Automorphisms for locally compact groups
1.
Let
G be a locally compact group and
Aut(G)
the group of
all its topological automorphisms with the Birkhoff topology.
A
neighborhood basis of the identity automorphism consists of sets
N(C,V) = {9EAut(G): e(x)EVx and e- 1 (x)EVx, all xEC},where C
is compact and
V is a neighborhood of the identity
is well known, Aut(G)
[1;p.57].
ry
of inner automorphisms.
locally compact
particular, if
G • As
In this article we are mainly
concerned with the topological properties of
Int(G)
of
is a Hausdorff topological group but not gen-
erally locally compact
group
e
Aut(G)
Aut(G)
and its sub-
We prove that for
G arbitra-
is a complete topological group.
G is also separable
Aut(G)
In
is a Polish group. As
far as we can determine this result is new; and of course, this fact
will be useful for the further study of
Aut(G).
Furthermore, we
give two new characterizations of the topology for
and 1.6.).
In Section 2
-subgroups (among them
Aut(G),
(1.1.
we turn to the question of when certain
Int(G) ) are closed in
Aut(G) , and several
equivalent conditions are given; for instance, Int(G)
G admits no nontrivial central sequences (2.2).
is closed iff
Applications to
more special classes of groups are also given, as well as to the
question of unimodularity of
Int(G) ,
(2.5).
We remark that there
is no separability assumption on the groups before 1.11.
1.1. Lemma.
= {T
where
The sets
w
'P1 ' • • • ' 'Pn; e:
E Aut(G)
cpj E Cc(G)
of the identity in
and
e > 0, form a basis for the neighborhoods
Aut(G)
a
- 2 -
Proof.
~1 ,
Let
n
••• ,~n E Cc(G)
U support (cp.) ,
F =
. 1
Icpi (x)- cpi (wx) I
such that
We claim
N(F,W) _sW
T(x)x- 1 E W
If
< e
for all
. rn ·e·
cp1, ••• ,,.n,
and T-1(x)x- 1 E W,
l~i (x)- cpi (T(x))l < e
( *)
l~i (x)- ~i (T(x))l
Let
I cpi (x)- ~i ( 'r(x)) I
1 ~ i! n
Thus, i'or
J
x
T E N(F,W).
1 ~ i ~ n.
Then for xEF,
,.-'\x) E F
Clearly if
~ F
x
and
'r(x)
~
we have the following sub cases:
and
,--1 (x) E F
(b)
T(x) E F
and
T- 1 (x) !_ F
(c)
'l"- 1 (x)
and
T(x)
(d)
T(X) 1. F
and
,.- 1 (x)
~
~ F,
~i (x) = ~i ( T(x)) = 0,
since in this case
F ,
so
then
< e ,
T(x) E F
F
~ F.
In each case (*) is satisfiede
Thus
T E N(F,W)
II~·cp. o,-- 1 1100 < e:, i.e.,
~
~
F c G be compact and
Conversely, let
in G
XT(x)- 1 E W,
i.e.
Similarly if
(a)
E F
w E W,
e
and
1_::i~n.
< e:,
Set
so
~~i (x)- cpi (,-- 1 (x))! < e:, 1_::i,:Sn.
then
x E G,
T- 1 (T(x))T(x)- 1 E W,
then
'f(x) E F;
be given.
W be a symmetric nbh. of
and let
~
~=
e > 0
and
implies
T
E W
cp1, ••• ,cpn;e:
•
W a neighborhood of e
•.
in
G.
u2.u-1 c
c
if'
Let
Let
u
w.
Let
and
be a compact neighborhood of
1jr E Cc(G)
1jr(u) >
(x1 , ••• ,xn}
{Uxi: 1 ~i_::n}
f,
covers
Indeed, suppose
j ,
V u E U.
and
F.
0 .::
in
G
Define
F
such that
'¥ ! 1 ' support ( 1lr)
(The existence of such a
be a finite subset of
We claim
for some
be such that
e
1jl
is clear.:
such that
wi E Cc(G) by 1jli (x)
W,1,
d• • ~ c N(F,W).
"'1'•••'"'n' 2
Then
x E Uxj
- 3 2
T(x) E U xj.
T (x
But then
w• s imilarly,
) x -1 E U2 xjx-1 c U2 U-1 c
I wj (x)- wj ( .,.- 1 (x)) I
t
<
.,.- 1 (x)x- 1 E W.
implies
This proves the
0
claim.
