C OMPOSITIO M ATHEMATICA
P. DE L A H ARPE
The Clifford algebra and the Spinor group of a Hilbert space
Compositio Mathematica, tome 25, n
o3 (1972), p. 245-261
<http://www.numdam.org/item?id=CM_1972__25_3_245_0>
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245
THE CLIFFORD ALGEBRA
AND THE SPINOR GROUP OF A HILBERT SPACE by
P. de la
Harpe
Wolters-Noordhoff Publishing Printed in the Netherlands
1. Introduction
Clifford
algebras
constitute an essential tool in thestudy
ofquadratic
forms.
However,
for infinite dimensionalproblems,
it seems oftennatural to ask for some extra
topological
structure on thesealgebras.
The
present
note studies such structures on anexample
which is theanalogue
for Hilbert spaces ofarbitrary
dimension of the standardalgebras Clk
for finite dimensional real vector spaces(the Ck’s
ofAtiyah- Bott-Shapiro [2]).
Inparticular,
wegive
anexplicit
construction of thecovering
group of the nuclearorthogonal
group; the main result appears in section 10. Thepresent
work reliesheavily
on a theorem due to Shaleand
Stinespring [23] and
ongeneral
results on Banach-Lie groups whichcan be found in Lazard
[19].
1 want to express my
gratitude
to J.Eells,
whoproposed
thisproblem
to me; to K. D.
Elworthy,
P. Stefan and R. F.Streater,
who have beenlistening
to mepatiently
when 1 was wrong as well as when 1 was(hope- fully) right
and who havepointed
out to me severalimprovements;
andto the ’Fonds national suisse pour la recherche
scientifique’,
for itsfinancial
support.
2. The Clifford
algebra
of aquadratic
formThis section is
only
to fix notations. For a more detailedexposition,
see
Atiyah-Bott-Shapiro [2],
Bourbaki[5] or Chevalley [6].
Let H be a real vector space of
arbitrary
dimension and letQ :
H - Rbe a
quadratic
form on H. TheClifford algebra of Q,
denotedby Cl(Q),
is a real
algebra
with unit eo, defined as aquotient
of the tensoralgebra
of H. There is a canonical
injection iQ :
H ~Cl(Q) by
which H can beidentified to a
subspace
ofCl(Q).
TheZ-graduation
on the tensoralgebra
of H induces a
Z2-graduation
onCl (Q),
and thecorresponding
decom-position
will be denotedby Cl(Q)
=Cl+(Q)~+Cl-(Q).
OnCl(Q),
thereis defined the
principal automorphism
a and theprincipal antiautomorphism
p.
Both ocand p
are involutions onCl(Q);
the restriction of a toCl+(Q)
is the
identity,
the restriction of a toCl-(Q)
is minus theidentity;
therestriction
of 03B2
to H is theidentity.
Let E be a
subspace
of H and letQE
be the restrictionof Q
to E. Theinjection E - H
induces aninjection Cl(QE) ~ Cl(Q) by
whichCl(QE)
can be identified to a
subalgebra
ofCl(Q).
Let(ei)ieI
be a basisof H,
and suppose I is
given
a total order. LetS(I)
be the setof strictly
increas-ing
finite sequences of elements ofI;
for s =(i1,···, ik) E S(I ), let es
be the
product
ei1··· eik inCl(Q);
then(eS)seS(I)
is a basis of the vectorspace
Cl(Q);
theidentity
eo ofCl(Q) corresponds
to theempty
sequence.We will write
S(k),
or even S when there is no risk ofconfusion,
insteadof
S({1,··· k})
for anystrictly positive integer
k.The rest of this note is concerned with the case in which H is a real
prehilbert
space where the scalarproduct
is denotedby (,),
and in whichQ
is the form x F-+81xlz (03B5 = ± 1).
We will denoteby
X H X* the com-position of 03B2
and oc if 03B5=-1(the
involutions aand p commute)
andthe
antiautomorphism 03B2
if e = + 1.PROPOSITION 1. Let H be a real
prehilbert
spaceof infinite
dimension.and let
Q be the form x H 03B5|x|2 as
above. ThenCl(Q)
andCl+(Q)
aresimple
and their centers both coincide with the scalarmultiples of
theidentity.
PROOF. Let E be a finite even dimensional
subspace
ofH ;
it is a classical result thatCl(QE)
issimple.
HenceCl(Q)
islocally simple,
i.e. any two of its elements are contained in somesimple subalgebra
of it. It follows thatCl(Q)
issimple.
The other affirmations of theproposition
areproved
the same way.
Proposition
1 is well-known:Doplicher-Powers [10],
Stormer[26].
