizing addi i e mapping on a on Neumann algeb a A . Then he e exis an elemen c E Z(A) and an addi i e mapping ( : A ---> Z(A) such ha F= L, + ( . He e and in he sequel, we will deno e by L, he le mul iplica ion x --> ax and by R . a he igh mul iplica ion x H xa . We will now e o mula e bo h he assump ion as well as he conclusion in e ms o ope a o equa ions . This will enable us o ob ain an ex ension o B esa 's esul o a bi a y C*-algeb as in Sec ion 4 . Obse e a i s ha e e y cen alizing addi i e mapping F on a C*- algeb a A is in ac commu ing, Le . [x, F(x)] = 0 o all x E A [9, P oposi ion 3 .1] . Replacing xby x + y he e o e gi es [x, F(y)] + [y, F(x)] = 0  (x, y E A) (2)  6F(y) - 6 y F = 0  o all y E A . Secondly, i F= L, + ~ whe e A is a C*-subalgeb a o a C*-algeb a B wi h cen aliza C,3 (A), c E CB(A) and ~ : A . -> CB(A), hen [x, F(y)] _ [x, cy] + [x, «y)] = [x, cy] o all x, yE A . Hence [x, F(y) - cy] = 0  (x, y E A) 5F(y)-cy = 0  o all yE A . Con e sely, i c E CB(A) sa is ies (3), hen ~ = F-L, de ines an addi i e mapping om A in o CB(A) . As a esul we a i e a he ollowing ques ion . (1 .4) Suppose ha F sa is ies (2) o all yE A . Is he e an elemen c E CB(A) o a `sui able' C*-algeb a B con aining A sa is ying (3) o all y E A? No e ha (3) p ecisely is a sys em o ope a o equa ions o he o o (1) pa ame ized by all elemen s in A . 74 8  M . MATHIEU 2 .4 . Gene a o s o dynamical semig oups . Le A be a uni al C*-algeb a . A bounded he mi ian-p ese ing linea ope a o L : A -> A wi h L(1) = 0 is called comple ely dissipa i e i , o allnEN, These ope a o s a e he gene a o s o no m-con inuous one-pa ame e semig oups (T ) ER + o uni al comple ely posi i e ope a o s T,, en A ; which desc ibe he i e e sible dynamics o open quan um sys ems, o , equi alen ly, se e as ansi ion ope a o s o non-commu a i e Ma ko p ocesses . In many conc e e si ua ions, hey a e buil om wo p o o- ypes : he comple ely posi i e ope a o s and he he mi ian-p ese ing gene alized inne de i a ions Sk , k . = R .k + Lk . . The con e se ques ion, when a gi en comple ely dissipa i e ope a o L can be decomposed in o L,(x*x) ? x * L,,(x) +L , ,(x * ) x  (x E M,, (A» . a e wo decomposi ions . Then, pu ing a = k1 - k 2 , we ha e ( 5 )  S .,a-+01-02=0 ., Thus, we may ask (1 .5)  Unde which condi ions does (5) imply ha  0? A mo e gene al ques ion would be which a in A" sol e he equa ion wi h 0 comple ely posi i e om A in o some possibly la ge C*-algeb a B and k E B was i s s udied by Co ini, Kossakowski and Suda shan [18] and Lindblad [21] ; and ela ed o cohomological p ope ies o A in [22] and [11] . I A C_ B(H), hen a decomposi ion (4) o L always exis s wi h OA CA" and k E A" . In gene al ; his decomposi ion will no be unique . The uniqueness p oblem can be e o mula ed in e ms o an ope a o equa ion as ollows . Suppose ha L = 01 + 4,,k, = 02 + 8kz,kz 3 . De ices All he abo e equa ions (1 .1) h ough (1 .5) can be subsumed unde he gene al o m (1) . To mo i a e ou ools o sol ing hem ; . le us How o SOLVE AN OP-ERATOR . EQUATION  74 9 u he mo e conside a special case o (1 .3) . Le A = B(H) and b E A be gi e . (1 .3')  Which a E A solee AI,,b = 0? In ou pa icula si ua ion, he answe is quickly eaclied . I L1I a, b = 0, hen axb~ = 0 o all x E A asid 1 E H . I b = 0, ob iously all a E A a e solu ions . I b 7~ 0 ; pick ~ E II wi h b~ :7~ 0 and no e ha b~ is cyclic o A, i .e . Ab~ = H, and hus a = 0 . Clea ly, his me hod only wo ks in he p esence o a Hilbe space on which A ac s' ansi i ely eno gh', e .g . i A is i educible . The algeb aic me hod p esen ed now wo ks wi hou unde lying space . I is con enien o eph ase (1 .3) using he ollowing concep . Fo e e y C*-algeb a A we le ú(A) be he algeb a o all ele nen a y ope a o s on A . We de ine a su jec i e algeb a llo o no phism (6)  0 : M(A) ® A11(A)"1> --, ú(A),  B(a (9 b) = 111,,b whe e A11(A) ® AJ(A)" deno es he algeb aic enso p oduc o M(A) wi h i s opposi e algeb a . The p oble n now is o de e mine hc ke nel o 0 . The ollowing was p e ed in [24, Pa 1 ; Co olla y 4 .4] . (7)  0 is injec i e i and onlg i A is p ime . Since p imi i e C*-algeb a5 a e p ime, i is emp ing e use ep esen- a ion heo y in o de o app oach hc gene al case o n he special one . Howe e , as i eme ged, he e may be p oblems in pu ing he 'local' in o ma ion oge he o ob ain a, 'global' pic u e . I , seems ad a, a,geous o iew he p ime C*-algebas as he building blocks, which esul s in ega ding a C*-algeb a as a se nip i ne algeb a a he i an a se-misim,ple one . In ac , simila ecliniques a d esul s as hose desc ibed below a e a ailable in he se ing o se nip i ne ings . The ideal s uc u e o a p ime algeb a is dis inguislied by he ac ha e e y non-ze o ideal is essen ial, Le . in e sec s cae] - ) o he non- ze o ideal non- i ially . This allows o "nie e a . ound o n one place o ano he ''' wi hin he C*-algeb a wi hou loss o in o ma ion . Fo an a bi a y C*-algeb a . A we he e o e deno e by 1, and 1, he collec ions o all essen ial and all closed essen ial ideals o A, espec i ely . No e ha he e a e di ec ed dow wa ds by inclusion, Le . 1 7 12 E 1, i nplies ha 1 1 nI 2 E1, . Fo e e y se nip i ne ing R,, he nul iplie ing AI(R) is de ined by i s uni e sal p ope y ha Id is an essen ial ideal in M(R) and he e is 75 0  M . MATHIEU a unique ex ension p o he inclusion p : R -4 M(R) which makes he ollowing diag am commu a i e ; whene e R is an ideal in ano he ing S, R  --'->  M (R) in o he wo ds, M(R) is he (abs ac ) idealize o R . Usually, AII(R) is cons uc ed ia double cen alize s o R . Mo eo e , p is injec i e i and only i R is essen ial in S . Now, i I, J E 1, and J C_ I, hen J will be an essen ial ideal in M(I) whence, om he abo e, he e is a unique injec i e *-homomo phism p» : M(I) -> M(J) making he ollowing diag am commu a i e J pes, M(J) We may desc ibe pjj as " es ic ing he double cen alize s" . By means o his, we ob ain a di ec ed sys em {M(I) ; p j, J C I} o C*-algebas and inclusions, and i s algeb aic di ec limi alglim M(I) along 1 ., will be deno ed by Qb(A) and called he bounded symme ic algeb a o quo- ien s o A . This is a p e-C*-algeb a wi h comple ion Qb(A) - = lim M(I) deno ed hence o h by MI,,(A) and called he local mul iplie algeb a o A . Fo each I E 1 e le P(I) deno e he Pede sen ideal o I [37, 5 .6] . Using he ac ha P(I) is *-in a ian , belongs o 1, and ha P(I)P(J) = P(I) 1 P(J) o all I, .I E 1, we de ine Q, (A) = alglim M(P(I)) along 1 e and obse e ha his de ini ion leads o