Surjective isometries on Grassmann spaces
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SURJECTIVE ISOMETRIES ON GRASSMANN SPACES
FERNANDA BOTELHO, JAMES JAMISON, AND LAJOS MOLN´
AR
Abs ac . Le Hbe a complex Hilbe space, na gi en posi i e in ege and le Pn(H) be he se o all
p ojec ions on Hwi h ank n. Unde he condi ion dim H ≥ 4n, we desc ibe he su jec i e isome ies
o Pn(H) wi h espec o he gap me ic ( he me ic induced by he ope a o no m).
1. In oduc ion
The s udy o isome ies o unc ion spaces o ope a o algeb as is an impo an esea ch a ea in
unc ional analysis wi h his o y da ing back o he 1930’s. The mos classical esul s in he a ea a e
he Banach-S one heo em desc ibing he su jec i e linea isome ies be ween Banach spaces o complex-
alued con inuous unc ions on compac Hausdo spaces and i s noncommu a i e ex ension o gene al
C∗-algeb as due o Kadison. This has he su p ising and ema kable consequence ha he su jec i e
linea isome ies be ween C∗-algeb as a e closely ela ed o algeb a isomo phisms. Mo e p ecisely, e e y
such su jec i e linea isome y is a Jo dan *-isomo phism ollowed by mul iplica ion by a ixed uni a y
elemen . We poin ou ha in he a o emen ioned undamen al esul s he unde lying s uc u es a e
algeb as and he isome ies a e assumed o be linea . Howe e , desc ip ions o isome ies o non-linea
s uc u es also play impo an oles in se e al a eas. We only men ion one example which is closely
ela ed wi h he subjec ma e o his pape . This is he amous Wigne ’s heo em ha desc ibes
he s uc u e o quan um mechanical symme y ans o ma ions and plays a undamen al ole in he
p obabilis ic aspec s o quan um heo y. To o mula e Wigne ’s heo em, le Hbe a complex Hilbe
space and deno e by P1(H) he se o all ank-1 p ojec ions on H. (The elemen s o P1(H) ep esen he
pu e s a es o a quan um sys em o which he Hilbe space His associa ed.) The ansi ion p obabili y
be ween he elemen s p, q ∈P1(H) is he quan i y pq, whe e s ands o he usual ace unc ional.
Wigne ’s heo em cha ac e izes all bijec i e maps o P1(H) which p ese e he ansi ion p obabili y.
Be o e gi ing he p ecise o mula ion we ema k ha he nume ical quan i y pq can be in e p e ed in
se e al ways. In ac , i is easily seen o be he squa e o he cosine o he angle be ween he anges o p
and q(as one-dimensional subspaces o H). On he o he hand, we also ha e kp−qk=√1− pq, whe e
k.ks ands o he ope a o no m (see, e.g., [7], p. 127). The e o e, he quan um mechanical symme y
ans o ma ions can be iewed ei he as bijec i e maps on he space o all lines in Hgoing h ough he
o igin which p ese e he angle be ween any wo lines o , al e na i ely, as bijec i e maps on P1(H) which
p ese e he no m dis ance be ween any wo ank 1 p ojec ions. Now we can o mula e Wigne ’s heo em
in he ollowing way.
Theo em 1.1. (c . [7], p. 12) The bijec i e ans o ma ion Φ:P1(H)→P1(H)sa is ies
kΦ(p)−Φ(q)k=kp−qk,∀p∈P1(H)
Da e: Oc obe 15, 2012.
2010 Ma hema ics Subjec Classi ica ion. P ima y 47B49, Seconda y 54E40.
Key wo ds and ph ases. Su jec i e isome y, G assmann space, Hilbe space p ojec ions, gap me ic.
The i s wo au ho s comple ed his wo k while ecei ing a P o essional De elopmen Awa d g an ed by he Uni e si y o
Memphis. The hi d au ho was suppo ed by he Hunga ian Scien i ic Resea ch Fund (OTKA) Reg.No. K81166 NK81402
and by he ”Lend¨ule ” P og am o he Hunga ian Academy o Sciences.
1
2 FERNANDA BOTELHO, JAMES JAMISON, AND LAJOS MOLN ´
AR
i and only i he e exis s ei he a uni a y o an an iuni a y ope a o Uon Hsuch ha
Φ(p) = UpU∗,∀p∈P1(H).
