Local automorphisms of the sets of states and effects on a Hilbert space

LOCAL AUTOMORPHISMS OF THE SETS OF STATES AND EFFECTS ON A HILBERT SPACE M´ ATY´ AS BARCZY AND MARIANN T ´ OTH Abs ac . We p o e ha e e y local au omo phism (a ine 1-local, o non-a ine 2-local) o he se s o all s a es on a Hilbe space is an au o- mo phism. We also p esen simila esul s conce ning he a ious au o- mo phisms o he se o all e ec s. 1. In oduc ion In wha ollows Hdeno es a sepa able in ini e dimensional complex Hilbe space and B(H) s ands o he algeb a o all bounded linea op- e a o s on H. The se So all s a es on His de ined as he se o all posi i e ope a o s on Hwi h ace 1, ha is, S={T∈B(H) : T≥0, T= 1}, whe e deno es he usual ace unc ional on B(H). The ope a o in e al E= [0, I] o all posi i e ope a o s on Hwhich a e bounded by he iden i y Iis called he e ec algeb a on H. The elemen s o Ea e called e ec s. These se s o ope a o s, ha is, he se o all s a es and he se o all e ec s, play impo an ole in he ma hema ical desc ip ion o quan um me- chanics (see, o example, [3, 7] and he e e ences he ein as well). Jus as wi h any algeb aic s uc u e, he s udy o he au omo phisms o hese se s when equipped wi h ce ain algeb aic s uc u es is o conside able impo - ance. So, wha algeb aic s uc u es can be gi en o he men ioned wo se s? As o S, i is a con ex subse o B(H) and one can conside i s a ine au omo phisms (also called S-au omo phisms [3] o mix u e-au omo phisms [7]) which a e he bijec i e maps φ:S→Ssa is ying (1) φ(λT + (1 −λ)S) = λφ(T) + (1 −λ)φ(S) Da e: Oc obe 22, 2005. 1991 Ma hema ics Subjec Classi ica ion. P ima y: 47B49, 47N50, 81Q10. Key wo ds and ph ases. S a es, e ec s, a ine au omo phism, e ec au omo phism. This esea ch was suppo ed by a g an om he Minis y o Educa ion, Hunga y, Reg. No. FKFP 0349/2000. 1 2 M ´ ATY ´ AS BARCZY AND MARIANN T ´ OTH o e e y T, S ∈Sand 0 ≤λ≤1. I is known (see, o example, [3]) ha e e y such φis o he o m (2) φ(T) = UTU∗(T∈S), whe e Uis ei he a uni a y o an an iuni a y ope a o on H( his la e no ion means a conjuga e-linea no m-p ese ing bijec ion o H). As o he se o all e ec s, we ha e se e al possibili ies how o in oduce algeb aic s uc u es on E. Fi s , Eis a con ex subse o B(H) and hence, jus as in he case o S, one can conside he a ine au omo phisms φo E. In ac , in he li e a u e i is e y common ha , in addi ion o ha φis an a ine bijec ion, hey also assume ha φis posi i e homogeneous as well ( ha is, φ sa is ies φ(λE) = λφ(E) o all E∈Eand 0 ≤λ≤1). In a ecen pape [11] Moln´a has de e mined all he a ine au omo phisms o he e ec algeb a in a on Neumann ac o wi hou his ex a condi ion on homogeni y. As a pa icula case o his esul we ob ain ha i φis an a ine bijec ion o E, hen φis ei he o he o m (3) φ(E) = UEU∗(E∈E), o o he o m (4) φ(E) = U(I−E)U∗(E∈E), whe e in bo h cases Uis ei he a uni a y o an an iuni a y ope a o on H. As a second possibili y, he e is also a so-called pa ial addi ion on E. Namely, i E, F ∈Eand E+F∈E, hen we de ine he sum o Eand Fby E+F. Now, we say ha he bijec i e map φ:E→Eis an e ec -au omo phism (in o he e minology, E-au omo phism [3]) i E+F≤Ii and only i φ(E) + φ(F)≤I and in