3. Hilbert
algebra
structure on theClk’s
andCl’k’s
In this
section, CI’,’
denotes the Cliffordalgebra
of thequadratic
formQ : x ~ 03B5|x|2
defined on the euclidean spaceRk (03B5
=±1).
LEMMA 2. There exists a
unique linear form
Âe onCl03B5,k such
that(i) 03BB03B5(e0)
= 1(ii) 2£(XY) = 2£(YX)
for allX,
YECl,,k (iii) 03BB03B5(03B1(X)) = 03BB03B5(X)
for all X ECI£,k.
PROOF. Existence: Let
(ei)1~i~k
be the canonical basis ofRk
and let(es)s~S
be the associated basis ofCl£,k.
Let X =¿seS XseS
be inCl03B5,k;
set
03BB03B5(X)
=Xo.
Then Â’enjoys properties (i)
to(iii).
Unicity:
Let es = e il...eip
be a basis element inCl’,’.
If p isodd,
03BB03B5(es)
must be zeroby (iii). If p
is even and not zero,by (ii),
so that03BB03B5(es)
= 0. Theunicity
follows..By
lemma 2 and a canonicalprocedure (see
e.g. Dieudonné[8], chap. XV, § 6),
one can define a scalarproduct
onCl’,’:
Equivalently,
for any basis(e,,),cs
ofCl03B5,k
associated to an orthonormal basis(ei)1~i~k
ofRk,
and if X =03A3s~S Xses
and Y =03A3s~S YS es
are twoelements of CIe,
k,
one can define(X, Y) = ¿seS
Xs Ys.PROPOSITION 3. The scalar
product (,)
and the involution X ~ X* turnCl03B5,k into a
real Hilbertalgebra.
PROOF. It sufhces to check the identities
X*, Y*> = Y, X)
andXZ, Y) = (X, YZ*>
for allX, Y,
Z ECl03B5,k,
andthey
followtrivially
from lemma 2.
NOTATIONS. Instead of
Cl03B5,k and Â’,
one writes as wellClk
and À when03B5 =-1,
andCl’k
and 03BB’ when e = + 1.REMARKS.
1. The definition of a real Hilbert
algebra
isanalogue
to that of acomplex
Hilbertalgebra
asgiven
forexample
in Diximer[9], chap. 1, §
5.2.
Complete
Hilbertalgebras
have been introducedby
Ambrose[1]
under the name of
H*-algebras.
The finite dimensionalCIe, k are clearly H*-algebras (see
however Ambrose’s remarkfollowing
his condition 3on page
366).
3. The canonical
injection
mentioned in section 2 is anisometry
ofthe euclidean space
Rk
into the Hilbertalgebra
Cl,,k,
so that it is still safe toidentify Rk
to asubspace
ofCl03B5,k.
4.
If j ~ k,
the inclusionCl,, ~ Cl03B5,k
inducedby
the inclusionRi --+ Rk
is anisometry
and the involution inCIe, j
is the restriction of the involution in CIe,k,
so that one can stillidentify Cl’,j
to asubalgebra
ofCl03B5,
k.
5. If
(ei)1~i~k
is an orthonormal basis ofRk,
then(es)s~S(k)
is anorthonormal
basis of CIe,k.
4. Hilbert
algebra
structure onCI(H)
andCl’(H)
In this
section,
H is an infinite dimensional realprehilbert
space andCl’(H)
is the Cliffordalgebra
of the formQ :
x Ho03B5lxlx (g =
+1).
Define the linear form Â’ on
Cl’(H)
as follows: Let X ECl’(H);
let Ebe a finite dimensional
subspace
of H such thatX ~ Cl (QE);
define03BB03B5(X)
to be the value of Â’ on X as calculated inCl(QE)’
whichclearly
does not
depend
on E. Then lemma 2 holds whenCl’,’
isreplaced by
Cl’(H).
ForX,
Y ECl’(H),
define as in section 3(X, Y>= Â’(XY*).
PROPOSITION 4. The scalar
product (,)
and the involution X H X* turnCl’(H)
into a real Hilbertalgebra.
PROOF. The
only
non-trivialpoint
to check is theseparate continuity
of the
multiplication
inCl’(H).
We will check thatsup(XY, XY)
00for all X E
Cl’(H),
where the supremum is taken over all Y’s inside the unit ball ofCl03B5(H).
Let X be
fixed,
and let Y be anarbitrary
element ofCl’(H).
Let E[resp. E (Y)]
be a finite dimensionalsubspace
of H such that X ECl(QE) [resp.
YC-Cl(QE(Y))].
