Jordan triple endomorphisms and isometries of unitary groups

JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS LAJOS MOLN´ AR Abs ac . In his pape we p esen he gene al o m o all con inuous endomo phisms o he g oup Uno n×ncomplex uni a y ma ices wi h espec o he Jo dan iple p oduc . These a e he con inuous maps φ:Un→Unwhich sa is y φ(V W V ) = φ(V)φ(W)φ(V), V, W ∈Un. The esul is applied o de e mine he s uc u e o ce ain isome ies o Un. These include he isome ies ela i e o any me ic gi en by a uni a ily in a ian no m on he space Mno all n×ncomplex ma ices and also he isome ies ela i e o any membe o a new class o me ics on Un ecen ly in oduced by Chau, Li, Poon and Sze [6]. 1. In oduc ion and s a emen o he main esul s The amous Mazu -Ulam heo em s a es ha e e y su jec i e isome y (i.e., su jec i e dis ance p ese ing mapping) be ween eal no med spaces is au oma ically a ine, i p ese es he ope a ion o con ex combina ion. Mo i a ed by his impo an esul , in he pape [9] Ha o i, Hi asawa, Miu a and Moln´a made a emp s o gene alize i o he noncommu a i e se ing, especially o me ic g oups and o ce ain subs uc u es o hem. The au ho s managed o ob ain esul s saying ha unde ce ain condi ions, he su jec i e isome ies o g oups equipped wi h ansla ion and in e se in a ian me ics locally p ese e an ope a ion called in e ed Jo dan iple p oduc . In some cases he local p ese a ion o ha ope a ion can be shown o ex end globally. These esul s demons a e ha in he conside ed cases he su jec i e isome ies ha e a ce ain ema kable algeb aic p ope y, hey a e some so o isomo phisms be ween he unde lying g oups. In [10] he esul s gi en in [9] we e u ilized o desc ibe he s uc u e o su jec i e isome ies o he uni a y g oup o an a bi a y complex Hilbe space ela i e o he me ic induced by he usual ope a o no m. In [15] Moln´a and ˇ Sem l de e mined he s uc u e o su jec i e isome ies o he uni a y g oup o a complex in ini e dimensional sepa able Hilbe space wi h espec o 2010 Ma hema ics Subjec Classi ica ion. P ima y: 15A60, 15A86. Seconda y: 47B49. Key wo ds and ph ases. Uni a y g oup, isome ies, uni a ily in a ian no m, Jo dan iple p oduc . The au ho was suppo ed by he ”Lend¨ule ” P og am (LP2012-46/2012) o he Hun- ga ian Academy o Sciences and by he Hunga ian Scien i ic Resea ch Fund (OTKA) Reg.No. K81166 NK81402. 1 2 LAJOS MOLN´ AR any uni a ly in a ian uni o m no m on he ull ope a o algeb a o e he unde lying Hilbe space. By employing di e en analy ical ools bu using he same algeb aic p ope ies o su jec i e isome ies be ween g oups ha we e ob ained in [9], Ha o i and Moln´a p esen ed esul s in [11] on he s uc u e o su jec i e isome ies ( ela i e o he usual no m) o uni a y g oups in C∗-algeb as and in on Neumann algeb as. An in e es ing ecen esul on he o m o ce ain isome ies o he special o hogonal g oup is o appea in [1]. In [15] he p oblem o desc ibing he su jec i e isome ies o he uni a y g oup unde uni a ily in