Deno e by B(H) he C∗-algeb a o all bounded ope a o s on H. I is well known ha he *-au omo -
phisms o B(H) a e all o he o m A7→ UAU∗, o some uni a y ope a o Uon H, and i s *-an iau o-
mo phisms a e all o he o m A7→ UA∗U∗, o some an iuni a y ope a o Uon H.
Consequen ly, he heo em abo e says ha e e y su jec i e isome y o P1(H) ex ends ei he o a
*-au omo phism o o a *-an iau omo phism o he algeb a B(H).
We p oceed by poin ing ou he ac ha P1(H) is a pa icula G assmann space. In ac , one usually
de ines a G assmann space as he collec ion o all subspaces o a Hilbe space wi h a ixed ini e dimension.
Howe e , closed subspaces and p ojec ions (sel -adjoin idempo en s) a e in a one- o-one co espondence.
In his pape we p e e o conside he spaces Pn(H) o all p ojec ions on Ho ank n(nis a gi en
posi i e in ege ) as G assmann spaces.
The ope a o no m de ines a me ic on Pn(H) which is usually called he gap me ic. This me ic
was i s in es iga ed by B. Sz˝oke al i-Nagy and independen ly by M.G. K ein and M.A. K asnoselski
unde he name “ape u e” (see [1], Sec ion 34). The gap me ic has a wide ange o applica ions om
pu e ma hema ics o enginee ing. One can easily ind a la ge numbe o e e ences demons a ing his
b oad applicabili y, among o he s, we lis he ollowing ields: pe u ba ion heo y o linea ope a o s,
pe u ba ion analysis o in a ian subspaces, op imiza ion, obus con ol, mul i- a iable con ol, sys em
iden i ica ion, and signal p ocessing.
The goal o he p esen pape is o de e mine and desc ibe he su jec i e isome ies o he G assmann
space Pn(H) wi h espec o he gap me ic. Obse e ha in ligh o Theo em 1.1 ou main esul , which
ollows, can be iewed as an ex ension o Wigne ’s heo em om P1(H) o he case o Pn(H).
Theo em 1.2. Le Hbe a complex Hilbe space, na gi en posi i e in ege , and dim H ≥ 4n. Assume
ha he su jec i e map Φ:Pn(H)→Pn(H)is an isome y wi h espec o he gap me ic, i.e.,
kΦ(p)−Φ(q)k=kp−qk,∀p∈Pn(H).
Then he e exis s ei he a uni a y o an an iuni a y ope a o Uon Hsuch ha
Φ(p) = UpU∗,∀p∈Pn(H).
Consequen ly, jus as in he case o P1(H), we ob ain ha e e y su jec i e isome y o he G assmann
space Pn(H) unde he gap me ic ex ends ei he o a *-au omo phism o o a *-an iau omo phism o
he ull ope a o algeb a B(H) on H.
We ema k ha in [8] (al e na i ely, see Sec ion 2.1 in [7]) he hi d au ho p esen ed an ex ension
o Wigne ’s heo em o he space o highe ank p ojec ions. In [8], Moln´a conside ed (no necessa ily
su jec i e) ans o ma ions on Pn(H) which p ese e he collec ion o so-called p incipal angles be ween
he elemen s o Pn(H). The ans o ma ions conside ed in [8] p ese e he comple e sys em o p incipal
angles (an n- uple o scala s) bu Theo em 1.2 deals wi h ans o ma ions ha p ese e only one o hose
p incipal angles, namely, he la ges one. In he las sec ion o his pape we shall discuss his u he .
A ew wo ds ollow abou he scheme o he p oo o Theo em 1.2. The main ing edien is he use
o a non-commu a i e Mazu -Ulam ype esul on he local algeb aic beha io o su jec i e isome ies
be ween subs uc u es o me ic g oups. In ac , his will imply ha he isome ies we conside he e
p ese e he ela ion o commu a i i y be ween he elemen s o Pn(H). Nex , using a cha ac e iza ion
o o hogonali y o ank-np ojec ions in ol ing he ela ion o commu a i i y and he gap opology, we
show ha he o hogonali y o he elemen s o Pn(H) is p ese ed unde any su jec i e isome y o Pn(H).
Finally, we comple e he p oo by applying a nice esul due o Gy˝o y and ˇ
Sem l desc ibing he s uc u e
o o hogonali y p ese ing bijec ions o Pn(H).