Let(e1,···, en)
be an orthonormal basis ofE+E(Y)
such that(e1,···, ek)
is an orthonormal basis of E. Write X =03A3s~S(k) Xses
and Y =03A3s~S(n) YSes.
Theinequality
can now be
easily proved by
induction on the number of non zero co-ordinates XS of
X ;
hence the mapPROPOSITION 5.
(i)
Theinjection
H -Cl£(H)
is anisometry.
(ii)
For anysubspace E
ofH,
theinjection Cl(QE) ~ Cl£(H)
is anisometry (QE
as in section2).
(iii)
E is dense in H if andonly
ifC’(QE)
is dense inCl’(H).
The
proofs
are trivial.NOTATIONS.
Let D
be thecompletion
of theprehilbert
spaceCl’(H);
the norm onD will
be denotedby ~~2.
When we shall want toemphasize
the Hilbertalgebra
structure ofCl’(H),
we shall use the morecomplete
notation(lij, CI£(H), *).
The involution a[resp. fi,
X i-+X*]
extendsuniquely
toan
isometric
involutionon D
which willagain
be denotedby
the samesymbol.
Note that as vectorspaces S) =1= Cl03B5(H).
The maximal Hilbert
algebra of
bounded elements in(S), CIE(H), *)
will be denoted
by (D, Cl03B52(H), *) (see
Dixmier[9], chap I, § 5,
no 3 forthe
definition).
For any A eCl03B52(H), LA
will denote the boundedoperator
on D
which extends the leftmultiplication by
A onCl2(H);
one has(LA)*
=L(A*). Similarly
for theright multiplication RA.
For any X ED, Li
will denote the(not necessarily continuous) operator on D
withdomain
Cl2(H)
definedby Li(A)
=RA(X)
for all A ECl2(H). Similarly
for the
right multiplication R’X.
Instead of
(D, Cl03B52(H), *),
one writes as well(D, C12(H), *)
when8
= -1,
and(D, Cl’2(H), 03B2)
when 8 = + 1.5. The involutive Lie
algebras spin (H; Co)
andspin H ; C2)
In this
section,
H is an infinite dimensional real Hilbert space andCl’(H)
is the Cliffordalgebra
of the formQ :
x HoEIX12 (g
= +1).
Let
o(H ; Co)
be the Liealgebra
of allskew-adjoints
finite rank opera- tors on H. This Liealgebra
issimple,
and is the ’classicalcompact
Liealgebra
of finite rankoperators
onH,
oftype
BD’ as defined in[14].
Its
completion
withrespect
to the Hilbert-Schmidtnorm Il 112
is thecompact L*-algebra
denotedby o(H; C2).
The purpose of this section isto
identify
these two Liealgebras
as sub Liealgebras
of the ad hoc Cliffordalgebras.
PROPOSITION 6. Let
spin (H ; Co) be
thesubspace of Cl’(H)
spanby
the elements
of the form [x, y]
where x, y are in H. Then:(i) spin (H; Co)
is a sub Liealgebra of Cl03B5(H).
(ii)
X+X*= 0 for
all X ~spin (H ; Co ).
(iii) If X E spin (H; Co ),
thenXy - yX is in H for all y in H;
the operatordefined
in this way onH,
denotedby Dpo(X),
is ino(H; C0).
(iv)
The mapis an
isomorphism of
involutive Liealgebras.
PROOF.
Step
one.Suppose
for convenience that 03B5 = - 1 and let x, y, z, t E H.Then the two relations
can be
proved
as in finite dimensions(see
Jacobson[16], chap. VII, § 6,
formulas
(29)
and(30)).
The first oneclearly implies
the first half of(iii)
and the second one
implies (i);
statement(ii)
can bereadily
checked.Step
two. Let X Espin (H; Co); proposition
4 says inparticular
that theadjoint
of theoperator Dpo(X)
isDpo(X*).
It follows from(ii)
thatDpo(X)
isskew-adjoint,
hence that it is an elementof o(H ; Co ),
whichends the
proof
of(iii).
Step
three. The mapDpo
of(iv)
isclearly
ahomomorphism
ofinvolutive Lie
algebras.
LetX E spin (H; Co )
and let(ei)ic-N*
be anorthonormal basis of H such that X can be
expressed
as a finite sum:Considered as an element of
D,
the norm of X isgiven by
Now the matrix
representation
ofD03C10(X)
withrespect
to the basis(ei)ieN*
of H isHence the Hilbert-Schmidt norm of
Dpo(X)
isgiven by
It follows that
Dp o
isinjective.
Step four.
