Section 10 Locally Uniform Transformation Groups

156

10.

SECTION 10

LOCALLY UNIFORM TRANSFORMATION GROUPS

In classical Lie group theory, i t is often the case that analytic properties follow from algebraic-topological conditions. For example, every closed subgroup of a finitedimensional real Lie group is a Lie group in the relative topology and every continuous homomorphism between real Lie groups is analytic. Lie groups can also be characterized in algebraic-topological terms as locally compact groups having "no small subgroups" [ 7 7 ] . In the following two sections it is shown that similar results hold also in the infinitedimensional case of Ranach Lie groups. The basic idea is to consider actions on Banach manifolds which satisfy a strong continuity condition. 10.1 DEFINITION. Suppose r is a continuous action of a topological group G on a Banach manifold M over K A local representation of r with respect to a chart (P,p,Z) of M about o is a continuous mapping

.

10.1.1 where S i s a neighborhood of e E G and D domain in Zc : = ZOKC such that every g E S

is a bounded satisfies

10.1.2 and

whenever

m

E

p-l(D)

and

r(g,m)

E

P

.

10.2 LEMMA. Let r# : S + Om(D,Z C be a local representation of r with respect to the chart (P,p,Z) M about o Then r#(e) = idD and

.

for all z in a neighborhood of gh belong to S

.

0 E D

whenever

g,h

of

and

LOCALLY UNIFORM TRANSFORMATION GROUPS

157

The first assertion follows from 1.11. Now suppose Then 10.1.2 implies n : = r(h,m) E p-'(D) and g,h,gh E S r(g,n) = r(gh,m) E p-I(D) for all m in a neighborhood of o E P. Hence 10.1.3 implies

PROOF.

.

0.E.D.

Now the assertion follows from 1.11, 10.3

DEFINITION.

group G on if for every M about o with respect

A

continuous action

r

of a topological

a Ranach manifold M is called locally uniform point o E M there exists a chart (P,p,Z) of such that r has a local representation r# to this chart.

PROPOSITION. Every analytic action r of a Ranach Lie group G on a Banach manifold M is locally uniform. 10.4

PROOF.

Choose a chart

(P,p,Z) of

M

about

o

,

an open

neighborhood S of e E G and a bounded domain D in Z c -1 containing 0 , such that r(S,o) c p (D) and there exists an analytic mapping F : S ~ +D zC satisfying F(g,pm) = p(r(g,rn)) whenever (g,m) E Sxp-l(D) and r(g,m) E P We may assume that there exists a convex open neighborhood T of 0 E g such that exp : T + S is bianalytic and

.

ZED where D1 denotes the first partial derivative with respect Then r#(g)(z) := F(g,z) defines a continuous to X E T C S + O,(D,Z ) which is a local representation mapping r# : Q.E.D. of r

.

.

LEMMA.

.

C

Let r# * S + O,(D,Z ) be a local representation of r with respect to a chart (P,p,Z) of M about o and let C be a convex open neighborhood of 0 E D such that R : = dist(C,aD) > 0 Suppose g E S 10.5

.

158

SECTION 1 0

satisfies

j < k

for

1

gj

,

1 < j < k

for

S

E

where

I

r#(gj)-id

d

<

R

.

and

In

< d/2

Then every

1 -(r (gk,z)-z) - (r#(g,z)-z) k #

I

<

d

10.5.1 C

z

E

satisfies

I

r # (g,z)-z

I

and

PROOF. For (g,z) E SxD define F(g,z) : = r (g,z)-z # Then 10.5.1 implies suppose z E C

.

W

:= r#(g,z)

C1 : = u R / , ( C )

E

.

0 < j < k

,

1.17.1

Now

By 1.11, we have

since this identity holds in a neighborhood of according to 10.2. It follows that

For

.

0

E

C n p(P)

implies

.

since C1 is convex and satisfies dist(Cl,aD) > R / 2 This implies the first assertion, and the second assertion is an O.E.D. immediate consequence. 10.6

COROLLARY.