1 • 2. By Braccmrl.er [ 1 J there is a continuous (modular) homomorphism
A : Aut (G) .... JR+
A(a)- 1
where
dx
with the property
JG foa- 1 (x)dx JGf(x)dx,
is a fixed Haar measure.
it is easy to see that
algebra
G
for
=
L1 (G) •
""'
a
becomes
Denote by
A.
f E Cc(G) ,
Defining
an automorphism of the group
the 1 eft regular representation of
as well as the left regular representation of
Viewing
that
""
a
a,
a
E Aut(G),
L1 (G)
on
A.(L 1 (G)),
as an automorphism of
L 2 (G).
we show
can be extended to an automorphism of the von Neumann alge-
bra of the left regular representation,
a
We define a unitary operator
U 9g = A( 9 )-i go a-1
U ,
a
Qi(G)
= A.(L 1 (G))
E Aut (G) ,
11
=>..(G)".
by
,
A straight forward calculation shows
= u 9 A.(f) u 9- 1 •
A.(e(f))
The unitary implementation
for
T E fA?(G)
S(T)
1.3. Lemma.
(g E L 2 (G)) ..
9
~> u 9
allows us to define
B(T)
by
=
9
s- 1
U T U
The map
a E Aut(G)
~> Uag E L2 (G)
is continuous
- 4- -
Proof.
Let
neighborhood
G E Cc(G)
U1
of
e
e > 0
and
in
G and set
Lemma 1.1. there is a neighborhood
a. E N(C,U)
c C
iJ.
N(C, U)
giJc:o < e /2~(K) i
u1
U = u- 1 c
If
•
U•support (g) c K.
x E
neighborhood
Then if
Ua g-
1.4-.
a.
support (g)
and x E support (goa.- 1 ),
E Aut(G)
t
A
there is a
so that for
a. E N1 ,
..
a. E N,
g[l~
,::
JKII
ua. g-
a.
gl~ d!J.(x)
a.
u h-hll 2 ,:: II u h- u
Our next aim is to study
€ /1J.(K)
yt.
we have
h E L2 (G) is arbitrary, e > 0,
II Ua. g - gll 2 < e , a E N , then
II
We can assume
ua. g- glb = II A(a )-t goa. - 1 - gllc:o <
a.
support ( U g- g) c K
ll
If
so that
1
N = N1 nN(C,U).
II
If
Aut(G)
By continuity of
1
Since
in
a. E N(C, U)
of the identity
N1
G.
IA(a.)- 2 -11 < E:J2IIg!lc:oll(K) 2
Set
U1 • support (g) • By
'
is a left Haar measure on
and
then
K =
Fix a compact
implies
II goa. -1 where
be giveno
let
ua g- gllc:o
g E Cc(G)
a.
gll2+1l u
Aut(G)
II
,::
1-1CK)
with
< e2 •
llg-hiJ 2 <e.
g-gll2+llg-hll2 < 3e.
by embedding it in
and we shall prove that the embedding is topological
if
0
Aut(c1((G)),
Aut(~(G))
is provided with the appropriate topology, namely the uniform-weak
topology, and a neighborhood base at the identity
t
E Aut(~ (G))
is given by
(a EAut(O?(G)): j((o:.-t.)ut1 , cpi)j<e,cpi EOi((G)*' 1,::i,::n},
where
0( 1
denotes the unit ball in
dual, U((G) * , is the Fourier algebra
fR (G)
..
e>O,
Recall that the pre-
A( G) , [3]..
Let
- 5 -
W
= {a. E Aut(G) : llep1. -cp1 aa.!l < e:, 1 ~i ~n} , cp1. E A(G) ,
cp1 '• • • 'cpn; e:
11·11
where
Proof.
a. E Aut(G);
denotes the norm in
First note
(a(T),cp) = (T,cpaa.),
i.e., at(cp) = cpoa. :
(a(A.(f)),cp) = A(a)- 1
{>..(f) : f E L1 (G)}
Since
all
T E
f E L1 (G),
we have
the claim follows.
Taking the supremum over
we get
and the lemma follows.
1.6. Proposition.
The sets
Hence the embedding
cp E A(G)'
0
form a base at the identity
W
, cp1. E A(G) and e: > 0 ,
cp1' •• • ,cpn;e:
t E Aut(G) for the Birkhoff topology.