Let Y Eo (H; C0). Again by writing
Y as a matrix withfinitely
many non zeroentries,
withrespect
to an ad hoc orthonormal basis ofH,
one caneasily
findX E spin (H; Co )
such thatD03C10(X)
= Y.This ends the
proof
of(iv)..
COROLLARY 7. Let
spin (H; C2)
be the closureof spin (H; Co )
inD.
Thenspin (H ; C2 ) is a
realL*-algebra,
the mapDpo
has aunique
continuousextension
which is an
isomorphism of
involutive Liealgebras,
and theequality
holds
for
all X Espin (H; C2) -
The
proof
isimmediate;
for the notion of a realL*-algebra,
see[14]
and the references
given
there.REMARK.
Sections 2 to 5
(and
sections 7 and 8below)
hold in the same way whenH =
Hp~+Hq
is theorthogonal
sum of twoprehilbert
spaces(Hilbert
,spaces in section
5)
and whenQ
is the formThen the Clifford
algebra
ofQ,
sayClp,q(H),
is a Hilbertalgebra
forthe scalar
product
defined as above and for the involution definedby
X* =
03B3(03B2(X)) = f3(y(X)),
where y is theautomorphism
of the Clifford.algebra
which extends(the
dimension of Hp is p, that of Hq is q, pand q
are eitherpositive integers
or~).
Inproposition 6,
thesubspace generated by
the[x, y]’s
is now a Lie
algebra spin (H,
p, q;C0),
andDp o
is anisomorphism
ofinvolutive Lie
algebras
where the last Lie
algebra
is defined as in[14]. Corollary
7 holds withthe
corresponding
modification.Clearly,
thecomplexification
ofClp,q(H)
does notdepend
of thepair (p, q)
butonly
of the sump + q
=dim(H) ~ N~{~}.
Thissuggests
that the consideration of the C1P’
q(H)’s
ingeneral
is not essential froma
physical point
of view(canonical
anticommutationrelations;
see forexample
Guichardet[11 ] and Slawny [24]). Moreover,
thecomplexity
insections 6 and 9-10 below would be very much increased if
arbitrary pairs (p, q)
were to be dealt with.Consequently,
we restrict ourselves to the cases(p, q) = (dimH, O)
and
(p, q) = (O, dimH).
About theClp,q(H)’s
whenp + q
isfinite,
seehowever Karoubi
[17].
6.
Bogoliubov automorphisms
In this
section,
H is a Hilbert space.DEFINITION. Let
(D, Cl03B52(H), *)
be as in section 4. Anautomorphism
rof this Hilbert
algebra
is anorthogonal
map ~ :D
-D
whose restriction toCl2(H)
is anautomorphism
of involutivealgebra
which preserves theidentity;
the group of all theseç’s
is denotedby Aut(Cl03B52(H)).
ABogoliubov automorphism
of(D, Cl03B52(H), *)
is anautomorphism
9 such .that~(H)
=H ;
thecorresponding
group is denotedby Bog(Cl03B52(H)).
Let
O(H)
be the group of allorthogonal operators
on H. Let U EO(H);
by
the universalproperty
of Cliffordalgebras,
U extends to an automor-phism CI(U)
ofCl03B5(H),
which isclearly
a*-automorphism.
If ÀE: is definedas in the
beginning
of section4,
then the linear form Â’ -Cl(U)
stillenjoys properties (i)
to(iii)
of lemma2;
hence Â’ -Cl(U) = À8
andCl(U)
defines an
orthogonal operator
onD,
which willagain
be denotedby
thesame
symbol.
It is moreoverelementary
to check thatCl(U)
is a *-automorphism
of thealgebra
of bounded elementsCl03B52(H).
In otherwords,
there is a canonicalgroup-isomorphism
LEMMA 8. Let U be in
O(H).
Thenthe following
areequivalent:
(i )
There exists an invertible element u ECl03B52(H)
such thatCl(U) =
Lu . Ru - 1 .
(ii)
There exists aunique (up
tomultiplication by
ascalar)
invertibleelement u E
Cl03B52(H)
such thatCl( U) - Lu - Ru-1,
and one has eitheru E
Cl2 + (H),
or u ECl03B5-2 (H).
PROOF. It is easy to check that the center of
Cl03B52(H)
is the same as thatof
ClE(H), namely
that it consists of themultiples
of theidentity only (proposition 1).
Hence theonly thing
to prove is the secondpart
of(ii).