This pape is o ganized as ollows. In Sec ion 2 we e iew all no a ion used h oughou his pape and
also collec he esul s needed o o hcoming a gumen s used in ou p oo s. We gi e he de ails o he
SURJECTIVE ISOMETRIES ON GRASSMANN SPACES 3
p oo o ou main esul in Sec ion 3. In Sec ion 4 we p esen some ema ks and b ie ly discuss su jec i e
isome ies o Pn(H) unde some o he me ics.
2. Backg ound and no a ion
In wha ollows he symbol H ep esen s a ini e o in ini e dimensional complex Hilbe space. Fo a
gi en posi i e in ege n, we deno e by Pn(H) he se o all p ojec ions on Hwi h ank equal o n. The
me ic de ined on Pn(H) is called he “gap me ic” and gi en by dg(p, q) = kp−qk,p, q ∈Pn(H), whe e
k.kdeno es he usual ope a o no m.
Th oughou his pape , Φ : Pn(H)→Pn(H) is a gi en su jec i e isome y, i.e., a su jec i e map wi h
he p ope y ha
kΦ(p)−Φ(q)k=kp−qk,∀p∈Pn(H).
We ecall ha o he gap me ic, he dis ance be ween any wo p ojec ions o di e en ank is equal
o 1. In ac , his ollows immedia ely om he ollowing olk esul , alid e en in he con ex o gene al
C∗-algeb as. I s p oo is omi ed since i only equi es s anda d C∗-algeb a echniques and elemen a y
compu a ions. By a symme y in a C∗-algeb a we mean any sel -adjoin uni a y elemen (o sel -adjoin
in olu ion).
P oposi ion 2.1. I p, q a e p ojec ions in a C∗-algeb a Awi h uni 1such ha kp−qk<1, hen he
elemen u= (1 −(p−q)2)−1/2(p+q−1) is a symme y in e wining pand q, i.e., upu =q.
We can ex end ou o iginal su jec i e isome y Φ : Pn(H)→Pn(H) om he se Pn(H) o he se o
all p ojec ions on Hby simply de ining Φ(q) = q o e e y p ojec ion qwi h ank di e en om n. I is
now appa en ha his ex ension yields a su jec i e isome y on he se o all p ojec ions on H.
The se o all p ojec ions a e in a bijec i e co espondence wi h he se o all symme ies in B(H)
ha we deno e by S(H). In ac , his co espondence is gi en by associa ing wi h any p ojec ion p
he symme y ade ined by a= 1 −2p(1 ep esen s he iden i y ope a o on H). Clea ly, he map Φ
de e mines a ans o ma ion Ψ : S(H)→S(H) gi en by
Ψ(a)=1−2Φ 1−a
2.
I is i ial o check ha Ψ is a su jec i e isome y o S(H) wi h espec o he me ic de ined om he
ope a o no m, i.e., kΨ(a)−Ψ(b)k=ka−bkholds o all pai s a, b in S(H).
A Mazu -Ulam ype esul plays a undamen al ole in he p oo o ou main esul . The s a emen o
his esul , he o hcoming P oposi ion 2.3, needs some p elimina y de ini ions ha we o mula e i s .
Le Gbe a g oup wi h uni 1. We call a subse Xo Ga wis ed subg oup i 1 ∈Xand ba−1b∈X, o
all a, b ∈X. I dis a me ic on G, we say ha i is ansla ion and in e se in a ian i , o all a, b, c, d ∈G,
we ha e d(cad, cbd) = d(a, b) and d(a−1, b−1) = d(a, b), espec i ely.
I is clea ha he uni a y g oup U(H) o all uni a y ope a o s on he Hilbe space Hequipped wi h
he me ic de e mined by he ope a o no m is a me ic g oup wi h ansla ion and in e se in a ian
me ic and S(H) is a wis ed subg oup o U(H).