his case we ha e φ(E+F) = φ(E) + φ(F). I is known [3], [7] ha he e ec -au omo phisms a e exac ly he maps o he o m (5) φ(E) = UEU∗(E∈E), whe e Uis ei he a uni a y o an an iuni a y ope a o on H. Finally, Ehas also a mul iplica i e s uc u e, namely one can conside he p oduc ABA on i (A, B ∈E). This ope a ion on Ewas in oduced in [11] and i was p o ed he e ha i dim H≥3, hen he au omo phisms o Ewi h espec o his mul iplica ion a e he same as in (5). The aim o his pape is o p esen some esul s on he local beha iou o he au omo phisms o he se s o s a es and e ec s appea ing abo e. The e a e ela i ely new in es iga ions on local maps o ope a o algeb as [5, 6, 10, 14] (also see he e e ences he ein). A linea map ψon an algeb a is called a linea (1-)local au omo phism i a e e y poin o he unde lying algeb a ψ coincides wi h an au omo phism o he algeb a (which au omo phism may, LOCAL AUTOMORPHISMS OF THE SETS OF STATES AND EFFECTS 3 o cou se, di e om poin o poin ). I is a ema kable ac conce ning he algeb a in ques ion i i hen ollows ha ψis necessa ily an au omo phism. Following he concep o 2-local au omo phisms due o ˇ Sem l [16], we say ha a map ψon any algeb aic s uc u e (no linea i y is assumed) is a 2- local au omo phism i a e e y pai o poin s o he unde lying s uc u e ψ coincides wi h an au omo phism (which au omo phism may depend on he pai o poin s in ques ion). Hence, we d op he assump ion on he linea i y o ψ, bu ins ead we equi e ha i beha es as an au omo phism a e e y pai o poin s. He e, he p oblem is ha whe he a 2-local au omo phism ψis necessa ily an au omo phism? Recen esul o ha kind can be ound in [13], [15]. I e e y linea (1-)local au omo phism o an algeb a is an au omo phism o e e y 2-local au omo phism o an algeb aic s uc u e is an au omo phism, hen we can say ha he au omo phisms o hose s uc u es a e, in a ce ain sense, comple ely de e mined by hei local ac ions. The aim o his pape is o ob ain esul s o ha kind conce ning he au omo phism g oups o Sand E. We no e ha he e a e se e al o he algeb aic s uc u es which also appea in ela ion o he ma hema ical o - mula ions o quan um mechanics. These a e, o ins ance, he o homodula pose o all p ojec ions on H, he Jo dan algeb a o all bounded sel -adjoin ope a o s on Hand he C∗-algeb a o all bounded ope a o s B(H). The local au omo phisms o hese la e s uc u es we e in es iga ed in [2], [13], [16]. 2. Resul s Ou i s esul which ollows says ha e e y a ine (1-)local au omo - phism o Sis an au omo phism. Theo em 1. Le φ:S→Sbe an a ine ans o ma ion ( ha is, a unc- ion sa is ying (1)) wi h he p ope y ha o e e y T∈S he e is an a ine au omo phism φTo Ssuch ha φ(T) = φT(T). Then φis an a ine au o- mo phism o S. P oo . Deno e by C1(H) he Banach algeb a o all ace-class ope a o s on H. Since φis an a ine ans o ma ion on S, i can be uniquely ex ended o a linea ans o ma ion Φ : C1(H)→C1(H). In ac , i Ais a nonze o posi i e ace-class ope a o , hen le Φ1(A) = (A)φ¡A (A)¢. We se Φ1(0) = 0. I B∈C1(H) is sel adjoin , hen B=B+−B−, whe e B+=B+|B| 2∈C1(H), B−=|B|−B 2∈C1(H) a e he posi i e and nega i e pa s o B, espec i ely. De ine Φ2(B) = Φ1(B+)−Φ1(B−). 