Aut(G) ~ Aut(L$((G) is topological.
We show first that the topology generated by the sets
W
is weaker than that of
cp1 ' o • o 'epn; e:
cp E A(G) , a. E Aut(G) ,
Writing
cp E A(G),
= (A.(f),cp•a.).
6\)(G) ,
T E ~1 •
((U'-t)W1,ep) ,;.jJcpoa.-a.l!'
ll cp -
T E d((G),
T = >..(f),
is dense in
((a-t )T,cp) = (T,cp•a.-cp),
Proof.
If
JG foa.- 1 (x) cp(x) dx
Now
0? 1
A(G).
cp 0 a. II = sup I<T TEw1
a(T)
sup I<CT-a(T))f,g)l
TE~1
-1
= su:e. I<CT-lf'Tua.
TE w'(1
)f,g)j
By Lemma 1 • 5 ,
•
f, g E L2 (G) ,
cp = (f*gl"oJ)V,
llcp-cpoa.ll =
' cp >I
Aut (G) •
we have
for
- 6 -1
I
-1
= sup (elf
TE 6?1
~ sup
I (elf
T - Tlf
-1
)f, tf
T- T )f, tf
-1
-1
g) I
g) I + su~
TE Uk1
TE(R1
-1
I ( eT -
uo.
T )f '1f1
-1
g) I
Now
all
By continuity of the map
e: > 0 ,
llf-lf
1
e
fll 2 1lgll 2 < ~'
<elf
-1
all
-1
T E
a? 1
-1
G•
Since
0
~
1Jr
(Lemma 1. 3) given
E Aut(G)
g)l
=
g)l
-1
I<Tf,g-tf
g)l
-1
gl12 '
•
N2
of
t
E Aut(G)
gll 2 < e:;2.
Letting
N = N1 ('\ N2 ,
F c G be compact and
U be a compact neighborhood of
llcp- cpoc:.l]
we get
< e:.
e
such that
w.
~ 1, 1jr(u)
Define
so that
W a neighborhood of e
A(G) is a regular algebra, there exists
3.2] o
so that
-1
Let
if~ u-1 c
t
f
Furthermore,
g!l2 ~ \1~112\lg-Ua.
Conversely, let
in
of
-1
-1
g)- (Tf, Ua.
Again there is a neighborhood
tf
Ua.
N1
-1
Tf, Uo:.
~ llTfll2llg-1fl
.
,..._.:>
a. E N1 •
-1
!lfll 2 llg-
1
•
g)
I<Tf,g)-(Tf,tf
all
Qt 1
-1
T- T)f ,ua.
= j(lf
=
+-> a.-
a.
there is a neighborhood
-1
T E
for
= 1
u E U,
and
support
~ E
AeG)
(~) c
-
with
if
[3; Lemma
{x1 , ••• ,xn} c F be so that (Ux. : 1 < i < n} covers F.
~
1
'¥· (y) = ljr(yx.- ), 1 <i <n. We claim W~n
rn • 1 c N(F,W).
Let
~
Indeed, suppose
l.
T
-
'~"1 ' " • • ''~"n'
-
EW
cp1, ••• ,cpn; 1
-
and let
x EF
o
Then
x E UxJ.
- 7
for some j •
lv j
o
11111 joT -111 jll < 1
-r(x) - lit j (x) I < 1 •
where
T(x)
Now
But for
x = uxj , u E U •
E
ifxj •
~
Hence
implies
!lilt j aT - ljt j lb < 1 ,
111-(x) = $.(ux.) =
J
J
J
T(x) E support ( $ j) , or
X
E UxJ.,
that
SO
~jt(u) = 1,
But then
-r(x)x-1 E u2 x .x- 1 E u2u- 1
J
W.
c
In addition
so the same argument as above yields
..., - 1 (x) E
Wx.
0
1.7. Corollary. Suppose G has small neighborhoods of the identity,
invariant under inner automorphisms (i.e., G E [SIN]).
the group
Int(G)
as a subgroup of
Aut(@(G)) , the pointwise-weak
and uniform-weak topologies coincide on
Proof.
As is well known,
Int(G).
G E [SIN]
if and only if
Qt(G)
is
The conclusion follows from [7; Propo-
a finite von Neumann algebra.
I]
sition 3.7].