a ian no ms in he ini e dimensional case was le as an open p oblem, see [15, 4. Rema ks, examples, open p oblems]. One o he aims o his pape is o gi e a solu ion o ha p oblem. On he o he hand, below we de e mine he s uc u e o all isome ies o he uni a y g oup Un ela i e o a new class o me ics on Un ha has been in oduced by Chau, Li, Poon and Sze e y ecen ly [6]. As can be suspec ed om he discussion abo e he isome ies we a e go- ing o conside u n o be isomo phisms unde a ce ain algeb aic ope a ion on Un. Indeed, his ope a ion is he in e ed Jo dan iple p oduc ha we de ine by he o mula V W −1V. Mo phisms wi h espec o his p oduc a e e y closely ela ed o mo phisms wi h espec o a much mo e common and impo an ope a ion which is called he Jo dan iple p oduc . This is de ined by he o mula V WV . T ans o ma ions p ese ing his ope a ion o alike ope a ions a e ex ensi ely in es iga ed in ing heo y and i s applica- ions. The second main aim o his pape is o ob ain he ull desc ip ion o all con inuous Jo dan iple endomo phisms o he g oup Un. We begin wi h p esen ing he no a ion and de ini ions ha we shall use h oughou he pape . We deno e by Mn he space o all n×ncomplex ma ices, by Hn he space o all sel -adjoin elemen s o Mnand by Un he g oup o all uni a y elemen s o Mn. A uni a y ma ix is called a symme y i i is sel -adjoin (i has eigen alues ±1). I is well-known ha e e y uni a y ma ix is he exponen o a skew-symme ic ma ix, i.e., e e y U∈Uncan be w i en o he o m U=eiH wi h some H∈Hn. In wha ollows k.k deno es he usual ope a o no m (o , in ano he wo ds, spec al no m) on Mn(i.e., kAkis he squa e- oo o he la ges eigen alue o he posi i e semi-de ini e ma ix A∗A). I no speci ied o he wise, when we speak o me ical o opological p ope ies ela ed o Unwe always mean he me ic induced by he no m k.k. In wha ollows Is ands o he iden i y ma ix, deno es he anspose o ma ices, T is he usual ace unc ional, and e e s o complex conjuga e. Recall ha a no m N(.) on Mnis called uni a ily in a ian i N(UAV ) = N(A) holds o all A∈Mn,U, V ∈Un. In wha ollows we assume ha n≥2 (in he case n= 1 he esul s below ollow om classical ma hema ical analysis). Ou i s main esul which gi es he comple e desc ip ion o Jo dan iple endomo phisms o Un eads as ollows. JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 3 Theo em 1. Le φ:Un→Unbe a con inuous map which is a Jo dan iple endomo phism, i.e., assume ha φsa is ies φ(V WV ) = φ(V)φ(W)φ(V), V, W ∈Un. Then he e exis a uni a y ma ix U∈Un, an in ege k, a numbe c∈ {−1,1}, and a se {P1, . . . , Pn}o mu ually o hogonal ank-one p ojec ions in Mn, a se {k1, . . . , kn}o in ege s and a se {c1, . . . , cn} ⊂ {−1,1}such ha φis o one o he ollowing o ms: (j1) φ(V) = c(de V)kUV U−1,V∈Un; (j2) φ(V) = c(de V)kUV −1U−1,V∈Un; (j3) φ(V) = c(de V)kUV U−1,V∈Un; (j4) φ(V) = c(de V)kUV U−1,V∈Un; (j5) φ(V) = Pn j=1 cj(de V)kjPj,V∈Un. F om he abo e esul one can immedia ely deduce he s uc u e o all con inuous