4 M ´ ATY ´ AS BARCZY AND MARIANN T ´ OTH I Cis an a bi a y ace-class ope a o , hen C=C1+iC2, whe e C1, C2 a e he eal and he imagina y pa s o C espec i ely, bo h o which belong o C1(H). We inally se Φ(C) = Φ2(C1) + iΦ2(C2). I needs only elemen a y calcula ions o e i y ha Φ is a linea ex ension o φ. Since φis a local au omo phism o S, by (2) we ob ain ha Φ is a posi i e linea map on C1(H) sending ank-one p ojec ions o ank-one p ojec ions. The o m o such maps can be easily de i ed om [4, Theo em 3.1, p. 21]. Namely, i ollows om ha esul ha ei he he e is an isome y Uon H such ha Φ(A) = UAU∗(A∈C1(H)), o he e is an an iisome y ( ha is, a conjuga e-linea no m-p ese ing map) Von Hsuch ha Φ(A) = V A∗V∗(A∈C1(H)), o he e is a ank-one p ojec ion Pon Hsuch ha Φ(A) = ( A)P(A∈C1(H)). Using he local p ope y o φwe can easily in e ha his hi d possibili y is un enable. Wi hou se ious loss o gene ali y we can assume ha he e exis s an an iisome y Von Hsuch ha (6) φ(T) = V TV ∗(T∈S). Pick an elemen To Swi h dense ange. By he local o m o φwe see ha φ(T) mus ha e dense ange oo. This gi es us ha he ope a o Vin (6) is in ac an iuni a y. This comple es he p oo o ou asse ion. ¤ The nex s a emen asse s ha e e y 2-local au omo phism o he space o all s a es is an au omo phism. We no e ha in he p oo h., .ideno es he inne p oduc on H. Because o ou ma hema ical educa ion we suppose ha his o m is linea in he i s a iable and conjuga e-linea in he second a iable. Fo any x, y ∈H, he symbol |xihy|s ands o he ope a o de ined by |xihy|z=hz, yix(z∈H). Theo em 2. Le φ:S→Sbe any ans o ma ion wi h he p ope y ha o e e y T, S ∈S he e is an a ine au omo phism φT,S o Ssuch ha φ(T) = φT,S(T)and φ(S) = φT,S(S). Then φis an a ine au omo phism o S. P oo . Le φ:S→Sbe as abo e and le P, Q be ank-one p ojec ions on H. The local p ope y o φimplies ha he e exis s an ei he uni a y o an iuni a y ope a o Uon Hsuch ha φ(P) = UPU∗, φ(Q) = UQU∗. In bo h cases we ob ain ha (7) (φ(P)φ(Q)) = (UPU∗UQU∗) = (UPQU∗) = (PQ). LOCAL AUTOMORPHISMS OF THE SETS OF STATES AND EFFECTS 5 By he o m o he au omo phisms o Swe see ha he es ic ion o φ on o he se P1(H) o all ank-one p ojec ions on Hmaps P1(H) in o i sel and by (7) i p ese e s he so-called ansi ion p obabili ies (see [3, 2.1, p. 923]). We now apply a non-su jec i e a ian o Wigne ’s classical heo em on ans o ma ions o P1(H) p ese ing ansi ion p obabili ies. Acco ding o he main esul in [17] he e is ei he an isome y o an an iisome y V on Hsuch ha (8) φ(P) = V PV ∗(P∈P1(H)). We should ema k he e ha he esul in [17] applied abo e is abou ec o - o- ec o ans o ma ions p ese ing he modulus o he inne p oduc be- ween ec o s. Bu , jus as wi h he o iginal Wigne ’s heo em, i is easy o see ha his is jus an equi alen o mula ion o ou p oblem conce ning ans o ma ions on P1(H) which p ese e he ansi ion p obabili ies. We nex show ha Vis su jec i e. Le (λn) be a sequence o pai wise di e en posi i e numbe s wi h sum 1. Pick an o hono mal basis in Hand le Pndeno e he ank-one p ojec ion on o he subspace o Hgene a ed by he n h basis ec o . We ix he ope a o T=X n λnPn. By he local p ope y o φwe can clea ly assume ha φ(T) = T o his pa icula T. Now, le n0∈Nbe a bi a y. By he