One o he main ools in he p oo o Theo em 1.2 is a gene al Mazu -Ulam ype heo em on he
local algeb aic beha io o he su jec i e isome ies o wis ed subg oups o g oups wi h ansla ion and
in e se in a ian me ics. The o iginal Mazu -Ulam heo em s a es ha e e y su jec i e isome y be ween
no med eal-linea spaces is au oma ically a ine. Mo i a ed by a mi aculous p oo o he Mazu -Ulam
heo em gi en by V¨ais¨al¨a [11], he au ho s in [5], p esen ed esul s conce ning su jec i e isome ies o
gene al non-commu a i e me ic g oups showing ha hose ans o ma ions (unde gi en condi ions)
locally p ese e he in e ed Jo dan iple p oduc ba−1b. Ou a gumen in he p oo o Theo em 1.2
elies on a esul o he same ype. Howe e , he esul needed in ou p oo canno be deduced om he
esul s p esen ed in [5], hence we gi e i s o mula ion in ou o hcoming P oposi ion 2.3 and also include
i s p oo . Fi s , we p o e a p elimina y lemma (c . Lemma 2.3 in [5]).
4 FERNANDA BOTELHO, JAMES JAMISON, AND LAJOS MOLN ´
AR
Lemma 2.2. Le Mbe a bounded me ic space, ϕ:M→Ma su jec i e isome y. Assume c∈Mand
k > 1is a cons an such ha
d(ϕ(x), x)≥kd(x, c),
o e e y x∈M. Then T(c) = c, o e e y su jec i e isome y T:M→M.
P oo . Le
λ= sup{d(T(c), c) : T:M→Mis a su jec i e isome y}.
Clea ly, λ < ∞. Selec a su jec i e isome y T:M→Mand conside ˜
T=T−1◦ϕ◦T, which is also a
su jec i e isome y o M. We ha e
λ≥d(T−1(ϕ(T(c))), c) = d(ϕ(T(c)), T(c)≥kd(T(c), c).
Since his holds o e e y su jec i e isome y T:M→M, we ob ain λ≥kλ which implies λ= 0.This
comple es he p oo .
We in oduce some addi ional no a ion. Gi en Xa subse o a me ic g oup, and wo elemen s a, b in
X, we deno e by La,b he ollowing subse o X:
La,b ={x∈X:d(x, a) = d(x, ba−1b) = d(a, b)}.
We now o mula e he esul which plays an impo an ole in he p oo o Theo em 1.2.
P oposi ion 2.3. Le Xbe a wis ed subg oup o a me ic g oup Gwi h ansla ion and in e se in a ian
me ic d. Le a, b ∈Xand le k > 1be such ha
(1) d(bx−1b, x)≥kd(x, b),
o all x∈La,b. I T:X→Xis a su jec i e isome y and he e exis s c∈Xsuch ha c(Ta)−1c=
T(ba−1b), hen c(Tb)−1c=Tb.
P oo . De ine
H={y∈X:d(y, T (a)) = d(y, T(ba−1b)) = d(a, b)}.
One can easily check ha H=T(La,b). The maps ϕ, ψ :X→Xde ined by ϕ( ) = b −1b,ψ( ) = c −1c,
∈Xa e su jec i e isome ies o X. I is s aigh o wa d o check ha ϕ(La,b) = La,b and ψ(H) = H.
The e o e, he ans o ma ion ˜
T=T−1◦ψ◦T, when es ic ed o La,b,is a su jec i e isome y. Since
La,b is a bounded me ic space, applying Lemma 2.2 we ob ain ˜
T(b) = b. This yields c(Tb)−1c=Tb and
comple es he p oo .
We no e ha a simila esul om [5] has been applied in [6] o de e mine he su jec i e isome ies o
he uni a y g oup U(H).
As he las p elimina y s ep we in oduce he ollowing no ion. Le Xbe a wis ed subg oup o a
g oup, T:X→Xa ans o ma ion, and aand belemen s o X. We say ha Tis (a, b)-mul iplica i e i
T(ba−1b) = T(b)T(a)−1T(b).
3. P oo o he main heo em
In his sec ion we p esen he de ailed p oo o ou main heo em.
Recall om Sec ion 2 ha he o iginal su jec i e isome y Φ : Pn(H)→Pn(H) is now ex ended o he
whole se o p ojec ions on H. We deno e he ex ension which is also a su jec i e isome y by he same
symbol Φ, and Ψ s ands o he co esponding su jec i e isome y o he space S(H) o all symme ies in
B(H).
We i s e o mula e P oposi ion 2.1 o symme ies. This esul will be used se e al imes h oughou
he pape .
SURJECTIVE ISOMETRIES ON GRASSMANN SPACES 5
P oposi ion 3.1. I a, b a e symme ies in a uni al C∗-algeb a Asuch ha ka−bk<2 hen he elemen
s= (1 −((a−b)/2)2)−1/2((a+b)/2) is a symme y and we ha e sas =b.