Suppose

is a topological group r on a connected Then G has no small subgroups, i.e., manifold M exists a neighborhood S of e E G such that every of G contained in S is trivial. G

a faithful locally uniform action

.

PROOF.

Consider a local representation of r with respect to a chart ( P , p , Z ) choose C C D as in 10.5. Then

r# : of

M

admitting Ranach there subgroup

S + om(D,ZC)

about

o

and

LOCALLY UNIFORM TRANSFORMATION GROUPS T := { g

E

:

S

159

(r#(g)-idlD < R/4 }

.

is a closed neighborhood of e E G Now suppose H Then 10.5 implies subgroup of G contained in T

.

is a

.

k > 1 Hence r (g) = idC , showing that r(g,m) = m for all m in a # neighborhood of o E P Since M is connected, 3.1 implies for all

g

r(g) = idM

and a l l integers

H

E

.

.

Since

r

is faithful, we get

H = {e}

.

O.E.D.

10.7

COROLLARY.

A Ranach Lie group

G

has no small

subgroups. PROOF.

G

Since

is locally connected, the identity component

of G is an open subgroup. Hence we may assume that G is connected. The faithful left translation action of G on M :=

G

is analytic and hence locally uniform by 10.4.

Now O.E.D.

apply 10.6.

A s the main result of this section, it will be shown that

for real Ranach Lie groups, the converse of 10.4 is also true, i.e., every locally uniform action of a real Ranach Lie group case

G G =

is in fact analytic. R

.

r

We consider first the special

is a locally uniform action of Consider a local representation r # : S + O O D ( D , Z L ) of r with respect to a chart (P,p,Z) of M about o and let C be a convex open neighborhood of 0 E D such that R := dist(C,aD) > 0 Then the limit 10.8 THEOREM. Suppose R on a Ranach manifold

M

.

.

h := lim (r (t)-id)/t # t+O exists and there exists IaxC 3 (t,z)

0

+

C

> 0 such that

r#(t,z)

defines an analytic mapping.

10.8.1

Om(C,Z )

E

E

2

C

IaC S

and

SECTION 10

160

.

PROOF'. Put d := R/3 and define Ck : = Ukd(C) By 1.17, af(z) : = f ' ( z ) defines a continuous linear mapping

a

C : om(D,2 ) -+ A :=

Hence there exists

T

>

C

om(C2, L(2

such that

0

.

))

and

IT C S

\r#(t)-idI,, < d

10.8.2

and (r#(t)'-e

A

1

c2

< 1

10.8.3

for all t E IT , where e ( z ) = id denotes the unit element A Define of the associative Ranach algebra A

.

where

0

< T <

T

.

Then af =

IT r#(t)'dt 0

is invertible, since 10.8.3 Define

h

E.

C

om(C2,2

A

E

IT-1 oaf-e

implies

by

)

h(w) : = af(w)

-1

(r#(T,w)-w)

A

I

c2

< 1

.

. 10.8.4

Now suppose (t,z) E IrxC , Then 10.8.2 implies If Is1 is small, we have w := r#(t,z) E C 2 r#(s,C2) C D and hence 10.2 implies r (t,w ) = r (s+t,w) # s # where w S : = r#(s,w) It follows that

.

.

=

Is+T r#(t,w)dt - IT 0

S

-

Is

0

r#(t,w)dt =

r#(t,w)dt = IS(r#(T+t,w) 0

-

,

Is+T r # (t,w)dt

T

r#(t,w))dt

.

Therefore 1 lim -(f(ws)-f(w)) s+o

= r#(T,w)

- w ,

10.8.5

LOCALLY UNIFORM TRANSFORMATION GROUPS

Define a continuous mapping 1

A(u) : = Then

0

A : C2

f'(w+t(u-w))dt

A(w) = f'(w) e G & ( Z C )

.

+ L(Z

C

)

161

by

.