Note that the above can just as well be stated for
groups
Then viewing
where
B c Aut(G)
is a subgroup.
Also, the corollary is
not too surprising in view of the fact that for
point-open and Birkhoff topologies of
[SIN]B-
Aut(G)
[SIN]-groups
agree on
the
Int(G)
[6; Satz 1 .. 6].
1 .. 8.
We say that
G is an
[FIA]B- group
(see [5]).
compact subgroup of
Aut(G)
quence of 1.6
G E [FIAJ:B
subgroup of
that
Aut(~(G))
if
B is a relatively
It is now a trivial conse-
if and only if
B , viewed as a
endowed with the uniform-weak topology, is
- 8 relatively compact.
Cf. (4; Theorem 2.4].
By [4; Corollary 1.6],
the pointwise-weak topology may be substituted for the uniform-weak
topologyo
1.9.
Next we show in an elementary way that for an arbitrary locally
compact group G,
Aut(G)
is a complete topological group (in its
two-sided uniformity).
Theorem.
Let
G be a locally compact group, then
Aut(G)
is
complete with respect to its two-sided uniformity.
Proof.
Let
(a.'V)
Aut(G) ~ oCCL2 (G))
be a Cauchy net in
Aut(G).
Since
a.
.,_> W,
is continuous in the strong operator topology, it
is also weakly continuous. Now W E .1. (L2 (G)) 1 ( = unit ball of
. 2
2
J (L
(G))); also the weak and ultraweak topology coincide on r£ (L (G)) 1
2
a.'V
and JC(L (G)) 1 is compact in this topology. Thus (U ) has a
point of accumulation U E cL(L 2 (G)) 1 ; let (a.~) be a subnet such
a.
2
that U ~ ~ U weakly. Then for f,g E L (G)
a. 'V
a. 'V
a.
a.
((U -U)f,g) = ((U -U ~)f,g)+((U ~-U)f,g)
=
a.-10.
a.-1
a.
(f-U v ~f, U 'V g) +((U IJ.-U)f,g)
a. -1 a.
a.
~ llf-U v llfll 2 1!g!l 2 +((U ll-U)f,g) (~J.,v)>
since
in Aut(G) •
operator topology.
V E oL'(L 2 (G)) 1 •
Let
a.
I (U
av
U
vU
in the weak
-1
Similarly ua.v
We claim V
be such that for
v0
Thus
0
v
=
u- 1 •
>- v 0
v Vf- UVf, g) I < e , and
converges weakly to some
Let
f , g E L 2 (G) ,
t:
> 0 •
- 9 v1
Choose
~
v
such that
v1
implies
a -1
a. -1
I (U v f- Vf U 'Yo g) I < e: •
v, 1J. .:> v 0
Then for
v1 ,
and
we have
a.-1
I (U IJ. u v f - UVf 'g) I
a.
a.
< I (U IJ.
a.-1
a.
a.
u v f- u IJ. Vf' g) I + I (U IJ. Vf - UVf' g) I
a.
I (U
where
1-L Vf,
a. -1
!<u
a
v f-U
g) I < e •
Also
-1
a.'J
1-lvr, g)l = l (U
-1
-1
-1
a.
f- Vf, U IJ. g) l
-1
a.
a.v
a.
< I (U v f- Vf 'u 0 g) I + I (U 'J f- Vf
-1
_4
a.v
11u
< e: +
a.IJ.
f..: VfU 2 lJu
a
-1
< e: + 2llfll 2 11u IJ. g-
so that
a. -1
a.
I (U IJ. u
'J
'
a.
-1
u IJ.
-1
o.v
g- u
0
g) I
-1
a.v 0
g- u
gll 2
-1
av
u
0
gll 2
< 2e ,
f- UVf 'g) I < 3e: •
But
a.
a. -1
(U \.1 U v
f, g)
=
a. a.,-1
(U 1-1 v f,g)
(IJ.,V)
>
(f 'g) '
hence
= (f,g)
(UVf,g)
thus
V -
u-1
(Uf,g)
=
•
all
'
2
f ,g E L (G) ;
In a dd ~· t ~on,
·
-1
a.v
lim(U f,g)
v
=
a.v
lim(f,U
g) = (f,Vg),
v
and we have u 1 = U* , so U is unitary.
o.'J
converges strongly to u:
argument now shows U
so
V = U* ,
a.v
2
a.v
IIU f- Ufll2 = (U f
a.v
'u
A standard
o.v
f)- (Uf 'u f)
o.v
a.v
a.v
-(U f,Uf)+(Uf,Uf) = 2(f,f)-(Uf,U f)-(U f,Uf) -> 0.