The nex s a emen es ablishes a mul iplica i e p ope y o Ψ o symme ies ha a e close in dis ance.
In he p oo we shall use ha he no m k.kis uni a ily in a ian and also ha s−1=sholds o each
s∈S(H).
Lemma 3.2. I a, b ∈S(H)a e such ha ka−bk<√3
2 hen Ψis (a, b)-mul iplica i e.
P oo . We i s obse e ha he e exis s , a posi i e numbe less han 1
2,such ha ka−bk<√3
2−.
Then we apply P oposi ion 2.3. We show ha he e exis s k > 1 such ha
kbxb −xk ≥ kkx−bk, o e e y x∈La,b.
Gi en x∈La,b, we ha e
kbxb −xk=k(xb)2−1k= sup
λ∈σ(xb)|λ2−1|= sup
λ∈σ(xb)|λ−1||λ+ 1|.
Since he elemen xb is a uni a y ope a o on H, we ha e σ(xb)⊆ {λ∈C:|λ|= 1}.The e o e, gi en λ
in he spec um o xb we conclude ha
|λ−1|≤kxb −1k=kx−bk≤kx−ak+ka−bk= 2ka−bk<√3−2.
By he Py hago ean heo em |λ+ 1|2+|λ−1|2= 4,hence |λ+ 1|2>4−(√3−2)2>1.We se
k=q4−(√3−2)2>1.Then
sup
λ∈σ(xb)|λ−1||λ+ 1| ≥ kkxb −1k=kkx−bk.
Consequen ly, he condi ion displayed in (1) holds o e e y x∈La,b.
Since kΨ(a)−Ψ(bab)k=ka−babk≤ka−bk+kb−babk= 2ka−bk<√3<2, by P oposi ion 3.1 we
ha e ha
c=1
1−Ψ(a)−Ψ(bab)
22
Ψ(a) + Ψ(bab)
2
is a symme y and
(2) cΨ(a)c= Ψ(bab).
6 FERNANDA BOTELHO, JAMES JAMISON, AND LAJOS MOLN ´
AR
Applying P oposi ion 2.3, we ob ain ha cΨ(b)c= Ψ(b). This implies ha ccommu es wi h Ψ(b). We
now show ha he dis ance be ween cand Ψ(b) is less han 2. In ac ,
kc−Ψ(b)k=
=
1
1−Ψ(a)−Ψ(bab)
22−1
Ψ(a) + Ψ(bab)
2+Ψ(a)−Ψ(b)
2+Ψ(bab)−Ψ(b)
2
≤
1
1−Ψ(a)−Ψ(bab)
22−1
Ψ(a) + Ψ(bab)
2
+
Ψ(a)−Ψ(b)
2
+
Ψ(bab)−Ψ(b)
2
≤
1
q1−(√3
2)2−1
+√3
4+√3
4<2.
I is easy o see ha he di e ence o wo commu ing p ojec ions has no m 1 unless hey coincide. Since
he symme ies cand Ψ(b) commu e and hei no m-dis ance is less han 2, we conclude ha c= Ψ(b).
Consequen ly, (2) becomes Ψ(bab) = Ψ(b)Ψ(a)Ψ(b), i.e., Ψ is (a, b)-mul iplica i e. This comple es he
p oo .
Ou goal is o inc ease he bound in he Lemma 3.2 in o de o ha e he mul iplica i e p ope y o Ψ
on a la ge domain. To p o e his, we need he ollowing echnical esul which is a pa icula case o
Lemma 7 in [6]. I s p oo is elemen a y and we include i o sake o comple eness.
P oposi ion 3.3. Le Xbe a wis ed subg oup o a g oup, and T:X→Xa mapping. Le x0, x1, x2, x3
and x4be elemen s in Xsuch ha
x2=x1x−1
0x1, x3=x2x−1
1x2,and x4=x3x−1
2x3.
I Tis (xi, xi+1)-mul iplica i e o i= 0,1,2 hen Tis (x0, x2)-mul iplica i e.
P oo . Since Tis (xi, xi+1)-mul iplica i e, we ha e
T(x2) = T(x1x−1
0x1) = T(x1)T(x0)−1T(x1)
T(x3) = T(x2x−1
1x2) = T(x2)T(x1)−1T(x2)
T(x4) = T(x3x−1
2x3) = T(x3)T(x2)−1T(x3).