local p ope y o φonce again, we can choose a uni a y o an iuni a y ope a o Uon Hsuch ha X n λnPn=φ(X n λnPn) = U·X n λnPn·U∗=X n λnUPnU∗ and φ(Pn0) = UPn0U∗. I ollows om he i s equa ion, ha UPnU∗=Pn o e e y n∈N, so φ(Pn0) = Pn0.Since n0was a bi a y, we can in e ha φ(Pn) = Pn(n∈N). Bu his la e equali y implies ha he ange o Vin (8) con ains he ange o all Pn, ha is, i con ains an o hono mal basis and his gi es us ha V is su jec i e. I emains o p o e ha φ(S) = V SV ∗holds o e e y S∈S. Clea ly, wi hou se ious loss o gene ali y we can assume ha Vabo e is uni a y. Le S∈Sand pick an a bi a y uni ec o x∈H. Taking in o accoun he o m (8) o φon P1(H) and he local p ope y o φ, we ob ain on he one hand ha φ(|xihx|)φ(S)φ(|xihx|) = (V·|xihx|·V∗)φ(S)(V·|xihx|·V∗) = hφ(S)V x, V xi·|V xihV x| and, on he o he hand, ha he e is an ei he uni a y o an iuni a y ope - a o Won Hsuch ha φ(|xihx|)φ(S)φ(|xihx|) = (W·|xihx|·W∗)(WSW∗)(W·|xihx|·W∗) = 6 M ´ ATY ´ AS BARCZY AND MARIANN T ´ OTH hWSW∗W x, Wxi·|WxihWx|=hSx, xi·|WxihWx|. This gi es us ha hV∗φ(S)V x, xi=hSx, xi. Since his holds o e e y uni ec o x∈H, we conclude ha V∗φ(S)V=S. The e o e, we ha e φ(S) = V SV ∗ o e e y S∈Sand his comple es he p oo . ¤ Since on E he e a e se e al algeb aic s uc u es we decided o begin wi h he 2-local au omo phisms. Theo em 3. Le φ:E→Ebe a ans o ma ion wi h he p ope y ha o e e y E, F ∈E he e exis s an ei he uni a y o an iuni a y ope a o UE,F on Hsuch ha φ(E) = UE,F EU∗ E,F and φ(F) = UE,F FU∗ E,F . Then he e exis s an ei he uni a y o an iuni a y ope a o Uon Hsuch ha φ(E) = UEU∗(E∈E). P oo . I is clea ha he es ic ion o φon o he o homodula pose Po all p ojec ions is a 2-local au omo phism o P. Now, [13, P oposi ion] ells us ha e e y 2-local au omo phism o Pis an au omo phism. The e o e, he e exis s an ei he uni a y o an iuni a y ope a o Uon Hsuch ha φ(P) = UPU∗ o e e y p ojec ion P. Clea ly, we can suppose ha φ(P) = P o e e y P∈P. By he 2-local p ope y o φwe easily deduce φ(λP) = λP (λ∈[0,1]). This p ope y also gi es us ha E≤Fi and only i φ(E)≤φ(F). The e o e, o any E∈Ewe ha e λP ≤Ei and only i λP =φ(λP)≤φ(E) (P∈P,λ∈[0,1]). Now, le P1, . . . , Pnbe pai wise o hogonal spec al p ojec ions o Eand λ1, . . . , λn∈[0,1] such ha PiλiPi≤E. Then we ha e λiPi≤φ(E) o e e y i= 1, . . . , n. As Picommu es wi h E, by he 2-local p ope y o φ we ob ain ha Pi=φ(Pi) commu es wi h φ(E). Mul iplying he inequali y λiPi≤φ(E) by Pi om bo h sides, we ha e λiPi≤Piφ(E)Pi. By he commu a i i y o Piand φ(E) we ha e ha Piand pφ(E) also commu e. We can compu e X i λiPi≤X i Piφ(E)Pi=X i Piφ(E) = pφ(E)(X i Pi)pφ(E)≤pφ(E)Ipφ(E) = φ(E). App oxima ing Ewi h ope a o s o he o m PiλiPiwe ob ain ha E≤ φ(E). In e changing he ole o Eand φ(E) in he a gumen abo e, we also ge φ(E)≤E. The e o e, φ(E) = E o e e y E∈Eand his comple es he p oo . ¤ We now ea he a ious (1-)local au omo phisms o E. Theo em 4. Le φ:E→Ebe an a ine ans o ma ion wi h he p ope y ha o e e y E∈E he e exis s an a ine au omo phism φEo Esuch ha φ(E) = φE(E). Then φis an a ine au omo phism o E. LOCAL AUTOMORPHISMS OF THE SETS OF STATES AND EFFECTS 7 P oo . By he o m o he a ine au omo phisms o Ei ollows ha ei he φ(0) = 0 o