But as U =
(-I)U(-I),
one hasCl(U) =
a -Cl(U) .
oc; ifCl(U) = LuRu-,,
then a.Cl(U).
oc =L03B1(u)R03B1(u)-1
anda(u)
must be amultiple
of u; as 03B1 is an
involution,
it follows thata(u) =
+ u.DEFINITION. Let
Cl(U)
be aBogoliubov automorphism
of(D, Cl2(H), *).
ThenCl(U)
is said to be inner if it satisfies conditions(i)
and(ii)
of lemma8,
and outer if not. Notationsbeing
as in(ii), Cl(U)
is said to be even if u ECl03B5+2 (H)
and odd if u EC12’- (H).
COROLLARY 9. Let
Spin(H; vN)
be the setof
allunitary
even elementsu E
Cl03B5+2(H)
such thatuHu-1 =
H. ThenSpin(H; vN)
is asubgroup
inthe
(abstract)
groupof
all invertible elementsof Cl03B52(H).
For u E
Spin(H; vN),
let03C103C5N(u)
be theorthogonal operator
Then the
image of
PvN is a normalsubgroup of O(H),
and itskernel,
consisting of
eo and - eo, isisomorphic
toZ2 .
Theproof
is trivial.Corollary
9 indicates thatSpin(H; vN)
mustsupport
thecovering
group of some
topological subgroup
ofO(H).
As one would like to usethe
general theory
of Banach-Lie groups, the Hilbert space structure onCl03B52(H)
is notconvenient,
because it is nevercomplete
in the infinitedimensional case. In order to remove this
impediment,
we furnishCl03B52(H)
with aC*-algebra
structure in section7;
we consider an ad hocsub-C*-algebra
and the ad hocsubgroup
ofSpin (H; vN);
afterhaving proved
a local result in section 8 and recalled a theorem in section9,
we can establish the main result of this work in section 10.
7.
C*-algebra
structure on the Cliffordalgebra
In this
section,
H is aprehilbert
space as in section 4.Consider the Hilbert
algebra (jj, Cl03B52(H), *),
and for each X ~Cl03B52(H)
let
LX
be theoperator on D
defined as in section 4. Define a new norm onC12’(H) by
Let
Ch (H)
be the involutivealgebra Cl03B52(H)
furnished with the norm~~~.
PROPOSITION 10.
Cl03B5~(H) is a
realC*-algebra.
PROOF. The
only point
to check is thecompleteness
ofCl03B5~(H).
Let(Xn)n~N be a Cauchy
sequence inCl03B5~(H);
it is a fortiori aCauchy
sequence with
respect
to thenorm ~~2,
so that it converges withrespect to ~~2
towards an element X inS).
Let M be a bound for(~Xn~~)n~N;
for any Y ~ Cl03B52(H) :
for every n E N. Hence
and X E
Cl03B5~(H).
It must yet be shown that
(Xn)neN
converges towards X withrespect
to
/1 /100.
Chose 03B4 > 0 and letY E Cl03B5~(H) with ~Y~ ~
1. As(Xn)n ~ N
is a
Cauchy
sequence withrespect to ~~~,
there is aninteger
no inde-pendent
of Y suchthat ~(Xn-Xm)Y~2 ~ 03B4 for n, m >
no. As(Xn)neN
converges towards X with
respect to ~~2
and as themultiplication
isseparately
continuous inCl03B52(H), ~(X-Xm)Y~2~03B4
for m ~ no. As no.is
independent
of Y:for m ~ n0.
Let
Cl03B51(H)
be theC*-algebra
which is the closure ofCl’(H)
inCl03B5~(H).
It will be called theC*-algebra of
theform
x Ho03B5|x|1
on H.Sections 2 to 4
imply straightforwardly
a certain number ofproperties
of
Cl03B51(H)
collected now for convenience in the case - = - l.SCHOLIE 11. Let H be an infinite dimensional real
prehilbert
space and letCh (H)
be the C*-Cliffordalgebra
of the formx ~ |x|2
on H.Then:
(i) Ch (H)
is aZ2-graded
realC*-algebra
with unit eo.(ii) Cl1(H)
andCl+1(H)
aretopologically simple
and their commoncenter is
equal
toReo.
(iii)
Theinjection
H -Cl1(H)
is anisometry.
(iv)
If(en)n~N* is
an orthonormal basis of H and ifCl’
is the standard Cliffordalgebra
of the spaceRk =
span{e1, ···, ek}
inH,
then theunion Cl’ of the
Clk,s
is dense inCh (H).
(v)
If E is a densesubspace
ofH,
then the inclusion E ~ H induces anisomorphism
onto theimage Cfi (E) - Ch (H).
