TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 170, August 1972
INVARIANTMEANSON A CLASS OF VON NEUMANNALGEBRAS
BY
P.F.RENAUD
ABSTRACT. For G a locally
compact
group with associated
algebra
VN(G) we prove the existence
of an invariant
mean on
mean is shown to be unique if and only if G is discrete.
1. Definitions
of [3].
group
Let
and preliminaries.
G be a locally
algebra
may
tive-definite
as the space
functions
subalgebra
Fourier
algebra
with identity
to left Haar measure
in general
e. Let
and let
of all finite
linear
the notation
L .(G)
C*(G)
combinations
on G. With the dual
norm
Banach
Fourier-Stieltjes
is a commutative
closed
group
we follow
be the
be the envel-
of L r(G). Denote by B(G) the dual space of C*(G).
be realized
B(G)
compact
of G with respect
oping C*-algebra
Throughout
von Neumann
VN(G). This
algebra—the
A(G) generated
of G. Let
by elements
and pointwise
posi-
multiplication,
algebra
of G. The
support
is called
with compact
P = \u £ A(G): u is positive
Then B(G)
of continuous
definite
and
the
\\u\\ = u(e) =
l!.
For
x a complex-valued
(a.e.
defined)
function
on G, denote
by x, x the
functions
x(g) = x(g~
),
x (g) = x(g~
) where
denotes
complex
conjugation.
If f £ L j(G), denote by L, the bounded operator on LAG) defined by L,x =
/ * x (where
* denotes
An important
convolution).
property
of A(G)
is that it is precisely
the set of functions
of
the form x *y" where x, y £ L 2(G) [3, p. 218].
Let
ators
A(G)
VN(G)
L , where
under
denote
the map
the w*- and weak
the von Neumann
f £ L ^(G).
Then
algebra
VN(G)
on L JG)
may be regarded
(T, u) = (Tx, y) it u £ A(G)
operator
topologies
VN(G) define the operator
on VN(G)
functional
Received
by the editors
of
For
u £ A(G),
T £
v £ A(G).
722on VN(G) is called
and that
||zzT|| < ||«|| ||T||.
a 7Z2ea72if
June 28, 1971.
AMS 1970 subject classfications.
43A40.
Key words and phrases.
algebra,
invariant
mean.
space
u = y * x. In particular,
coincide.
VN(G) is an A(G>module
A linear
by the oper-
uT £ VN(G) by
(uT, v) = (T, uv),
It follows readily that
with
generated
as the dual
Locally
Primary 22D15, 43A15; Secondary 22D25, 43A07,
compact
group,
von Neumann
algebra,
Fourier
Copyright © 1972, American Mathematical Society
285
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286
P. F. RENAUD
[August
(i) 772(T)> O if 7 > O, and
(ii) 722(7)= 1 where
722is called
7 is the identity
an invariant
operator.
mean if it satisfies
(iii) m(uT) = 272(F) for all
(i), (ii), and
T £ VN(G), u £ P, or equivalently
(iii') m(uT) = u(e)m(T) tot all T £ VN(G), u £ A(G).
A linear
functional
772satisfying
(iii) will be called
invariant.
If G is Abelian with dual group G then A(G) is isomorphic
VN(G)
to 7-^(0).
G is Abelian
bility).
group
The existence
and hence
The main
G, there
amenable
(see
of this
paper
purpose
exists
an invariant
and only if G is discrete.
an arbitrary
2. The discrete
Theorem
case
on L^G)
to L y(G) and
is guaranteed
account
is to show that for every
mean on VN(G)
the property
the discrete
means
[4] for a comprehensive
In a certain
group enjoy
tion we treat
of invariant
sense
therefore,
of amenability.
of amena-
locally
and that this
mean
the "dual
since
compact
is unique
structures"
For convenience
if
of
of exposi-
separately.
case.
1. Let
G be a discrete
group.
Then
there
exists
a unique
invariant
mean on VN(G).
Proof.
The function
L 2(G) and since
tional
8
8e * 8
defined
= 8g, then
by 8 (g) =1
8g is also
if g = e and
in A(G).