Jo dan iple au omo phisms o Un. Co olla y 2. Le φ:Un→Unbe a con inuous Jo dan iple au omo phism, i.e., a con inuous bijec i e map which sa is ies φ(V WV ) = φ(V)φ(W)φ(V), V, W ∈Un. Then he e exis a uni a y ma ix U∈Unand a numbe c∈ {−1,1}such ha φis o one o he ollowing o ms: (a1) φ(V) = cUV U−1, V ∈Un; (a2) φ(V) = cUV −1U−1, V ∈Un; (a3) φ(V) = cUV U−1, V ∈Un; (a4) φ(V) = cUV U−1, V ∈Un. As ou second main aim in his pape , in he nex heo em we de e mine he s uc u e o all isome ies o he uni a y g oup Unwi h espec o any uni a ily in a ian no m gi en on Mn. Theo em 3. Le N(.)be a uni a ily in a ian no m on Mn. I φ:Un→Un is an isome y, i.e., φis a map which sa is ies N(φ(V)−φ(W)) = N(V−W), V, W ∈Un, hen he e exis s a pai U, U0∈Uno uni a y ma ices such ha φis o one o he ollowing o ms: (i1) φ(V) = UV U0, V ∈Un; (i2) φ(V) = UV −1U0, V ∈Un; (i3) φ(V) = UV U0, V ∈Un; (i4) φ(V) = UV U0, V ∈Un. In ou ou h heo em we de e mine he isome ies o Unwi h espec o a ecen ly de ined class o in e es ing me ics on Un. Mo i a ed by conside - a ions in quan um in o ma ion p ocessing, in [5] Chau in oduced a ce ain amily o me ics on Un. In he pape [6] Chau, Li, Poon and Sze ha e ex ended his class signi ican ly and p esen ed a numbe o i s in e es ing 4 LAJOS MOLN´ AR p ope ies. I is a ema kable ac ha s a ing om a e y much di e en o igin in [2], An ezana, La o onda and Va ela ha e been led p ac ically o he same class o dis ances on Un. As o he de ini ion o he me ics in ques ion, we i s ema k he ollow- ing. To any V∈Un he e co esponds a unique sel -adjoin ma ix H∈Hn wi h spec um in ]−π, π] such ha V= exp(iH). Indeed, Hcan be ob ained in he ollowing way. Applying an app op ia e uni a y simila i y ans o - ma ion, Vis ans o med in o a diagonal ma ix. The diagonal elemen s o his ma ix a e complex numbe s o modulus 1. Fo each such diagonal elemen ake he co esponding unique angle ha belongs o ] −π, π]. F om he so ob ained angles o m he co esponding diagonal ma ix and inally ans o m i wi h he in e se o he p e iously men ioned uni a y simila i y ans o ma ion. Wha we ge is jus he sel -adjoin ma ix H ha we ha e been looking o . Fo empo a y use, in his pape we call his sel -adjoin ma ix H he angula ma ix o V. Now, gi en a uni a ily in a ian no m N(.) on Mn, o any pai V, W ∈Uno uni a y ma ices pick he angula ma ix Ho V W−1and de ine dN(V, W) = N(H). I has been p o en in [6] ha dNis a me ic on Unand se e al in e es ing p ope ies o dNha e been de i ed. In ou las esul we de e mine he s uc u e o he co esponding isome- ies o Un. Theo em 4. Le N(.)be a uni a ily in a ian no m on Mn. The s uc u e o he isome ies o Unwi h espec o he me ic dNde ined abo e is exac ly he same as in Theo em 3. Rema k 5.A his poin le us ema k he ollowing. The esul s in he p e ious s a emen s can all be e e sed, meaning ha all ans o ma ions o any o he o ms which appea in he conclusions a e in ac isome ies, con inuous Jo dan iple au omo phisms, and con inuous