We no ice ha
x4=x3x−1
2x3=x2x−1
1x2x−1
2x2x−1
1x2
=x2x−1
1x2x−1
1x2=
=x1x−1
0x1x−1
1x1x−1
0x1x−1
1x1x−1
0x1
=x1x−1
0x1x−1
0x1x−1
0x1
=x2x−1
0x2.
SURJECTIVE ISOMETRIES ON GRASSMANN SPACES 7
Simila ly, one can show ha T(x4) = T(x2)T(x0)−1T(x2). We ha e T(x2x−1
0x2) = T(x2)T(x0)−1T(x2)
which comple es he p oo .
In he nex lemma we enla ge he domain o e which Ψ is mul iplica i e.
Lemma 3.4. I a, b ∈S(H)a e such ha ka−bk ≤ 1.2 hen Ψis (a, b)-mul iplica i e.
P oo . We de ine a symme y s ha in e wines aand b, i.e., sas =b. In ac , applying P oposi ion 3.1
we se
s=1
q1−a−b
22
a+b
2.
We know ha sis a symme y and sas =b. Mo eo e , we ha e
ks−ak=
1
q1−a−b
22−1
a+b
2+b−a
2
≤
1
q1−(1.2
2)2−1
+1.2
2=1
0.8−1+0.6<√3
2.
An applica ion o Lemma 3.2 implies ha Ψ is (a, s)-mul iplica i e, i.e., Ψ(b) = Ψ(sas) = Ψ(s)Ψ(a)Ψ(s).
We se x0=a,x1=s,x2=b=x1x0x1,x3=x2x1x2and x4=x3x2x3. One can easily check
ha ka−sk=kx0−x1k=kx1−x2k=kx2−x3k=kx3−x4k<√3
2. The e o e Ψ is (xi, xi+1)-
mul iplica i e wi h i= 0,1,2. An applica ion o P oposi ion 3.3 yields ha Ψ is (a, b)-mul iplica i e,
hence Ψ(bab) = Ψ(b)Ψ(a)Ψ(b).
We apply he same p ocedu e o e ine u he he cons an in Lemma 3.4.
Lemma 3.5. I a, b ∈S(H)a e such ha ka−bk ≤ √2 hen Ψis (a, b)-mul iplica i e.
P oo . Jus as in he p oo o he p e ious lemma, we de ine
s=1
q1−a−b
22
a+b
2.
Then sis a symme y ha in e wines aand b,b=sas. We compu e he dis ance be ween sand a,
ks−ak=
1
q1−a−b
22−1
a+b
2+b−a
2
≤(√2−1) + √2
2<1.2.
Lemma 3.4 asse s ha Ψ is (a, s)-mul iplica i e. Conside ing he elemen s x0=a,x1=s,x2=b,
x3=x2x1x2and x4=x3x2x3 he p oo now ollows a simila easoning p esen ed o he p e ious
lemma.
We now show ha he ans o ma ion Ψ p ese es he commu a i i y be ween hose symme ies ha
co espond o p ojec ions in Pn(H). This is he mos impo an s ep in ou p oo .
Lemma 3.6. Le p, q ∈Pn(H)and se a= 1 −2p, b = 1 −2q. I ab =ba hen Ψ(a)Ψ(b) = Ψ(b)Ψ(a).
P oo . Since aand bcommu e, so does he co esponding p ojec ions, p=1−a
2and q=1−b
2. The e o e
he e exis s an o hono mal basis o Hsuch ha pand qha e he ollowing block ma ix ep esen a ions
p=
I1OOO
O O2O O
O O I3O
O O O O4
, q =
O1O O O
O I2O O
O O I3O
O O O O4
.
8 FERNANDA BOTELHO, JAMES JAMISON, AND LAJOS MOLN ´
AR
Hence
a=
−I1O O O
O I2O O
O O −I3O
O O O I4
, b =
I1O O O
O−I2O O
O O −I3O
O O O I4
.
Since p, q a e o he same ank, he sizes o he iden i y ma ices I1and I2in he ep esen a ion abo e
a e he same. Hence we can o m he ollowing block ma ix
s=
O I1O O
I1O O O
O O −I3O
O O O I4
.
I is easy o check ha sis a symme y ha in e wines aand b, i.e., sas =b. Mo eo e , we ha e
ks−ak=
I1I1
I1−I1
=√2.