φ(0) = I. Suppose i s ha φ(0) = 0. Then one can p o e jus as in he p oo o Theo em 1 ha φcan be ex ended o a linea ans- o ma ion Φ : B(H)→B(H). By he local o m o φwe deduce ha Φ sends p ojec ions o p ojec ions. Since Φ is a posi i e linea ans o ma- ion, i is well-known o be no m-con inuous. Now, i needs only elemen- a y compu a ion o e i y ha Φ is a Jo dan *-endomo phism o B(H) (see, o example, he p oo o [8, Theo em 2]) We show ha φp ese es he ank-one p ojec ions. Le Pbe a ank-one p ojec ion. Suppose ha φ(P) = U(I−P)U∗ o some uni a y o an iuni a y ope a o Uon H. As o φ((1/2)P) we ha e wo possibili ies. Fi s assume ha he e exis s an ei he uni a y o an iuni a y ope a o Wsuch ha φ((1/2)P) = W((1/2)P)W∗. Since φ((1/2)P) = (1/2)φ(P) ( his ollows om he ac ha φis a ine and φ(0) = 0), we ha e (1/2)U(I−P)U∗=W((1/2)P)W∗ which is an ob ious con adic ion i one conside s he anks o he op- e a o s appea ing in ha equali y. So, i emains ha φ((1/2)P) = W(I−(1/2)P)W∗ o some uni a y o an iuni a y ope a o W. This leads o he equali y (1/2)U(I−P)U∗=W(I−(1/2)P)W∗ which also leads o a con adic ion since he spec um o he ope a o on he le -hand side is {0,1/2}while he spec um o he ope a o on he igh -hand side is {1/2,1}. The e o e, φ(P) mus be o he o m UP U∗ o some uni a y o an iuni a y ope a o Uand his gi es us ha Φ p ese es he ank-one p ojec ion. Mo eo e , simila ly as abo e one can e i y ha φ(I) = Iand his yields ha Φ(I) = I. So Φ is a Jo dan *-endomo phism o B(H) which p ese es he ank-one p ojec ions and i sends I o I. Simila ly as in he p oo o [1, Theo em 3] we ob ain ha Φ is implemen ed by ei he a uni a y o an an iuni a y ope a o . This comple es he p oo in he case when φ(0) = 0. I φ(0) = I, hen one can conside he ans o ma ion E7→ I−φ(E) o educe he p oo o he p e ious case. ¤ Theo em 5. Le φ:E→Ebe an addi i e ans o ma ion in he sense ha o e e y E, F ∈Ewi h E+F∈Ewe ha e φ(E+F) = φ(E) + φ(F). I o e e y E∈E he e exis s an e ec -au omo phism φEo Esuch ha φ(E) = φE(E), hen φis an e ec -au omo phism. P oo . We a e going o p o e ha φis an a ine ans o ma ion and φ(0) = 0. I his is done, hen one can simply e e o ou p e ious heo em. The ac ha φ(0) = 0 is i ial by he local p ope y o φ. ¿F om he addi i i y o φi ollows ha φis mono one inc easing. Once again, he addi i i y o φgi es us ha nφ(E) = φ(nE) o e e y n∈Nand E∈E wi h nE ∈E. Now simple calcula ion shows ha φ(E) = φ( E) o e e y posi i e a ional numbe and E∈Ewi h E ∈E. Finally, o any eal 8 M ´ ATY ´ AS BARCZY AND MARIANN T ´ OTH numbe 0 < α < 1 and E∈Ewe ha e φ(αE) = αφ(E). Indeed, we can cons uc wo sequences (βn), (γn) o a ional numbe s in [0,1] such ha , βn< α < γn o e e y nand bo h (βn) and (γn) con e ge o α. Using he mono oni y o φ, we ge βnφ(E) = φ(βnE)≤φ(αE)≤φ(γnE) = γnϕ(E), and aking limi s we ha e he desi ed equali y αφ(E) = φ(αE). I E, F ∈E a e a bi a y and λ∈[0,1], hen we can compu e φ(λE + (1 −λ)F) = φ(λE) + φ((1 −λ)F) = λφ(E) + (1 −λ)φ(F). This shows ha φis a ine and we a e done. ¤ Theo em 6. Le φ:E→Ebe a ans o ma ion sa is ying φ(EFE) = φ(E)φ(F)φ(E) (E, F ∈E). I o e e y E∈E he e exis s an au omo phism φEo Eo he o m (5) such ha φ(E) = φE(E), hen φis an au omo phism o Eo he