Jo dan iple en- domo phisms, espec i ely. To e i y hese one needs o apply only simple obse a ions. 2. P oo s In his sec ion, a e e i ying some auxilia y esul s, we p esen he p oo s o ou main heo ems. Ou i s lemma ha ollows s a es ha he con inuous Jo dan iple endomo phisms o Una e all Lipschi z unc ions. The esul could also be de i ed ollowing he a gumen gi en in [13] (p. 177, Sa z 1) ela ing o g oup endomo phisms o linea g oups. Fo he sake o comple eness below we p esen a mo e di ec and simple p oo in he case o Jo dan iple endomo phisms o Un. Lemma 6. Le φ:Un→Unbe a con inuous Jo dan iple endomo phism. Assume φ(I) = I. Then φis a Lipschi z unc ion. JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 5 P oo . We begin wi h he ollowing obse a ion. Fo an a bi a y V∈Un le Hbe he angula ma ix o Vand deno e he ope a o no m kHko H by m(V). Appa en ly, we ha e he inequali ies kV−Ik ≤ m(V)≤2kV−Ik. Mo eo e , i m(V)< π and kis a posi i e in ege such ha k m(V)< π, hen we ha e m(Vk) = k m(V). Tu ning o he p oo o he lemma, we i s asse ha he e exis s a posi i e eal numbe Lsuch ha kφ(U)−Ik ≤ LkU−Ikholds o all U∈Un. Assume on he con a y ha we ha e a sequence (Uk) in Unand a sequence (ck) o posi i e in ege s such ha ck→ ∞ and (1) kφ(Uk)−Ik> ckkUk−Ik holds o e e y k∈N. Since Unis a compac me ic space, (Uk) has a con e gen subsequence. Wi hou se ious loss o gene ali y we may and do assume ha al eady he o iginal sequence (Uk) is con e gen . I i s limi we e di e en om I, by (1) we would ha e kφ(Uk)−Ik→∞which con adic s kφ(Uk)−Ik ≤ 2. The e o e, Uk→Iand φ(Uk)→Ias k→ ∞. We can also assume ha kφ(Uk)−Ik=k, k<1/2 holds o all k∈N. Choose posi i e in ege s lksuch ha 1/(lk+ 1) ≤k<1/lk. Clea ly, lk≥2. Since k> ckkUk−Ik, we ha e kUk−Ik< k/ckand his implies ha m(Uk)≤2kUk−Ik<(2k)/ck. On he o he hand, we ha e 2klk ck <2 ck < π. The e o e, we in e m(Ulk k) = lkm(Uk)<2/ckwhich implies kUlk k−Ik ≤ m(Ulk k)<2/ck. Consequen ly, Ulk k→Iand since Uk→Ialso holds, we ha e Ulk+1 k→Ias k→ ∞. We con inue wi h he inequali ies m(φ(Uk)) ≤2kφ(Uk)−Ik= 2k and 2k(lk+ 1) <2(lk+ 1)/lk< π, whe e in he las inequali y we ha e used lk≥2. These imply ha m(φ(Uk)lk+1) = (lk+ 1)m(φ(Uk)). 6 LAJOS MOLN´ AR Hence we compu e 1 = (1/k)kφ(Uk)−Ik ≤ (lk+ 1)kφ(Uk)−Ik ≤(lk+ 1)m(φ(Uk)) = m(φ(Uk)lk+1)≤2kφ(Ulk+1 k)−Ik. Consequen ly, φ(Ulk+1 k)6→ Iand his con adic s Ulk+1 k→I. The e o e, we ha e a posi i e eal numbe Lsuch ha kφ(U)−Ik ≤ LkU−Ikholds o e e y U∈Un. To comple e he p oo pick a bi a y uni a ies W, W 0∈Un. We can choose V∈Unsuch ha V2=W0and hen ind U∈Unsuch ha V UV =W. We in e kφ(W)−φ(W0)k=kφ(V UV )−φ(V2)k=kφ(V)φ(U)φ(V)−φ(V)Iφ(V)k =kφ(U)−Ik ≤ LkU−Ik=LkV UV −V2k=LkW−W0k. This p o es ha φis a Lipschi z unc ion.  