Now, Lemma 3.5 implies ha Ψ is (a, s)-mul iplica i e. Se x0=a,x1=s,x2=b=x1x0x1,x3=x2x1x2
and x4=x3x2x3. One can easily check ha ka−sk=kx0−x1k=kx1−x2k=kx2−x3k=kx3−x4k=√2.
Lemma 3.5 implies ha Ψ is (xi, xi+1)-mul iplica i e o i= 0,1,2. An applica ion o P oposi ion 3.3
yields ha Ψ is (a, b)-mul iplica i e, Ψ(bab) = Ψ(b)Ψ(a)Ψ(b).Mo eo e , he commu a i i y o aand b
implies ha bab =aand hus we ha e Ψ(a) = Ψ(b)Ψ(a)Ψ(b). This implies ha Ψ(b)Ψ(a) = Ψ(a)Ψ(b)
and comple es he p oo .
We obse e ha he p e ious lemma implies ha ou o iginal ans o ma ion Φ : Pn(H)→Pn(H)
maps commu ing p ojec ions o commu ing p ojec ions. This was a consequence o he ac ha Φ is
a su jec i e isome y on Pn(H). Howe e , Φ−1is also a su jec i e isome y and hence i has he same
p ese ing p ope y. Consequen ly, we ob ain ha Φ p ese es commu a i i y in bo h di ec ions.
We nex p esen a cha ac e iza ion o o hogonali y among ank-np ojec ions as a s epping s one o
he p oo ha Φ p ese es o hogonali y in bo h di ec ions. Gi en wo p ojec ions pand qin Pn(H) he
symbol {p, q}c ep esen s he ( ela i e) commu an o {p, q}in Pn(H), i.e., he se o all p ojec ions in
Pn(H) ha commu e wi h bo h pand q. A amily {pi}io p ojec ions on His called a esolu ion o he
iden i y i Pipi= 1 and any wo dis inc elemen s a e o hogonal, i.e., pipj= 0 o i6=j.
P oposi ion 3.7. Le Hbe a Hilbe space ei he in ini e dimensional o o ini e dimension wi h dim H ≥
4n. Fo any wo commu ing p ojec ions p, q in Pn(H)we ha e ha pand qa e o hogonal i and only i
he se {p, q}cas a subspace o he me ic space Pn(H)has a pa hwise connec ed componen Ksuch ha
he maximal numbe o pai wise commu ing p ojec ions o ank nin Kcis exac ly 2n
n.
P oo . We conside wo commu ing p ojec ions pand qin Pn(H). We de ine he ollowing esolu ion o
he iden i y ob ained om p, q:
P={p−pq, pq, q −pq, 1−(p+q−pq)}.
Le be a p ojec ion o ank n ha commu es wi h pand q, i.e., ∈ {p, q}c.Then =P4
i=1 piwi h
p1=p−pq,p2=pq,p3=q−pq and p4= 1 −(p+q−pq).Se ing ni= ank( pi) we associa e wi h a
4- uple (n1, n2, n3, n4) o nonnega i e in ege s such ha n1+n2+n3+n4=n. We call (n1, n2, n3, n4)
he ank ep esen a ion o ela i e o P. We obse e ha any wo p ojec ions in {p, q}ca e pa hwise
connec ed h ough a pa h in {p, q}ci and only i hey ha e he same ank ep esen a ions ela i e o P.
To see his, one may ecall P oposi ion 2.1, and in okes he ollowing wo ac s: P ojec ions o he same
ank a e uni a ily equi alen , and he uni a y g oup is pa hwise connec ed in he ope a o no m.
Gi en a p ojec ion p0in Pn(H) wi h ange con ained in he ange o p4, hen p0is o hogonal o bo h
pand qand i s ank ep esen a ion ela i e o Pis (0,0,0, n). We say ha his p ojec ion is suppo ed in
SURJECTIVE ISOMETRIES ON GRASSMANN SPACES 9
p4. Le Kbe he componen o {p, q}cwhich consis s o all p ojec ions in Pn(H) suppo ed in p4. Then
Kcis he se o all ank-np ojec ions ha commu e wi h all ank-np ojec ions which a e suppo ed
in p4. The ank o p4is g ea e han n. Selec ing an elemen in Kc, commu es wi h e e y ank-n
p ojec ion suppo ed in p4. Then also commu es wi h e e y ank-1 p ojec ion suppo ed in p4. I
implies a he easily ha he ange o p4is an eigenspace o . Hence p4= 0. This shows ha Kcis
equal o he se o all elemen s o Pn(H) which a e o hogonal o p4. The maximal numbe o commu ing
elemen s o his se is clea ly ank(1−p4)
n. I p, q a e o hogonal, hen his numbe is 2n
nand we ob ain
ha he condi ion o o hogonali y s a ed in he p oposi ion is necessa y.