'J
- 10 -
It remains to show that
of
A. (G)
U A.(x)u- 1
1->
(and thus of G ) o
Cauchy net in
say
A.(x)
Fix
x E G,
defines an automorphism
clearly
(a.v(x))
is a
G and (since G is complete) converges to an element,
a. (x) E G •
Then
a.
a. -1
U 'VA.(x)U v = A.(a.v(x)) -> A.(a.(x))
'V
weakly,
and
-1
a.'V
a.'V
1
U A.(x)U
-> U A.(x)u'V
A.(a. (x)) = U A.(x)u-' 1 •
I. e.
weakly.
To prove
a
is a homomorphism,
A.(a.(xy)) = UA.(xy)u- 1 = (UA.(x)u- 1 )(UA.(y)tr 1 ) =A.(a.(x))A.(a.(y)) =
A.(cx.(x)a(y));
so
a.(xy) = a.(x)a.(y).
Also
A.(a.(x- 1 )) = U A.(x- 1 )tr 1 = U A.(x)- 1 u- 1 = (U A.(x)u- 1 )- 1
A.(o.(x)- 1 ) ,
= A.(a(x) )-1 =
o.(x- 1 ) = o.(x)- 1 •
i.e.
To prove continuity of
a ,
let
.... X
in
0
G.
Then
->
!l
in the weak operator topology.
of
G into
A.(G),
where
logy ([4; Lemma 2.2]).
continuous, and we have
But
x
~>
A.(x)
A.(G) c oL(L2 (G))
Thus
o.(x ) .... o.(x ) •
!l
0
a. E Aut (G) ,
so that
is a homeomorphism
carries the weak topoSimilarly,
Aut (G)
o.- 1
is
is complete ..
0
1.10.
Remark.
Since by 1.6
the complete group
Aut(G)
[7; Proposition 3.5], it would be
Aut( c:'R(G)) ,
natural to prove completeness of
Aut (Ji> (G)) •
(a.v)
Aut(G)
by showing it is closed in
Actually, such a proof can be given, utili zing the
profound duality theory in [9].
a net
is topologically embedded in
in
weak topology.
Aut(G)
We sketch the argument.
such that
By duality theory
algebra with comultiplication
Civ .... y E Aut( ~(G))
ut (G)
Consider
in uniform
is a Hopf - von Neumann
5: C/t(G) .... dt(G) ® (/{(G)
which is a
- 11 -
a- weakly continuous isomorphism given by
where
k E L2 (Gx G),
Wk(s,t) = k(s,st),
5 (T) =
w- 1 (T ® 1 )W , T E cR.(G ),
s,t E G,
[9; Section 4].
Furthermore, one has
(TE ~(G):6(T)=T®T}- {0} = (TE ot(G):T=A(s), for some sEG}.
Notice that
Aut(G)
corresponds to the subgroup
(a.EAut(t'l\>(G)):6(a).(s)) = a.A.(s)®a.).(s), all sEG}
av .... y EAilt(uY(G))
Since
s E G;
continuity of
6(av A.(s)) = av X.(s)®av A.(s),
and
6
y = a
for some
Corollary.
1o11.
Aut(G)
If
all
gives
6 ( y (A ( s))) = y (). ( s)) ® y ().. ( s) ) ,
Thus
o
a. E Aut(G)
all
s E G•
0
o
G is a separable locally compact group, then
is a Polish topological group.
00
Proof.
G = n~ 1 Fn ,
Indeed, if
is a neighborhood base at
borhood base at
1.
E Aut(G),
and by 1.9. it is complete.
2.
e E G,
Fn
then
so that
compact, and if
{Um}mEJN
{N(Fn, Um) }n m is a neigh-
Aut(G)
'
is metrizable [8],
0
We proceed now to applications of the Theorem in 1.9.
we turn to the question of when certain subgroups of
closed.·
Aut(G)
First
are
The following result contains a group theoretical analog.
to [2; Theorem 3.1].