In he nex auxilia y esul we show ha e e y con inuous Jo dan iple endomo phism o Unwhich is uni al (i.e., maps I o I) gi es ise o a linea ans o ma ion on Hn. The use o one-pa ame e g oups in he p oo ha o igina es om [12] has al eady been exploi ed in he pape s [11] and [1]. Lemma 7. Le φ:Un→Unbe a con inuous Jo dan iple endomo phism wi h φ(I) = I. Then he e exis s a linea ans o ma ion :Hn→Hnsuch ha φ(ei A) = ei (A), ∈R, A ∈Hn. Mo eo e , sa is ies (V AV ) = φ(V) (A)φ(V) o e e y A∈Hnand symme y V∈Un. P oo . Since φis a uni al Jo dan iple endomo phism, i is easy o check ha φ(Vk) = φ(V)kholds o e e y V∈Unand posi i e in ege k. We show ha φp ese es he in e se ope a ion. To p o e his, le W∈Unbe such ha W2=V. We compu e φ(W)φ(V−1)φ(W) = φ(WV −1W) = φ(I) = I which implies ha φ(V−1) = φ(W)−2=φ(W2)−1=φ(V)−1. The e o e, we ob ain ha φ(Vk) = φ(V)kholds o e e y in ege kand V∈Un. In he es o he pape we shall use se e al imes ha , in pa icula , φmaps symme ies o symme ies. In he nex s ep, ollowing an a gumen simila o he p oo o Theo em 7 in [11] we show ha φmaps one-pa ame e uni a y g oups o one-pa ame e uni a y g oups. Pick an a bi a y sel -adjoin ma ix T∈Hnand de ine ST:R→Unby ST( ) = φ(ei T ), ∈R. JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 7 We asse ha STis a con inuous one-pa ame e uni a y g oup in Mn. Since φis con inuous, we only need o p o e ha ST( + 0) = ST( )ST( 0) holds o e e y pai , 0o eal numbe s. Fi s selec a ional numbe s and 0 such ha =k mand 0=k0 m0wi h in ege s k, k0, m, m0. We compu e ST( + 0) = φ(eikm0+k0m mm0T) = φ(ei1 mm0T)km0+k0m =φ(ei1 mm0T)km0φ(ei1 mm0T)k0m=ST( )ST( 0). Since φis con inuous, we deduce ha ST( + 0) = ST( )ST( 0) holds o e e y pai , 0o eal numbe s. By S one’s heo em we ob ain ha he e exis s a unique sel -adjoin ma ix (T)∈Hn( he gene a o o he one-pa ame e uni a y g oup ST) such ha φ(ei T ) = ST( ) = ei (T), ∈R. We nex p o e ha :Hn→Hnis in ac a linea ans o ma ion. Pick A, B, C ∈Hn. We compu e ei( /2)Aei Bei( /2)A−ei C i =(ei( /2)A−I)ei Bei( /2)A+ (ei B −I)ei( /2)A+ (ei( /2)A−I)−(ei C −I) i →A/2 + B+A/2−C=A+B−C as →0. I ollows ha lim →0 ei( /2)Aei Bei( /2)A−ei C i = 0 ⇐⇒ C=A+B. I C=A+B, hen using he Lipschi z p ope y o φp o en in Lemma 6 we ha e ei( /2) (A)ei (B)ei( /2) (A)−ei (C) i =φ(ei( /2)A)φ(ei B)φ(ei( /2)A))−φ(ei C) i =φ(ei( /2)Aei Bei( /2)A))−φ(ei C) i →0 as →0. On he o he hand, jus as abo e we deduce ei( /2) (A)ei (B)ei( /2) (A)−ei (C) i → (A) + (B)− (C). This gi es us ha (A) + (B)− (A+B) = 0, i.e., is addi i e. The homogenei y o is i ial o see. Indeed, we ha e ei λ (A)=φ(ei λA) = ei (λA) o e e y , λ ∈Rwhich implies λ (A) = (λA). Consequen ly, is a linea ans o ma ion on Hn. 8 LAJOS MOLN´ AR To ob ain he las s a emen o he esul we compu e ei φ(V) (A)φ(V)=φ(V)ei (A)φ(V) =φ(V)φ(ei A)φ(V) = φ(V ei AV) = ei (V AV ). Since his holds o e e y ∈Rwe easily ge he desi ed equali y (V AV ) = φ(V) (A)φ(V) o e e y A∈Hnand symme y V∈Un. In wha ollows Tdeno es he ci cle g oup which is jus he uni a y g oup in he one-dimensional case. The nex auxilia y esul desc ibes he s uc- u e o con inuous Jo dan iple unc ionals on Un. I can be iewed also as a cha ac e iza ion o he de e minan unc ion on he uni