, and [28; 8.14.31

implies

It follows that A(ws) f(w S )-f(w) = A(ws)(ws-w) if I s 1 is small and 10.8.4 and 10.8.5 imply

E

G L ( ZC )

It follows that the mapping ITxC 3 (t,z)

is differentiable in equa t ion

+

t

r # (t,z)

E

C2

and satisfies the differential

a r (t,z)

h(r (t,z))

=

10.8.6

#

at

.

with initial condition r#(O,z) = z By 5.7, there exists a local analytic flow F : IU xC + C 2 , where 0 < u < T , with infinitesimal generator

The uniqueness theorem for solutions of ordinary differential equations [28; 10.5.21 implies that r# 1 IU xC = F is analytic. Since 10.8.6 implies r (t,z)-z = #

it follows from [28; 8.5.41

for all

t

E

Ih'lC < +1 exists.

IT

since

0

,

that

r#(s,C) C C 1

by 10.8.2.

Since

by 1.13, it follows that the limit 10.8.1

COROLLARY. Banach manifold 10.9

,

It h(r#(s,z))ds

Q.E.D.

Every locally uniform action of M is analytic.

R

on a

SECTION 10

162

DEFINITION. For a locally uniform action r topological group G on a Ranach manifold M , let

10.10

of a

denote the set of all complete analytic vector fields M for which there exists a continuous homomorphism R 3 t + gt E G satisfying r(gt)

=

.

exp(tX)

X

on

10.10.1

We say that the continuous 1-parameter subgroup (g,) of G satisfying 10.10.1 is associated with X It is clear that tX E g M whenever (t,X) E R x g M'

.

10.11 COROLLARY. Suppose r is a locally uniform action of Consider a a topological group G on a Ranach manifold M local representation S + Om(D,Z") of r with respect r# to a chart (P,p,Z) of M about o and let C be a convex

.

--

open neighborhood of 0 there exists a mapping

E

D

P # : 9,

such that

.+

dist(C,aD) > 0

.

Then

0JC,ZC)

uniquely determined by the identity (P#X)OP = xp

I

10.11.1

.

holding on p -1 ( C ) For (t,X) E R x g H , we have p#(tX) = top (X) If (gt) is a continuous 1-parameter # subgroup of G associated with X , then (t,z) + r#(gt,z) defines a local analytic flow on C with infinitesimal genera tor

.

X# : = P#(X)

a

.

PROOF. Since C is connected, 1.11 implies that P # is uniquely determined by 10.11.1. Now suppose X E g, and let (gt) be a continuous 1-parameter subgroup of G associated with X Then 10.8 shows that the limit

.

LOCALLY UNIFORM TRANSFORMATION GROUPS

163

p#X : = lirn (r#(gt)-id)/t E c)oD(C,ZC ) t+O Since r#(gt,pm) = p(r(gt,m)) = p(exp(tX)(m))

exists. all m E p-l(C) choice of 10.12

,

(9,)

it follows that

p#X

does not depend on the O.E.D.

and satisfies 10.11.1.

COROLLARY.

For all

X,Y

E

gM

,

for

we have

( X +Y )id = lim (r ( g h )-id)/t # # # t t t+O

E

O-(C,Z C )

and [ X ,Y lid = lim (r#(gth,g_,h-,)-id)/t2

#

#

t+ 0

E

om(C,Z C )

Here ( g t ) and ( h t ) are continuous 1-parameter subgroups of G associated with X and Y , respectively. Put R := dist(C,aD) , d := R/6 and define C k : = ukd(C) We may assume that Ir#(g)-idl,, < d for all Let T > 0 satisfy gshtgslhtl E S whenever g E S For (t,z) E I x C put s,t,s',t' E I T T wt := r#(ht,z) E C 1 and zt : = r#(gt,wt) E C2 Then the mean value theorem [28; 8.6.21 and 5.12.3 imply PROOF.

.

.

.

.

where the limits

and

exist according to 10.8.

and

Since [28; 8.5.41

implies

.