2.1. Proposition.
B
a subgroup of
Let
G be a separable locally compact group, and
Aut(G) •
Suppose there is a Polish group
a continuous surjective homomorphism
are equivalent.
w : H .... B •
H
and
Then the following
- 12 -
a)
B is closed in Aut(G).
b)
w: H ... B
c)
For any neighborhood
cp1 ,. o
is open onto its range B.
cpn E Cc (G)
• ,
V
and
llcpi •w(h)- cp1 ll00 < e
of the identity in
e > 0
1 ~i~n,
Same statement as c) with
algebra
A(G)
are both Polish.
h E H,
Cc(G)
=> h E V•(kerw)
replaced by the Fourier
(and its norm 11·11 ) •
a) => b):
Proof.
such that, for all
and
llcpi •w(h- 1 )- cpillco < e,
d)
H there exist
IT
B
is closed in
Aut( G)
then
and B
H
Observe then that a continuous homomorphism between
two Polish groups is open [2; Lemma 3.4]
b) => c):
1.1
Put
K = kerw
o
Since
that given a neighborhood
cp 1 ,o •• ,cpn E Cc(G)
functions
V
and
w is open it follows from Lemma
of the identity in
e >
0
H there are
so that
,...,
W
n B c w(V) •
Now w. can be lifted to a map w of
cp1 ' • • • 'cpn; e:
H/K ... B, so that the diagram commutes and
is a homeomorphism.
w
Thus
w(h) E W
implies
cp1 ' • • • 'cpn; e:
w(h) E w(V) = w(VK) , hence w(hK) E w(VK) ,
H/K
i~
H
w
> B
c) <==> d)
d)
=> a):
the sets
Aut(G) •
hEhKCVK.
is clear in view of Proposition 1.6.
By 1.6
there is a sequence
(cp )
n
from
A(G)
such that
W =W
rn .1/n
form a base for the identity in
n
cp1, ••• ,..,..n,
Let {Vn} be a countable base for the identity in H.
By d) , given
h E VnK.
so that
Let
n
there is an
9 E B-
m(n)
so that
and choose a sequence
w(h) E Wm(n)
(a.n)
from
implies
B
so
- 12 -
a)
B
is closed in
Aut(G).
b)
w: H ... B
c)
For any neighborhood
is open onto its range B.
cp1 , • o. , cpn E Cc (G)
V
e: > 0
and
cpi lt::o < e
II cpi ow(h) -
of the identity in
1 ~ i ~n ,
Same statement as c) with
algebra
A(G)
are both Polish.
Cc(G)
(and its norm
a) => b):
Proof.
If
such that, for all
h E H,
and
llcpi ow(h- 1 )- cpillco < e ,
d)
H there exist
B
=> h E V• (ker w)
replaced by the Fourier
II ·II ) •
is closed in
Aut( G)
then
and B
H
Observe then that a continuous homomorphism between
two Polish groups is open [2; Lemma 3.4]
b) => c):
1.1
K = ker w •
Put
Since
that given a neighborhood
functions
cp1 ,.o.,cpn E Cc(G)
V
and
w is open it follows from Lemma
of the identity in
e >
0
H there are
so that
,..,
W
n B c w{V). Now w can be lifted to a map w of
cp1 ' o • • 'cpn; e:
H/K -+ B, so that the diagram commutes and
is a homeomorphism.
w
Thus
w(h) E W
implies
cp1, ••• ,cpn;e:
w(h) E w(V) = w(VK) , hence w(hK) E w(VK) ,
H/K
i~
H
c)
d)
> B
w
<~>
d)
=> a):
hEhKCVK.
is clear in view of Proposition 1.6.
By 1.6
the sets
wn :
Aut(G) •
Let
By d) , given
h E VnK.
so that
Let
there is a sequence
(cpn)
from
A(G)
such that
w
M •1/n
form a base for the identity in
cp1 ' • • • ' .....n'
{Vn} be a countable base for the identity in H.
n
there is an
9 E B-
m(n)
so that
and choose a sequence
w(h) E Wm(n)
(a.n)
from
implies
B
so
- 13 -
w-
1 ca. )
E W
Setting
n
n+j n
m(n) for j > 0.
-1
v
This is to say that
= hnK, we have h n+J.hn o K c nK, j > 0
Since H/K is
(hnK) is Cauchy in the left uniformity of H/K o
a. n
that
- e
a.- 1 a.
and
0
locally compact, it is complete, and
e
w(h) = w(hK) =
A sequence
2.2.