REMARK. Similar statements hold for the C*-Clifford
algebras Clp,q1(H),
as indicated at the end of section 5.8. The involutive Banach-Lie
algebra spin (H ; C1 )
and the Banach-Lie group
Spin (H ; C1 )
From now on, H is
always
aseparable
Hilbert space. Thecompletion
of the Lie
algebra o(H ; Co)
defined in section 5 withrespect
to the nuclearnorm ~~1
is an involutive(compact)
real Banach-Liealgebra
denoted
by o(H; C1).
PROPOSITION 12. Let
spin (H; C1 ) be
the closureof spin (H; Co )
inCl’(H).
Thenspin (H; C1 )
is an involutive real Banach-Liealgebra,
themap
Dp o (see proposition 6)
has aunique
continuous extentionwhich is an
isomorphism of
involutive Liealgebras,
and theequality
holds
for
all X Espin (H; C1).
PROOF. It is
clearly
sufficient to showthat ~D03C10(X)~1
=4~X~~
forall X e
spin (H; C0).
We consider the case E = -1..Step
one. Let(en)n~N
be an orthonormal basis of H and let X be anélément in
spin(H; C0)
of the form03A3kn=1 03BBn[e2n-1, e2n].
From theproof
of
proposition 4: ~X~~ ~ 203A3kn=1|03BBn|;
fromstep
three in theproof
ofproposition 6: ~D03C10(X)~1
=803A3kn=1|03BBn|
·Hence ~D03C10(X)~1 ~ 4~X~~.
Consider now X as an element of the
complexified
C*-Cliffordalgebra
Cl1(H)c.
For each n~ {1,···, k},
define(it
issupposed
that none of theÀn’s
iszero),
and let e be theproduct
03A0kn=1e03C3(n).
Anelementary computation
then shows that Xe =2i(03A3kn=1 |03BBn|)e.
Hence2i(03A3kn=1|03BBn|)
is aneigenvalue
for theoperator
of leftmultiplication by
X in theC*-algebra Cl1(H)c.
It follows that thenorm of X in
Ch (H)e
is at least aslarge
as the modulus of2i(03A3|03BBn|),
hence
that ~X~~ ~
203A3kn=1 /Ànl,
hencethat ~D03C10(X)~1
=4~X~~.
Step
two. Let X be any element ofspin (H; Co).
Then there exists anorthonormal basis of H with
respect
to which theoperator D03C10(X)
hasthe matrix
representation
where
À1, ... , 03BBk
are non zero real numbers.By proposition 6(iv),
itfollows that the element X is
equal
to03A3kn=1 03BBn[e2n-1, e2n],
so that theargument
of step oneapplies.
The case E = + 1 is similar.Let
Cl03B51(H)
be the involutive real Banach-Liealgebra
definedby
the
C*-algebra Cl03B51(H),
and letCl03B51(H)inv
be the group of invertibleelements in
Cl03B51(H),
which is a Banach-Lie group with Liealgebra
Cl03B51(H).
LetSpin (H; C1)
be the group ofunitary
even elementsu ~ Cl03B5+1(H)
such thatuHu-1 =
H.PROPOSITION 13. The group
Spin(H; C1) defined
above has aunique struc-
ture
of sub
Banach-Lie groupof Cl03B51(H)inv,
with Liealgebra spin (H; C1).
PROOF. It is evident that exp X E
Spin (H ; Ci)
for all X Espin (H ; Cl ).
On the other
hand,
the Liealgebra spin (H ; Ci)
is stableby Ad(u)
forany
uESpin(H;C1); indeed,
for anyXEspin(H;C1): Ad(u)(X) =
uXu-1;
if X is of theparticular
form[y, z] with y, z E H,
thenuXu-1 =
[uyu-1, uzu-1] ~ spin (H; C1); by linearity
andby continuity
of themultiplication
inCl03B51(H),
it follows thatuXu-1
is inspin(H; C1)
ingeneral. Proposition
13 follows now fromgeneral principles
in thetheory
of Banach-Lie groups(see
Lazard[19],
corollaire21.10). e
9. A result of Shale and
Stinespring
The group ofthose
*-automorphisms
cp ofCl03B51(H)
such that~(H)
= His denoted
by Bog(Cl03B51(H))
and isclearly isomorphic
to the groupBog(Cl03B52(H))
defined in section 6. The canonical groupisomorphism
O(H) ~ Bog(Cl03B51(H))
is
again
denotedby
Cl.The group of those
orthogonal operators
on H of the formidH +
Twhere T is trace-class is a Banach-Lie group denoted
by O(H; C1),
whose connected
component
is denotedby O+ (H; Cl ),
and whose Liealgebra
is thato(H; Ci)
defined in section 8(see [14]).