0 otherwise
Define
the linear
is in
func-
m on VN(G) by
«(T) = <7, 8J = (T8e, 8e).
It is immediate
Let
that
7 e VN(G),
272is a mean.
We show
u £ P.
u(e) = 1 we have
Since
that
222is invariant.
u8
=8
222(2.7)= <277, 8e) = (T, u8e) = <7, 8J
Thus
and hence
= 272(7).
272is invariant.
To show
uniqueness
suppose
that
272 is an invariant
mean on VN(G).
Let
7 £ VN(G), v £ A(G). We have
<<3e7, v) = (T, 8ev) = (7, v(e)8e)
and hence
<5e 7 = 772(7)7. But then since
772' (7)
which
proves
Remarks.
on VN(G)
case
1.
It is easy
so that
VN(G)
L^G)
has
3. The general
of "convergence
ship
= v(e)m(T) = (m(T)l, v)
277' is invariant
= 272' (8e 7) = 272' (m(T)l)
and 5 e £ P, '
=722(7),
uniqueness.
2. If G is Abelian
this
= vie) (7, 8J
to show
case.
of invariant
case,
mean,
normal
trace
and it is well known that in
namely
aid to the study
We introduce
272is a faithful
algebra.
G is compact
invariant
A useful
in this
von Neumann
and discrete,
a unique
to invariance".
to the existence
that
is a finite
of amenability
a similar
means.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
the Haar
idea
integral.
is the concept
and show its relation-
1972]
INVARIANT MEANS ON A CLASS OF VON NEUMANN ALGEBRAS
Definition
2. A net
\u \ C P is called
w- (norm) convergent
287
to invariance
if ¡or
all v £ P,
w-litn(uav
-
ua) = 0,
i.e.
lim (uav
-
ua,
T)
= 0
for all T £ VNÍG).
(lim ||zzai2 - z/J =
The following
groups
it is proved
Proposition
choose
u
Proof.
proves
the existence
of such
nets.
For compact
in [2].
3. Let
£ P such
norm convergent
that
proposition
\V \ be a neighbourhood
that spt uaisupporl
of uj
basis
at e and for each
C Va. Directed
a,
by inclusion,
[uj
is
to invariance.
Fix
v £ P, e > 0. Choose
\\v - v'\\ < e/2.
By [3, p. 208],
v
there
£ P with
exists
compact
some
support
K and such
u £ AÍG) with
K. Then (zz.- v)ie) = 0 so that by [3, p. 229] we can find an element
|| u — v — w|| < e/2
Va C U C\ K then
and
w identically
uau = ua and
zero
on some
uaw = 0. Hence
neighbourhood
for such
uig) = 1 on
w £ AÍG)
with
U of e. Now if
a's,
Wuav
- XJ < ll*> - "')|| + hav> - «J
< i/2 + Ua v' - uJ = e/2 + ü«>'
u — w)
< e/2 + || i/ - u - j¿z|| < e.
Hence
¡a^J is norm convergent
Using
this
Theorem
nonempty
N denote
3 there
of
mean. If u e P,T
A(G) as a subspace
exists
VNÍG)*
\Uß\ be a subnet
the set of invariant
a net
\uj
is w*-compact,
on VNÍG).
Then
N is a
of V/V(G)* in the usual
way.
By
tZ2*-convergent
C P, »-convergent
\uj
to invariance.
has a tz7*-accumulation
to 272. Each
zz^ is clearly
Since
point
the
ttz. Let
a mean so that
272is a
£ VNÍG) then
miuT) = lim (u„,
and since
means
set.
We consider
Proposition
ball
we can prove
4. Let
convex
Proof.
unit
result
to invariance.
\u À is w*-convergent
uT) = lim (u „u, T)
to invariance,
lim [(ußu,
T) - (uß, T)] = 0
so that
míuT)
and
222is an invariant
The following
discrete
mean.
proposition
= lim (uo,
Hence
gives
T) = 722(F)
N is nonempty.
a useful
That
distinction
N is convex is immediate.
between
groups.