a y g oup. Lemma 8. Le ϕ:Un→Tbe a con inuous Jo dan iple unc ional, i.e., assume ha ϕis con inuous and sa is ies ϕ(V WV ) = ϕ(V)ϕ(W)ϕ(V), V, W ∈Un. Then he e is an in ege kand a numbe c∈ {−1,1}such ha ϕ(V) = c(de V)k, V ∈Un. P oo . Clea ly, ϕ(I)3=ϕ(I) implying ha ϕ(I) = ±1. The e is no loss o gene ali y in assuming ha ϕ(I) = 1. Since, by he ans o ma ion λ7→ diag(λ, 1,...,1), he g oup Tembeds i ially in o Un, he unc ional ϕ: Un→Tgi es ise o a con inuous uni al Jo dan iple endomo phism o Un. Applying Lemma 7 o his ans o ma ion one can easily check ha he e is a linea unc ional l:Hn→Rsuch ha ϕ(ei A) = ei l(A), ∈R, A ∈Hn. By he second s a emen in Lemma 7 we u he ha e ha l(V AV ) = l(A) holds o all A∈Hnand symme y V∈Un. P oceeding u he , since lis a linea unc ional on he eal Hilbe space Hn, by Riesz ep esen a ion heo em he e is an elemen H∈Hnsuch ha l(A) = T (AH), A∈Hn. Then T (AH) = l(A) = l(V AV ) = T (V AV H) = T (AV HV ) holds o e e y A∈Hnwhich implies ha H=V HV o e e y symme y V∈Hn. Mul iplying by V, his gi es us ha Hcommu es wi h all symme- ies in Un. Since any symme y Vis o he o m V= 2P−Iwi h some p ojec ion, i ollows ha Hcommu es wi h e e y p ojec ion and we ob ain ha His necessa ily a scala mul iple o he iden i y. Le h∈Rbe such ha H=hI. We ha e ϕ(ei A) = ei h T (A), ∈R, A ∈Hn. Pick a ank-one p ojec ion P∈Mn. Since exp iπP is a symme y, i ollows ha i s image unde ϕis a numbe ha has squa e equal o 1. The e o e, exp(iπh T (P)) = exp(iπh) equals ±1 which yields ha his an in ege . Deno e i by k. We compu e ϕ(ei A) = ei k T (A)= (eT (i A))k= (de (ei A))k which shows ha ϕ(V) = (de V)kholds o e e y V∈Un. JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 9 In he case whe e ϕ(I) = −1 we apply he abo e a gumen o he Jo dan iple unc ional −ϕ. A e hese p elimina ies we a e now in a posi ion o p o e ou i s main heo em. The basic idea o he p oo is he use o a s uc u al esul con- ce ning commu a i i y p ese ing linea ans o ma ions o Hn. Tha esul holds only when n≥3. The wo-dimensional case equi es some special conside a ions. P oo o Theo em 1. Le φ:Un→Unbe a con inuous Jo dan iple endo- mo phism. Clea ly, we ha e φ(I) = φ(I)3which implies ha I=φ(I)2, i.e., φ(I) is a symme y. We ha e φ(V) = φ(IV I) = φ(I)φ(V)φ(I) o e e y V∈Un. Mul iplying by φ(I) om ei he side we ob ain ha φ(I) commu es wi h e e y elemen φ(V) o he ange o φ. De ining ψ(V) = φ(I)φ(V), he ans o ma ion ψ:Un→Unis a con inuous map which is easily seen o be a Jo dan iple endomo phism o Unwhich sends I o I. In wha ollows we assume ha al eady ou o iginal map φ ixes he iden i y I. By Lemma 7 we ha e a linea ans o ma ion :Hn→Hnsuch ha (2) φ(ei A) = ei (A), ∈R, A ∈Hn. We p o e ha p ese es commu a i i y meaning ha i A, B ∈Hna e such ha AB =BA, hen (A) (B) = (B) (A) holds, oo. Pick commu ing ma ices A, B ∈Hn. Then o e e y , s ∈Rwe ha e ei Aei2sBei A =eisBei2 AeisB implying φ(ei A)φ(ei2sB)φ(ei A) = φ(eisB)φ(ei2 A)φ(eisB) and hence ei (A)ei2s (B)ei (A)=eis (B)ei2 (A)eis (B). Fixing he eal a iable sand pu ing he complex a iable zin he place o i we ha e ha he equali y ez (A)ei2s (B)ez (A)=eis (B)e2z (A)eis (B) be ween ma ix alued holomo phic (en i e) unc ions o he a iable zholds along he y-axis. By he uniqueness heo em o holomo phic unc ions we in e ha he abo e equali y holds necessa ily on he whole complex plane. Nex , ixing zand inse ing he complex a iable win he place o is, he same easoning leads o ha he equali y ez (A)e2w (B)ez (A)=ew (B)e2z (A)ew (B) holds o all alues o he a iables z, w ∈C. In pa icula , o a bi a y eal numbe s , s se ing z= /2, w =s/2 we ha e (3) pe (A)es (B)pe (A)=pes (B)e (A)pes (B). 16 LAJOS MOLN´ AR small. The e o e, Unis compac also ela i e o he me ic dNand hence any isome y φ:Un→Unwi h espec o he me ic dNis necessa ily su jec i e. Le us check ha he condi ions in P oposi ion 9 a e sa is ied. We i s show ha dNis ansla ion and in e se in a ian . Indeed, dN(UW, V W) = dN(U, V ), U, V, W ∈Un holds i ially and om he equali y V−1U=V−1(UV −1)V we deduce ha dN(V−1, U−1) = dN(U, V ). The e o e, dNis in e se in- a ian and, since i is igh ansla ion in a ian , we ob ain ha i is le ansla ion in a ian , oo. We ha e no ed in he p oo o he p e ious heo em ha N(.) is a sym- me ic no m equi alen o k.k. Hence we ha e a posi i e scala csuch ha ck.k ≤ N(.)≤N(I)k.k. Se α=πc/4. Pick V, W ∈Unwi h dN(V, W)< α. Le X∈Unbe such ha dN(X, V ) = dN(X, WV −1W) = dN(V, W). Then we ha e dN(X, W)≤dN(X, V ) + dN(V, W)=2dN(V, W)<2α=πc/2. I ollows ha he angula ma ix Ho WX−1=eiH sa is ies N(H)< πc/2, which implies ha kHk< π/2. F om his we in e ha he angula ma ix o (WX−1W)X−1= (WX−1)2is jus 2H. This yields dN(WX−1W, X)=2dN(W, X). These show ha he condi ions in P oposi ion 9 a e ul illed (wi h cons an K= 2) and we conclude ha φ(V W−1V) = φ(V)φ(W)−1φ(V) holds o any V, W ∈Unwi h dN(V, W)< α. Nex , choose a posi i e numbe βsuch ha β < α/(2N(I)). Assume kV−Wk< β. Then kV W−1− Ik< β and i easily ollows ha he angula ma ix Ho V W−1sa is ies kHk<2β. Hence dN(V, W) = N(H)≤N(I)kHk< N(I)2β < α holds which u he implies he equali y (7) φ(V W−1V) = φ(V)φ(W)−1φ(V). Consequen ly, o any pai V, W ∈Unwi h kV−Wk< β (i.e., o elemen s close enough ela i e o he usual me ic) we ha e he abo e equali y. The a gumen gi en in he i s pa o he p oo o Theo em 8 in [10] is abou showing ha he abo e p ope y which ells us ha (7) holds lo- cally in Unin ac implies ha i holds also globally. We can employ ha a gumen he e oo and ob ain ha φsa is ies (7) o all pai s V, W ∈Un. Since φis an isome y wi h espec o he me ic dNwhich induces he same opology as k.k, i ollows ha φis con inuous in he ope a o no m. Applying P oposi ion 10 we ob ain he desi ed conclusion.  JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 17 Rema k 11.We inish wi h he ollowing open p oblem and ema ks. In Theo em 1 we ha e desc ibed he Jo dan iple endomo phisms o Un. A na u al p oblem a ises which asks o he s uc u e o all con inuous Jo dan iple endomo phisms om Unin o ano he uni a y g oup Um. O cou se, i m≤n, ou esul can be applied (Umembeds in o Un), bu wha happens i m>n? Recall ha in ou p oo abo e we ha e hea ily used he s uc u e o commu a i i y p ese ing linea maps o Hnwhich s a emen is no longe alid be ween di e en spaces. We ema k ha he me hods wha we applied in [15] o de e mine he Jo dan iple au omo phisms o he uni a y g oup o a complex sepa able in ini e dimensional Hilbe space can mos p obably be modi ied o he ini e dimensional se ing whe e n≥3. Howe e , because o he essen ial use o he s uc u e o commu a i i y p ese ing non-linea maps in [15], he wo-dimensional case would ce ainly emain unco e ed. Obse e ha he app oach we ha e ollowed in he p esen pape has p o ided esul also in ha low-dimensional case. Re e ences [1] T. Abe, S. Akiyama, and O. Ha o i, Isome ies o he special o hogonal g oup, p ep in . [2] J. An ezana, G. La o onda and A. Va ela, Op imal pa hs o symme ic ac ions in he uni a y g oup, p ep in , a Xi :1107.2439. [3] R. Bha ia, Ma ix Analysis, Sp inge -Ve lag, New Yo k Be lin Heidelbe g, 1997. [4] M. B in and G. S uck, In oduc ion o Dynamical Sys ems, Camb idge Uni . P ess, 2002. [5] H.F. Chau, Me ics on uni a y ma ices and hei applica ion o quan i ying he deg ee o non-commu a i i y be ween uni a y ma ices, Quan . In o m. Comp. 11 (2011), 721-740. [6] H.F. Chau, C.K. Li, Y.T Poon and N.S. Sze, Induced me ic and ma ix inequali ies on uni a y ma ices, J. Phys. A: Ma h. Theo . 45 (2012), 095201, 8 pp. [7] M.D. Choi, A.A. Ja a ian and H. Radja i, Linea maps p ese ing commu a i i y, Linea Algeb a Appl. 87 (1987), 227–241. [8] S. Gudde and G. Nagy, Sequen ially independen e ec s, P oc. Ame . Ma h. Soc. 130 (2002), 1125–1130. [9] O. Ha o i, G. Hi asawa, T. Miu a and L. Moln´a , Isome ies and maps compa ible wi h in e ed Jo dan iple p oduc s on g oups, Tokyo J. Ma h. 35 (2012), 385–410. [10] O. Ha o i and L. Moln´a , Isome ies o he uni a y g oup, P oc. Ame . Ma h. Soc. 140 (2012), 2141–2154. [11] O. Ha o i and L. Moln´a , Isome ies o he uni a y g oups and Thompson isome ies o he spaces o in e ible posi i e elemen s in C∗-algeb as, p ep in [12] S. Sakai, A cha ac e iza ion o W∗-algeb as, Paci ic J. Ma h. 6(1956), 763–773. [13] W. Maak, Fas pe iodische Funk ionen, Die G undleh en de Ma hema ischen Wis- senscha en in Einzelda s ellungen, Be lin, 1950. [14] L. Moln´a , Selec ed P ese e P oblems on Algeb aic S uc u es o Linea Ope a o s and on Func ion Spaces, Lec u e No es in Ma hema ics, Vol. 1895, Sp inge , 2007. [15] L. Moln´a and P. ˇ Sem l, T ans o ma ions o he uni a y g oup on a Hilbe space, J. Ma h. Anal. Appl. 388 (2012), 1205–1217. 18 LAJOS MOLN´ AR MTA-DE ”Lend¨ ule ” Func ional Analysis Resea ch G oup, Ins i u e o Ma hema ics, Uni e si y o Deb ecen, H-4010 Deb ecen, P.O. Box 12, Hun- ga y E-mail add ess:[email p o ec ed] URL:h p://www.ma h.unideb.hu/~molna l/