PROPOSITION 14. Let U E
O (H).
ThenCl(U) is
inner and evenif
andonly if U ~ O+ (H; Cl ).
INDICATIONS FOR THE PROOF.
Step
one. Let U eO+ (H; Cl ).
Then there existsY E o (H; C1 )
such thatU = exp Y
(it
is an easycorollary
ofproposition
II.15.B in[14],
of whichthe
proof
followsclosely
Putnam and Wintner[22]).
LetDpl
be as inproposition 12,
let X =(D03C11)-1(Y)
Espin (H; C1)
and let u = exp X ESpin(H; C1 ). Then,
for any z E H:where adX is as
usually
the mapIt follows that
Cl ( U )
=LuRu -1,
so thatCl(U)
is inner and even.Remark. Let U E
O(H)
be such thatCl(U)
is inner and even, and suppose moreover that one can chose u in the connectedcomponent
of the Banach-Lie groupSpin(H ; Ce)
such thatCl(U)
=LuRu-1.
Thenthere exists a finite number of elements
X1,···, Xn
inspin (H; C1)
suchthat u =
exp(X1) ·· exp(Xn); hence, by
astraighforward computation:
for all z E
H,
so that U EO+ (H; Cl ).
We will see later that the Banach-Lie groupSpin(H; C1 )
is indeedconnected,
but we do not know that at thispoint
of ourargument.
Hence we must call on:Step
two. Let U EO(H)
be such thatCl(U)
is inner and even; then U ~O+ (H; C1).
We refer to Shale andStinespring [23 ] or Slawny [24]
for the
proof
ofstep
two, which is a consequence of results about those canonical transformations which areimplementable
in all Fock represen- tations of thecomplex C*-algebra Cl03B5~(H)c (terminology
as in[24]).
Alternatively,
it seems that a directproof
ofstep
two could follow from lemmas 1 and 2 in Blattner[4]. e
10. The universal
covering
pi :Spin(H; C1)
~O+(H; C1 )
and the
homotopy type
of theSpin
groupCorollary
9 andproposition
14imply
that thediagram
is commutative and that pl is onto. As
Dpl
is a continuous(hence smooth) isomorphism by proposition 12,
and as each of theexponential
maps is a local
diffeomorphism
whose derivative at theorigin
of the Liealgebra
is theidentity,
it follows that pi issmooth,
that it is a localdiffeomorphism,
and that the derivative of p 1 at theidentity
ofSpin(H; C1)
isprecisely Dpl.
LEMMA 15. The Banach-Lie group
Spin(H; C1 )
is connected andsimply
connected.
PROOF. Let K be the
homomorphism
ofZ,
inSpin (H ; Ce)
whoseimage
consists of eoand - eo.
As the sequenceis exact, as p 1 is
smooth,
and as the fundamental group ofO+ (H; C1)
is
precisely Z2 ,
it is sufficient to check that the twopoints
eo and - eo in the kernelof pl
can be connectedby
an arc. But let xand y
be twoorthogonal
unit vectors inH;
then xy =[1 2 (x - y), 1 2 (x + y) ] ~ spin (H; C1)
and the continuous arc in
Spin(H; Ci) given by exp(txy) = (cos t)eo + (sin t)xy
connects eo(t
=0) to - eo (t = n). 8
We can now sum up the results obtained so far as
follows,
in the case03B5=-1:
THEOREM. Let
Cl,,,, (H)
be theC *- Clifford algebra of
thequadratic form x ~ -|x|2
on the real Hilbert spaceH,
which isseparable
andinfinite
dimensional. Let
Spin(H; C1)
be thesubgroup of
the groupof
invertibleelements
Ch (Hynv consisting of
those evenunitary
elements u ECl’ (H)
such that
uHu -1 -
H. Letspin (H; C1 )
be the closedsubalgebra of
theBanach-Lie
algebra defined by Cl1(H)
which isgenerated
as a vectorspace
by the products [x, y]
with x, y E H. Then there exists onSpin(H; C1 )
a
unique
structureof
sub Banach-Lie groupof Ch (Hynv
with Liealgebra spin (H; C1 );
this Banach-Lie group is connected in its owntopology and
the map
is the universal
covering of O+ (H; C1);
moreover,if ~~1
is the nuclearnorm on operators on H and
if 111100
is the normdefined
on theC*-algebra Ch (H), then ~D03C11 (X)~1
=411Xll00 for
all X Espin (H; C1).
Similar statements hold in the case 03B5 = + 1.