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discrete
and non-
288
P- F. RENAUD
Proposition
5. Let
m be an invariant
[August
mean on VN(G).
If m £ A(G) then
G is
discrete.
Proof.
Suppose
222(7) = (u, T)
that
for all
772£ A(G)
so that there
7 e VN(G).
Then
exists
some
u £ A(G)
such
that
(22, 7) = 272(7) = 272(727)= (vT, u) = (7, vu)
fot all 7 £ VN(G), v £ P. It follows that u = vu for all v £ P. If now G is nondiscrete
This
we can choose
contradicts
We now consider
then there
by establishing
In [l] Arens
multiplication
some
showed
that
u(g) / 0 and
u = vu. Hence
the problem
if G is nondiscrete
mence
g £ G such
the fact that
v £ P such that
of uniqueness.
Specifically
is more than one invariant
properties
that,
of the dual
given
on 73** which
a Banach
extends
27(g) = 0.
G is discrete.
space
algebra
multiplication
we want to show that
mean on VN(G).
We com-
VN(G)*.
B, it is possible
on B. In the case
to define
where
a
73 =
A(G), the idea is as follows, for 222£ VMM*, 7 £ VNÍG) define 772o 7 e VN(G) by
(222o 7, u) = (m, uT)
For
fot all
272,22 £ VN(G)*,
u £ A(G).
define
m on £ VN(G)*
by
(772072, 7) = (772, n oT)
tot all
7 e VN(G).
This
||m||»||
operation
o makes
VN(G)* a Banach
algebra
where
also
¡|tt2 o tj|| <
for all m, n £ VN(G)*.
Lemma
6. Let
m £ VN(G)*.
Then
m is invariant
if and only if m = v o m for
all v £ P.
Proof.
For 7 £ VN(G) we have
(v o 772, 7)
and the result
Corollary
= (v, ?72° 7)
= (272o 7, v) = (m, vT)
follows.
7. Let
m, n £ VN(G)*.
If m is invariant
then
so is mon.
Proof. For if v £ P then
v o (m o n) = (v ° 772) ° 72 = m ° n.
Let
H be a normal,
compact
Aie/TV) may be identified
which
are constant
subgroup
of G. Eymard
with the subalgebra
on the cosets
[3, p. 217]
of A(G) consisting
of 77. The adjoint
of this
Similarly
we may identify
subspace
constant
on the cosets
M oí L 2(G) of all functions
pT' on L 2(G) by p7'x = 7'x
if x e Al .It
that
follows
readily
pT'
€ VN(G) and that
shown
that
of all functions
identification
morphism 27 of VN(G) onto VN(G/H).
VN(G/H) define the operator
has
is a homo-
L (G/H) with the
of 77. For
7'
£
if x e Al and pT'x = 0
p is an isometric
embed-
ding of VN(G/H) into V7V(G).
For
u £ A(G),
f (T ) = (u, pT')
on VN(G/H) so that f(T') = (u,
With these
Proposition
variant
7¿e72
results
8. Let
continuous
linear
functional
u £ A(G/H).
we can prove
H be a normal,
mean in VN(G)* and
222o 72 is invariant
is an ultraweakly
T') for some uniquely defined
and
n
compact
an invariant
subgroup
element
\\m o n\\ = ||«'||.
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of G. Let
in VN(G/H)*.
m be an inLet
n = 77*77'.
1972]
INVARIANT MEANS ON A CLASS OF VON NEUMANN ALGEBRAS
Proof.
By Corollary
7, we know that
m on
is invariant.
Also
289
since
ll«°«ll<HI««ll
= ll^«'ll<IK1l
it suffices
to prove
that
||r72 o «|| > ||»'||.
Fix e> 0 and choose
7" e VNÍG/H) such that ||T'|| = 1 and \(n', T') \ >
||«'|| - f. If a £ AÍG), v' £ AÍG/H) (C AÍG)) then
(niupT'),
v') = (upT',
= (v'V,
so that
v')=
(pT',
uv') = (v'pT',
u')=
{u'T1, v')
u)
niupT ) = u T .