Con e sely, assume ha pand qa e no o hogonal. Then he ange o p∨q=p+q−pq = 1 −p4
has dimension s ic ly less ha 2n. We conside a componen Ko {p, q}clabeled wi h (n1, n2, n3, n4),
n46= 0. We can see ha e e y elemen o Kccommu es wi h he di e ence o any wo ank-n4
p ojec ions suppo ed in p4.This implies ha such a p ojec ion commu es wi h e e y ank-1 p ojec ion
suppo ed in p4. We in e ha he ange o p4is an eigenspace o and hence p4= 0. The e o e, he
elemen s o Kca e suppo ed in 1 −p4. Hence he maximal numbe o commu ing elemen s in Kcis
s ic ly less han 2n
n. Finally, we conside a componen Ko {p, q}clabeled wi h (n1, n2, n3,0). The
commu an o his componen con ains all p ojec ions which a e suppo ed in p4. Since he dimension o
His g ea e o equal o 4nand he ank o p∨qis less han 2n, he ank o p4is s ic ly g ea e han
2n. This implies ha he maximal numbe o pai wise commu ing p ojec ions in Kcis g ea e han 2n
n.
The p oo is comple e.
We a e now in a posi ion o p o e ou main heo em.
P oo o Theo em 1.2. P oposi ion 3.7 gi es a cha ac e iza ion o o hogonali y be ween he elemen s
o Pn(H) based on opological concep s and commu a i i y. Since he su jec i e isome y Φ : Pn(H)→
Pn(H) p ese es he opological p ope ies as well as he commu a i i y in bo h di ec ions, we ind ha
he bijec i e map Φ p ese es he o hogonali y in bo h di ec ions. We can now apply a esul o Gy˝o y [4]
and ˇ
Sem l [9] on he s uc u e o such ans o ma ions which ex ends Uhlho n’s amous heo em [10] om
one-dimensional subspaces o he case o highe dimensional subspaces. ˇ
Sem l p o ed his esul o in ini e
dimensional Hilbe spaces H, while Gy˝o y conside ed also he ini e dimensional case o dim H>3n.
In ei he case, he conclusion is ha any bijec i e map on Pn(H) which p ese es o hogonali y in bo h
di ec ions is implemen ed ei he by a uni a y o an an iuni a y ope a o . This means ha we ha e a
uni a y o an iuni a y ope a o Uon Hsuch ha Φ(p) = UpU∗,p∈Pn(H). The p oo o he heo em
is comple e.
4. Rema ks
We conclude he pape wi h a ew ema ks. Fi s , one may ask i con inuing he abo e p ocess o
“pumping up” he alue o he cons an in Lemma 3.2 as we did in Lemmas 3.4 and 3.5 we could inally
each he cons an 2, i.e., we could p o e ha o any pai a, b o symme ies wi h ka−bk ≤ 2 we ha e
ha Ψ is (a, b)-mul iplica i e. In ac , his would mean ha Ψ is a Jo dan iple au omo phism o S(H),
a bijec i e map sa is ying Ψ(bab) = Ψ(b)Ψ(a)Ψ(b) o all a, b ∈S(H). Howe e , we ha e ound ha his
p ocess leads o a limi which is less han 2 (using Maple we ha e ob ained ha his limi is app oxima ely
1.67857351).
Abo e we ha e desc ibed he isome ies o he G assmann space Pn(H) wi h espec o he gap me ic,
he me ic coming om he ope a o no m. Howe e , he e a e se e al o he me ics on Pn(H) which
a e used in di e en a eas o ma hema ics. Mos o hose me ics a e ela ed o he concep o p incipal
angles be ween highe dimensional subspaces o a Hilbe space. One way o de ine ha concep is he
ollowing. Le M, N be n-dimensional subspaces o Hand deno e by p, q he p ojec ions in Pn(H) ha
p ojec on o Mand N, espec i ely. Conside he dec easing sequence o eigen alues o he posi i e