S E C T I O N 10

164

< a 1 (r#(gs)-id( *lhYICl + Ir (h #

c2

the first assertion follows.

Since

[ X ,Y lid(z) = hi(z)hy(z) #

1.13 implies

#

[X ,Y ]id

1

; i lhYIC2

s

-

,

h;(z)hX(z)

.

om(C,Zc) For (t,z) v ut := r#(hqt,z) E C 1 , t : = r#(gVt,ut) E C 2 w t := r#(ht,vt) E C3 and zt : = r#(g,,wt) E C4 Taylor's formula [28; 8.14.31, applied to p = 2 imply #

#

E

I

I (r#(g,htg_,h-,,z)-z)/t2

E

.

,

I xC

put

T

Then and 5.12.2

- [X#,Y#lid(z) I

1

t + 0

For on

,

C

,

the last expression converges to

0

, uniformly O.E.D.

as a consequence of 5.12.5.

10.13 LEMMA. Suppose r is a locally uniform action of a Consider a topological group G on a Banach manifold M C local representation r S + Om(C,Z ) of r with respect # . to a chart of M about o and let B be a convex open neighborhood of 0 E C such that R := dist(R,aC) > 0 Then for any h E Om(C,Zc) , the vector field

.

.

X

:=

h(z)

a

E

T(C)

generates a local analytic flow rh : IxR t E I has the following property: Whenever knt

+

+- , gk

the limit

gn

E

E

S

G

+ C

such that every

and kn E R are sequences satisfying for all k < qn : = [knt] and such that

LOCALLY UNIFORM TRANSFORMATION GROUPS

h = lim kn(r#(gn)-id) n+exists, it follows that

Here

E

Om(C,ZC)

r h : ITxR1

+

C

Since

IhtIBl

.

and the mappings

ht : = (rh(t)-id)/t satisfy

.

and define R1 : = U d ( B ) By 5.7, u , >~ 0 such that X generates a

PROOF.

local analytic flow

10.13.1

< q

[q] denotes the greatest integer

Put d : = R/2 there exist constants

165

0/2

rh(t) = t*ht+id

Ih;IB,

I

<

0

E

om(Bl,Z C

and

, the mean value theorem implies

Irh(s,z)-rh(s,w)I

Iz-WI

10.13.2

(l+(slo)

.

for I s ( < 'I and z,w E B We may assume U T < d suppose g n E G and k E R are sequences as above. n tn : = t/qn E I T and hn : = (r#(gn)-id)/tn E o , ( C , Z C )

.

Now Put

.

Since

kn/tn

+

1

and therefore

Ih

fn : = r#(gn)

or

,

I

I

+ 0 for B1 For n < u for (almost) all := r (t ) , it follows that n h n

10.13.1

implies

Ifn-idl B1

We now show by induction that

< d/qn

Ih-h

.

.

k fn(R) C B1

n +

m

10.13.3

and

whenever 0 < k < qn , Th s is clear for k = 0 , and assuming the assertion for some k < qn , it follows that fk+l is well-defined on R and 10.13.3 implies n

166

S E C T I O N 10

+ (fn-idIB k < d/qn + kd/qn Since k+l < qn hand, we have

,

this implies

fk+'(B)C

R1

.

. On the other

and

O.E.D.

10.14 THEOREM. Suppose r is a locally uniform action of a Hanach Lie group G on a Ranach manifold M Then there exists a locally uniform action r* of the Lie algebra of G on M , called the differential of r , such that the following diagram commutes

.

4

A

r*

aut(M)

PROOF. F o r (t,X) E R x g I rx(t) := r(exp(tX)) defines a locally uniform action of R on M By 10.8, rx defines a global analytic flow on M Let r,(X) E aut(M) be the

.

.

LOCALLY UNIFORM TRANSFORMATION GROUPS

infinitesimal generator of

rX

commutes and

. .