(xn)
from
ad(x ) -> 1 in Aut(G) o
n
n
(zn) from the center Z(G)
Corolla£l.
Suppose
If
and let· {Vn}
Int(G)
k > k.
- J
we have ·xk
Let
E
j
en= ck.
~c~ 1
(k . )
J
o.f
G
~
n
Int(G)
is closed
G are trivial.
(xn
be a central sequence,
for the identity in G.
from JN
V.·Z(G), hence
kj
->eo
G is separable, then
e >0
and
n
~
By d) of 2.1. we
so that
such that
'
0
for
Q
E B.
such that
=> jjq>. Ad(x. ) - q> -II < €.
J
. Kj
J
J
J
hence
is trivial if there is a sequence
can find, .for each n E JN, cpn E A( G)
Choosing a sequence
e
and thus
is closed, let
be a base
hK E H/K ,
G is said to be central if
<==> all central sequences in
Proof.
w,
by continuity of
I?
hnK
1
ck
j
j
n < kj+ 1 ;
E
V.
J
for some
j = 1,2,3, ••••
ck. E Z(G) •
J
Then
J
7
e, and
(xn)
is trivial.
The converse is shown in the same way as d) => a)
in 2.1.
0
2.3.
Aut(G)
is
We remark that the class of groups for which
locally compact includes the compactly generated Lie groups [6; Sats
2.2].
For
Int(G)
Corollary.
Int(G)
Let
we have the following
G be separable and locally compact.
is locally compact <==> Int(G)
is closed.
Then
- 14 -
Proof..
If
Int(G)
is locally compact, it is necessarily
closed [8; Theorem 5.11].
take
G
=H
and
the other hand if
On
w =Ad,
Ad(g)x = gxg- 1 ,
where
Then by continuity of
Ad, Int(G)
Z(G) = center of G..
O
2.4..
Let
GF
be the closed normal subgroup of elements
G E [SIN] , GF
e
is open since any compact
is contained in
continuous homomorphism
subgroup
w(g)
w(G) c: Aut(GF) •
GF..
F
Clearly
GF
{ gx g -1· :
G/Z(G).
x
in
g E GJ •
G
If
Int(G)- invariant neigh-
Let
= Ad(g)IG
is closed,
x,g E G, in 2.1 ..
is homeomorphic with
having relatively compact conjugacy classes
borhood of
Int(G)
w : G .... Aut(GF)
, [8], and let
is an
be the
B be the
[SIN]B group, and
we have
Corollary..
Let
G be separable..
Then, with notation as above,
B is closed <=> B is compact <=> G/ker w is compact.
Proof..
closed,
The first equivalence is proved in [5]..
B is homeomorphic with
a) => b) ) so by compactness of
versely, if
w..
B is
G/ker w (The Proposition in 2 .. 1,
B , G/ker w must be compact.
G/kerw is compact then so is
uity of the lifted map
If
Con-
B = w(G/kerw) by contin-
0
Specializing the preceeding corollary even further we obtain
2 .. 5..
Corollary..
Int(G)-
Let
is compact..
G be a locally compact group and suppose
Then
Int(G)
is closed <==> G/Z(G) is compact
(Z(G) = the center of G) ..
Proof.
This follows immediately from the Corollary in 2.4 .. if
G is separable..
From [6]
Int(G)
is closed <==> Int(G) is compact ..
- 15 -
But
Int (G)
compact implies
5 .. 29], hence
sely
if
G/Z(G)
G/Z(G) ... Int(G)
2.6.
Corolla;z.
Int(G)
~
Int(G)
Ad : G ... Int (G)
G/Z(G) , and so
G/Z(G)
is compact, lifting Ad
we see that
Let
Int(G)
is open (8; Theorem
If
is compact, hence closed.
G be a separable locally compact group.
Int(G)
and
Int(G)
It is then easy to see
G is unimodular, we give a proof for completeness.
IJ.(cp) =
Z(G)
and
I
Using right invariance of
one verifies easily that
unimodular.
dx
Conversely, if
we show that
G/Z(G)
1J.
~
By assumption
c
y E G, then implies
dx
cpy(x)
= cp(yx)
) •
Thus
Int(G)
IJ.(cp)
= IJ.(cp) =
Thus
G is
is closed
1J.
is right-invariant.
cp(x)
= IJ.(cpy)
=
Int (G)
Since
JZ(G) cp(xz)dz
for all
JG/Z(G) Cf)Cx)a.X,
is unimodular..
is
cp E Cc(G),
is right-invariant:
JG/Z(G) q;Y.. Cx)dx = IJ.Ccp·)y
(here
is the center,
G/Z(G) ..