Let now
(ek)keN*
be an orthonormal basis of H. LetClk
be the Cliffordalgebra
of the formFor all k
e N*,
the inclusionClk ~ Cl1(H)
induces an inclusionSpin (k) ~ Spin (H; C1 ).
These define aninclusion j
of the inductive limitSpin (~)
of theSpin (k)’s
inSpin (H; C1).
PROPOSITION 16. The inclusion
Spin (~) Spin (H; C1)
is ahomotopy equivalence.
PROOF. Consider the commutative
diagram
All vertical lines are Serre
fibrations;
as i is ahomotopy equivalence (see
references in
[14],
section11.6),
the five lemmaimplies that j
is a weakhomotopy equivalence.
ButSpin(~)
andSO(~)
have both thehomotopy type
of an ANR(see
Hansen[13], corollary 6.4),
andSpin(H; C1 )
andO+(H; C1 )
are ANR(see
Palais[21],
th.4);
hence Whitehead lemmaapplies (see [21 ],
lemma6.6) and j
is ahomotopy equivalence.
11. Hilbert
spin
manifoldsLet M be a smooth real
manifold, locally diffeomorphic
to a Banachspace B.
(We
suppose Mparacompact,
connected and withoutboundary;
and B
separable
andhaving
smoothpartitions
ofunity;
it follows that M isseparable
and has smoothpartitions
ofunity.) Let 1 :
E ~ M be avector bundle over M whose fiber is a
separable
real Hilbert space H. A Riemannian nuclear structureon 03BE [resp.
an oriented Riemannian nuclear structure, aspin
nuclearstructure] is
a reduction of the structural groupof Z
toO(H; Ci) [resp. O+(H; Cl ), Spin(H ; Ci)],
and the bundle with such a structure will be denotedby 03BE [resp. 03BE+ , 03BESpin]. Any
bundle suchas 03BE being
trivialisable(Kuiper’s theorem),
such a reduction of structure isalways possible. However, if 03BE
hasalready
been furnished with one of these structures, it may not bepossible
to reduce the structural group further. Forexample:
PROPOSITION 17.
Let 03BE
be a Riemannian nuclear vector bundle over M.Then ç is
orientableif
andonly if
thefirst Stiefel Whitney
classw1(03BE)
EH1(M; Z2)
vanishes.Proposition
17 isstandard;
infact,
much morecomplete
results in this direction have beenproved by
Koschorke[18] (his propositions
6.2. and6.3).
Apossible
methodof proof is
that used below forproposition
18.PROPOSITION 18.
Let 03BE+ be an
oriented Riemannian nuclear vector bundle over M.Then 03BE+
has(at least)
onespin
structureif
andonly if
thesecond
Stiefel- Whitney
classw2(03BE +)
EH2(M; Z2 )
vanishes.PROOF. The exact sequence of
topological
groupsinduces an exact sequence of the
cohomology
sets(see
Hirzebruch[15],
3.1 and
2.10.1)
and ç+
can be considered as an elementof H1(M; O+(H; Ci)).
Hence itis sufficient to prove that v = w2.
By naturality,
it is sufficient to do sowhen M is the
classifying
spaceBo+
of the groupO+(H; Cl ).
As theusual inclusions
SO(~) ~ O+(H; C1 )
arehomotopy equivalences,
H1(Bo+ ; Z2 )
=Z2 . Consequently, depending
of its value on theclassify- ing
bundle overBo +,
the map v is eitheridentically
zero, or is the class w2. In order to exclude the firstalternative,
it is sufficient to make surethat there exists at least one manifold M and one
bundle 03BE +
over Mwhich has no
spin
structure. Suchexamples being
wellknown,
it followsthat v = w2. This
result,
in finitedimensions,
is due toHaefliger [12].
Proposition
18 does notdepend
on theexplicit
construction ofSpin(H; C1 ),
butonly
on its universalproperty
ascovering
group ofO+(H; Cl ).
Infact,
thisproposition
is apreliminary step
for the follow-ing project,
thecarrying
out of which will indeeddepend
on ourexplicit
construction:
investigate
theproperties
of the differentialoperator
defined overSpin(H; Cl )-manifolds
as the Diracoperator
is definedover finite dimensional
spin
manifolds(see
forexample
Milnor[20]
andAtiyah-Singer [3],
section5).
Motivations forstudying
infinite dimensio- nalelliptic operators
can be found in Dalec’kii[7].
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(Oblatum 28-VI-1971 & 1-VI-1972) Mathematics Institute University of Warwick Coventry (G.B.)
Present professional address:
Institut de Mathématiques Collège Propédeutique de Dorigny Université de Lausanne
1015 Lausanne (Suisse)