167

Then the diagram 10.14.1

r*(tX) = t r*(X) In order to show that is a homomorphism of real Lie algebras, C consider a local representation r # * S + O o J ( D , Z ) of r Let B with respect to a chart (P,p,Z) of M about o r* : g + T ( M )

-

.

be a convex open neighborhood of 0 E D such that R : = dist(B,aD) > 0 Put d : = R / 2 and C := Ud(13) Given X,Y E g , define may assume r(S,o) C p-'(C)

.

9,

exp(tX)

:=

, qt

:=

.

exp(tY)

and put

X1 : = X+Y

E

.

We

T(M)

X2 : = [X,Y] E T ( M ) , h : = ( X# +Y # )id E om(C,Z C ) and IC ) h2 := [X ,Y ]id E Om(C,Z For t > 0 and s := t1j2 #

define

.

#

, ,

and 2

n'

: = 's/n

+s/n +-s/n '-s/n

By 10.13, the vector fields

generate local analytic flows r : I T x R + C having the j property stated in 10.13. Using the Campbell-Hausdorff series [21; 11.71 we may assume that for k < n j , whenever that

since

is closed.

S

r#

contains all powers It1 < T We may also assume

S

.

Then 6 . 7 implies

is continuous,

exp(tX.1 3

E

S

and

By 10.12, we have

Therefore 10.13 implies

.

Hence r.(t) = r (exp(tX.)) on B whenever 0 C t < T By 3 # 3 differentiation, it follows that (X.p)(m) = h.(pm) for all 7

3

SECTION 10

168

.

m E p-l(B) showing that r* is an action of 9 on M * g M + O_(C,Zc) as in 10.11. Then r,(g)CgM Now define p# * and the mapping p#or, : g is real-linear since

holds on p-l(C) all g E S Let 0 E 4 such that gk : = exp(X/k)

.

.

-

.

r*

+

C Om(C,z )

10.14.2

is real-linear and the identity

We may assume that (r#(g)-idlD < d/4 for T be a star-like open neighborhood of exp(T) C S For X E T and k > 1 , put Then 10.5 implies

.

klr # (gk )-idlC c 2 r#(exp X)-idlC

6

d/2

.

.

Hence For k + , we get Ip#(r+X)lc c d/2 for all X E T 10.14.2 is a continuous mapping, show ng that the action r* O.E.D. of g on M is locally uniform. 10.15 THEOREM. An action r of a Banach Lie group G on a Banach manifold M is analytic if and only if r is locally uniform and its differential r* : g + aut(M) C J(M) is linear. By 10.4 and 6.12, every analytic action r is locally uniform and has a linear differential. Conversely, suppose that r is a locally uniform action whose differential r* is linear. Since the action r* of g on M is locally uniform by 10.14, 5 . 3 0 implies that r* is an analytic Lie O.E.D. algebra action. Now apply 5.32. PROOF.

10.16 COROLLARY. An action r of a real Ranach Lie group G on a Banach manifold M is analytic if and only if it is locally uniform. 10.17 COROLLARY. Suppose r is a locally uniform action of a topological group G on a Banach manifold M Let H be a real Lie group and let 4 : H + G be a continuous homomorphism, Then roe : H + Aut(M) defines an analytic

.

169

LOCALLY UNIFORM TRANSFORMATION GROUPS

action of

H

on

M

.

PROOF. Since r is locally uniform and 4 is continuous, the action ro$ is locally uniform and hence analytic. O.E.D.

10.18 COROLLARY. Every continuous homomorphism between real Lie groups is analytic.

4 :

H + G

PROOF. Apply 10.17 to the (analytic) left translat on act on Q.E.D. of G on G

.

10.19

COROLLARY.