Cc(G) ... C (G/Z(G)), cp ~ cp,
surjective (8, Theorem 15.21];
G.
It will then follow that Int(G)
is unimodular.
Define
the mapping
Z(G)
G is unimodular and
Int(G)
as above.
is a left Haar measure on
and the fact that
is unimodular. since
1J.
,
is even right invariant.
IJ.
and dX
dz
Let
JZ(G) cp(xz)dzdx
By the Weil integration formula
Let
respectively, and
G/Z(G)
the canonical map..
UG/Z(G)
is closed.
it is topologically isomorphic with
G/Z(G) , so that the latter is unimodular.
G --> G/Z(G)
Then
is unimodular, in particular it is closed,
so by the Proposition in 2.1
be Haar measures on
Conver-
to a continuous map
is unimodular <=> G is unimodular
Proof.
is compact.
0
- 16 -
2.7.
Corollary.
pose
Int(G)
Let
G be separable and locally compact and sup-
is closed.
Each of the following properties for
implies the same property for
a)
G
Int(G) •
There is a compact neighborhood of the identity element invariant under inner automorphisms ([IN]- property).
b)
There is a neighborhood basis of the identity consisting of
compact sets which are invariant under inner automorphisms
( [Slli] -property).
c)
All the conjugacy classes of
Proof.
G are precompact ([FC]--property).
We need only notice that in each case
requirec property.
G/Z(G)
0
The easy details will be omitted.
Next we give an example of a group
G
has the
for which
Int(G)-} Int(Gf,
using Corollary 2.5.
Fix an irrational number A , and let wA: 2Z 2 x 2Z 2 -~ ;
ni >un..,n2
wA((m1 ,m2 ),n1 ,n2 )) = e
, where 'll' is the circle group .. Let
G = 2Z 2 x 'lt.' with the product topology and group composition
2.8.
Example.
(m1 ,m2 ,s)•(n1 ,n2 ,t) = (m1 +n1 ,m2 +n2 ,e
G is a topological group with center
( ( 0, 0)) x 'F
is open, G has small
of the identity.
ni Am1n 2
st).
((O,O))x'Ir,
and since
Int (G)-invariant neighborhoods
Moreover, all the conjugacy classes of
G are pre-
compact, so by the Ascoli theorem for groups [6, Satz 1.7],
is compact.
and hence
However
Int(G)
G/Z(G)
is non-compact, being infinite discrete;
is not closed (Corollary 2.5).
ly, choose a sequence
(kn)
Int(G)-
from Zl
To see this directni A~
ni/2
e
-> e
,
such that
n
TTi t )
(
TTi m2 t /2
(
and put aA m1 ,m2 ,e
= m1 ,m2 ,e
) .. Then aA E Aut(G) and
is not inner. A routine calculation shows that Ad(O,kn,1) 11> aA
(by [6; Satz 1.6] it suffices to check
Ad(0,~,1)
11>aApointwise) ..
- 17 -
References
1)
J. Braconnier, Sur les groupes topologiques localement compact,
J. Math. Pur Appl. 27 (1948),1-85.
m1
2)
A. Connes, Almost periodic states and factors of type
J. Funct. Anal. 16 (1974), 415-445.
3)
P. Eymard, L'alg~bre de Fourier d'un groupe localement compact,
Bull. Soc. Math. France 92 (1964), 181-236.
4)
W. Green, Compact groups of automorphisms of von Neumann algebras,
Math. Scand. 37 (1975), 284-295.
5)
s.
6)
S. Grosser, 0. Loos, and M. Moskowitz, tiber Automorphismengruppen lokal-kompakter Gruppen und Derivationen von LieGruppen, Math. z. 114 (1970), 321-339.
7)
U. Haagerup, The standard form of von Neumann algebras,
Math. Scand. 37 (1975), 271-283.
8)
E. Hewitt and K. Ross, Abstract Harmonic Analysis, I,
Springer-Verlag.
9)
M. Takesaki and N. Tatsuuma, Duality and subgroups,
Annals of Math. 93 (1971), 344-364.
,
Grosser and M. Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1-40.