o m (5). P oo . Fi s we show ha φp ese es he p ojec ions and he o hogonali y be ween hem. Indeed, by he local p ope y o φwe deduce ha φsends p ojec ions o p ojec ions. Le P, Q ∈B(H) be p ojec ions such ha PQ = 0. Then we ha e 0 = φ(PQP) = φ(P)φ(Q)φ(P) which implies ha 0 = φ(P)φ(Q)φ(P) = (φ(Q)φ(P))∗(φ(Q)φ(P)). This gi es us ha φ(Q)φ(P) = 0. Nex obse e ha φalso p ese es he pa ial o de ing ≤be ween p o- jec ions. To see his, le P, Q ∈B(H) be p ojec ions and suppose ha P≤Q. Then we ha e P=QPQ and φ(P) = φ(QPQ) = φ(Q)φ(P)φ(Q) which yields ha he ange o φ(P) is included in he ange o φ(Q), ha is, φ(P)≤φ(Q). We p o e ha φis ini ely o hoaddi i e on he se o all ini e ank p ojec ions in B(H). To see his, le P, Q ∈P(H) be mu ually o hogonal ini e ank p ojec ions. By he o de p ese ing p ope y o φwe ha e φ(P), φ(Q)≤φ(P+Q). Since φp ese es o hogonali y, we in e ha φ(P) + φ(Q) is a p ojec ion and φ(P) + φ(Q)≤φ(P+Q). By he local o m o φwe ind ha φalso p ese es he ank o p ojec ions. Hence we compu e ank(φ(P) + φ(Q)) = ank(φ(P)) + ank(φ(Q)) = ank(P) + ank(Q) = ank(P+Q) = ank(φ(P+Q)) which yields ha φ(P) + φ(Q) = φ(P+Q). We now ex end φ om he se o all ini e ank p ojec ions o a Jo dan *-homomo phism o he algeb a F(H) o all ini e ank ope a o s in B(H). This can be done in a way e y simila o wha was ollowed in he p oo o [9, Theo em 1]. By he p ope ies o φ, his ex ension p ese es he ank-one LOCAL AUTOMORPHISMS OF THE SETS OF STATES AND EFFECTS 9 p ojec ions and he o hogonali y be ween hem. Now, [12, Lemma 2] ells us ha he e exis s an isome y o an iisome y U:H→Hsuch ha (9) φ(P) = UPU∗ holds o e e y ini e ank p ojec ion Pon H. I B∈[0, I] is an a bi a y ope a o , due o φ(I) = Iwe ha e (φ(B))2= φ(B)φ(I)φ(B) = φ(B2), which implies ha pφ(A) = φ(√A) holds o e e y A∈[0, I]. We p o e ha φis homogeneous in some sense. Le P∈B(H) be a ini e ank p ojec ion and λ∈[0,1]. By he local p ope y o φwe ge φ(P) = V PV ∗and φ(λP) = WλPW∗=λWPW∗, whe e Vand Wa e uni a y o an iuni a y ope a o s. Since λP ≤P, i ollows ha φ(λP)≤φ(P). The e o e, λWPW∗≤V PV ∗. As WPW∗and V PV ∗a e p ojec ions o he same ank, i ollows ha WPW∗=V PV ∗which implies ha φ(λP ) = λφ(P). We claim ha he ope a o Uappea ing in (9) is ei he uni a y o an- iuni a y. To e i y his, le Pi(i∈N) be pai wise o hogonal ank-one p ojec ions and λi∈[0,1] (i∈N) be scala s, such ha P∞ i=1 Pi=Iand P∞ i=1 λi<∞. Se A=P∞ i=1 λiPi. By he p ope ies o φwha we al eady know, we in e n X i=1 λiφ(Pi) = n X i=1 φ(λiPi) = n X i=1 φ(√APi√A) = n X i=1 φ(√A)φ(Pi)φ(√A) =pφ(A) n X i=1 φ(Pi)pφ(A)≤pφ(A)Ipφ(A) = φ(A). So we ha e P∞ i=1 λiφ(Pi)≤φ(A). By he local p ope y o φwe ge ha (φ(A)) = (A) = ( ∞ X i=1 λiPi) = ∞ X i=1 λi= ( ∞ X i=1 λiφ(Pi)), which implies ha (φ(A)−P∞ i=1 λiφ(Pi)) = 0. Since i he ace o a posi i e ope a o is 0 hen he ope a o in ques ion is necessa ily 0, i ollows ha ∞ X i=1 λiφ(Pi) = φ(A). The e o e, we ha e φ(A) = ∞ X i=1 λiUPiU∗=U( ∞ X i=1 λiPi)U∗=UAU∗. Since, by he local p ope y o φ, he ange o φ(A) mus be dense, we deduce ha Uis a uni a y o an iuni a y ope a o . Clea ly, wi hou any loss o gene ali y we now can assume ha φ(P) = P holds o e e y ini e ank p ojec ion P. We claim ha φis he iden i y on