A continuous homomorphism

4 :

G + H

between complex Lie groups is holomorphic if and only if its differential $ * : 4 + h is complex-linear. PROOF. Since 4 is real-analytic by 10.18, 4 is holomorphic if and only if Te(4) : Te(G) + Te(H) is complexO.E.D. linear. It will now be shown that, for locally compact groups and manifolds, every continuous action is already locally uniform. In general, the set C(M,N)

:=

{ f

: M + N

: f

continuous ]

of all continuous mappings between locally compact spaces M and N will be endowed with the compact-open topology [921 having a subbasis of open sets (K;Q) := { f

C(M,N)

E

: f(K) C Q

]

for all compact subsets K of M and all open subsets 0 of N In case N is a uniform space, the compact-open topology coincides with the topology of uniform convergence on

.

all compact subsets of

M

.

10.20 PROPOSITION. Every continuous action r of a topological group G on a locally compact complex manifold M is locally uniform.

170

SECTION 1 0

PROOF.

Since

r : GxM + M

is continuous and

M

is locally

compact, the homomorphism

is continuous [92]. Let (Q,p,Z) be a chart about o E. M and let P be an open connected neighborhood of o E Q such Then D := p ( P ) is that P is compact and contained in Q a bounded domain in 2 and S : = { g E G : r(g) E ( 7 ; Qn ) ( o ; P ) } is an open neighborhood of e E G For (g,m) E SxP , define O.E.D. r#(g)(pm) : = p(r(g,m))

.

.

.

Note that a connected Ranach manifold

M

over

K

is

locally compact if and only if M is finite-dimensional. For locally compact groups, we have the following somewhat deeper result. 10.21 THEOREM. Every continuous action r of a locally compact group G on a locally compact manifold M is locally uniform. PROOF. Choose a chart (O,p,Z) of M about o , a compact neighborhood T of e E G and a fundamental system { Bn : n E N } of connected open neighborhoods €3, of -1 0 E Zc such that On : = p (R,) satisfy r(T,Qn) C Q Then pn(f) := fop defines an injective continuous linear

.

mapping

We may assume that there exists a compact neighborhood K of p(Q) in Zc Ry Montel's Theorem [log; 1.61, the set

.

Kn

:=

{ f

E

C)(Rn,ZC

: f(Rn)

C K }

is compact. Since 4,(g) := pOr(g))Qn defines a continuous -1 (pnKn) mapping +n : T + C(Qn,Z C ) , it follows that Tn : = $n is a compact subset of T and an := p n-1 0 4 ~: Tn is a + Kn continuous mapping. For every g E T , K is a neighborhood

LOCALLY UNIFORM TRANSFORMATION GROUPS

.

of p(r(g,o)) E Zc Hence there exist such that the diagram

commutes.

Ry definition,

g

E

Tn

and

un

n

E

N

171

and

.

f = an(g)

f

Since

is a Raire space [ 9 2 ] and T = T, , Tn has an interior point h for some n Since m : = r(h,o) E Q

G

.

there exist open neighborhoods p(m)

E

N

of

m E

Q

and

C

Kn

E

,

of

such that there is a commuting diagram

Zc

where f is holomorphic. N o w choose a closed neighborhood S of e E T with hS C Tn and a connected open neighborhood B of 0 E Bn such that r(hg,o) E N and

.

Then r(g,o) E Q an(hg)(R) C C for all g E S r#(g) : = foan(hg) defines a continuous mapping C S + O(R,Z ) satisfying r# :

and

r#(g)(pm) = f(p(r for all

m

E

p -1 ( B )

.

Now

et

D

be a relatively compact P : = p -1

connected open neighborhood of 0 E R , define and replace S by { g E S : r(g,o) E P }

(D)

.

NOTES.

Q.E.D.

The concept of locally uniform (or "strongly continu-

o u s " ) action as well as the proof of Theorem 10.8 are due

to W. Kaup. The main results 10.14-10.19 appear in C137,1381. For a direct proof of 10.18, see C 2 1 ; Ch. 111, 5 8, nol, Thgor6me 1 1 . The technical results 10.5 and 10.13, of basic importance in this section and in Sections 11 and 13, are due to H. Cartan (cf. [log; Ch. 91 and C106; Ch. V,

5

2, Lemma 11).

In the more general setting of analytic spaces of finite dimension, Theorem 10.21 is proved in C801.