he symme ic algeb a o quo- ien s o A as de ined (sligh ly di e en ly) in ing heo y . l ollows ha Qb(A) embeds as a *-subalgeb a in o Q .,(A) and is in ac he bounded pa o Q,(A) [2, Theo em 1 .3] . A s onge ela ion be ween Qb(A) and Q, (A) p o ed in [3, Theo em 2] is ha Q, (A) is he cen al localiza ion o Qb(A) . Rema ks . The cons uc ion o M 1o ,(A) was i s pe o med by Ped- e sen [361 and Ellio [16] unde he name o essen ial mul iplie s . They used i o s udy ope a o equa ions o he o m 5 = ba,  aE Mlo, : (A), and How To SOLVE AN OPERATOR EQUATION  75 1 a = AJ ., i .,  u, E Ah o ,(A) uni a y, ha is, o ob ain inne ness o de i a ions 8 and *-au o no phis as a in A4h o ,(A) . In pa icula , Pede sen p o ed ha (8) always has a solu ion i A is sepa able [36, P oposi ion 2] . A abou he sa ne i ne, Kha chenko in oduced he sy mne ic ing o quo ien s o semip ime ings and used i in pa icula in Galois he- o y [19] ; [20] . This heme was u he pu sued by Pass nan [34], [35], Mon goine y [33], and o he s . I is o be seen in a long adi ion go- ing back o l e 30's in in es iga ing gene al ings o quo ien s, c . [40] . The basic idea - o enla ge a gi en 'domain' by addi ional 'nunnbe s' (=' ac ions', 'quo ien s') in o de o be able o sol e mo e equa ions - also se es as he mo i a . ion o ou app oa,ch o ope a o equa ions . In he la e 80's, M, o ,(A) was edisco e ed independen ly by A a [2], [3] a ld he au ho [26], [29] which hen launched a join esea ch p ojec on he s uc u e and applica ions o local mul iplie s [4], [5], [6] ; a co n- p ehe isi e accoun o his is o be gi en in [7] . We will now compile come o he basic p ope ies o Ah o ,(A) . P oposi ion 6 . Le , A be a C*-algeb a uwi h local nul iplie algeb a Ai1,, : (A) . (i) A is commu a i e i and only i A1h,,(A) i .s com nu a i e . (ii) A is p ime i and only i M1,(A) has i ial, cen e . (iii) Fo each I E Z, and each uni iza ion B o A we ha e A z ., ;([) = A4" ,, .(A) = NI,~ .(B) . (i ) Le , l be he p imi i e spec um o A .  I A is disc e e, hen H,,,,(A) = A4-(A) . ( ) I A is an AW*-al g geb a,, hen Ah o ,(A) =A . om (7) and (ii) in he abo e p oposi ion we see ha he ke nel o 0 is closely ela ed o he cen e Z = Z(A1h,,(A)) o 1VI lo ,(A) . I is he e o e impo an o analyse i s s uc u e . The ollowing was p o ed in [5, Theo em 1 and Co olla y 1] and can be iewed as a local e sion o he well-known Dauns-Ho inann heo em iden i ying he cen e Z(111(A)) o NI (A) wi h he algeb a, C(3Á) o all con inuous complex- alued unc ions on he S one-Cech compac i ica ion 3 l o l . P oposi ion 7 . Fo e e y C*-algeb a A, he cen e Z o A1h, (A) is an AW*-algeb a and can be i,den i ied wi h C( li E n1 0Í), whe e he in e se 758  , , . ;M .-,MATHIEU 5 . Conclusion We hope ha he- esul s desc ibed abó e may gi e some e iden e ha he local mul iplie álgeb a can se e as a `uni e sé', in which ope a o equa ions on C*-algeb as, a leas hose o he o o (1),``can~be sol ed' by a uni ied me hod . Re e en es l . ._ C . . A .,AKEMANN .ANDS . WRIGHT, Compác and weakly-compac de i a ions o  C*-algeb as, 'Paci ic J . Ma h . 85 (1979), 253-259 . 2 .  P . ARA, The ex ended cen oid o 'C*-algeb as ; A ch . .=Ma h . 1 54 (1990) 1 358-364 . 3 .  P . 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