Hence
(TZ ° pT',
u) = («, UpT')
= (27*«',
= <«', niupT'))
so that
upT')
= («', u'T')
= u'íe)(n',
T')
n o pT' = (n , T')l.
Therefore,
1(2220 72, pT')\
from which
= |(t72, « ° p7")|
we deduce
Corollary
VNÍG)*
and
VNÍG/H)*
of invariant
in VNÍG)*.
VNÍG/H)*
then
Finally
admits
more
mean on VNÍG). Define
by oin ) = m o n* n . By the previous
elements
T')|
> ||«'||
-e
than one invariant
VNÍG)*.
Let 272be an invariant
the subspace
| <«',
\\m o «|| > ||«'||.
9. // H is compact
mean then so does
Proof.
that
= |<7», <«', T')/>|=
elements
Consequently,
a(«|),
o: VNÍG/H)*
proposition,
in VNÍG/H)
o is an isometry
* into the subspace
if «1? «2 are distinct
ct(«2') are distinct
invariant
—>
invariant
means
from
of invariant
means
in
in VNÍG)*.
we shall need
Proposition
two distinct
10.
Let
invariant
K be an open subgroup
means
on VNÍK).
of G and suppose
Then there
are two distinct
that
there
are
invariant
means on VNÍG).
Remark.
Theorem
For
6].
G compact,
The proof
K a closed
carries
subgroup,
over to noncompact
this
result
groups
was proved
and is included
in [2,
for
completeness.
Proof.
Let cfe denote
map cfe is bounded
and since
that if 222' is an invariant
on VNÍG) suchthat
Let
vector
definite
functions
homomorphic
map,
cfe: AÍG) —» A(/C). The
K is open, cfe is onto [3, p. 215].
mean on VNÍK)
then there
exists
It suffices
an invariant
to show
mean
272
222'= m\<fe*VNÍK).
222 be an invariant
lattices
the restriction
[6, Theorem
in AÍK)
mean on VNÍK).
4.3] that there
such
that tz^-lim
It follows
exists
1/=
a net
772'. For
from a general
iz/
theorem
S of normalized
V a neighbourhood
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on
positive-
of e
290
P- F- RENAUD
in G, choose
uy £ P such
that
[August
spt uy C V. Let
v'a y = v'ai¿v\ H and observe
that
if 7 £ VN(K) then for all V,
lim {v'a v, T) = lim ya,
Choose
va£ P such
cumulation
point
that
uv\H
• T, = (272', uy\H
rf>va= v'a and define
of \va v\.
By the proof
• 7) = (722', T).
va v = vaz2</. Let
of Theorem
772be a ^-ac-
4, 272is an invariant
mean
on
VN(G). We show that m\cb*VN(K) = 722'.
For 7 £ VN(K), e > 0, choose
- (222', 7)|
< e/2. Choose
a
such that for a > aQ and all V |(t/
(a, V) > (a^,
V) such that
|<w, 0*7)
- (12^,
v, 7)
0**7) | <
e/2. Then
1(772,0*7)
-(772',
7>|
< 1(772,
0*7)-
(v aV, 0*7) I + \(vaiV,<p*T) ~(m',
< e/2 + |<027a_ v, 7) Hence
(722, p*7)
Proof.
our main
result.
11. If G is nondiscrete,
Let
A be a compact
VN(G)* has
neighbourhood
ated
by A is an open subgroup and applying
that
G is compactly
Theorem
that
an open,
applies
compact
We may therefore
Assume
a decreasing
neighbourhood
4, each
Cauchy
sequence
discrete
This
and this
We have already
on VN(G)
is a faithful
Theorem
12.
But
A(G)
of Sakai
implies
uncountably
remarked
trace
272on
basis
for its open
Let
\V 77 }°°
77=1, be
22 e P with
spt u
Applying
Proposition
Theorem
mean.
A(G)
algebra
is weakly
5 now shows
4 we see
It
[22 \ is a weak
that
sequen-
that
VN(G)
G is
must
means.
that if G is discrete
At the other
and
words
of the von Neumann
to show that
C
and by the proof of
= 222. In other
many invariant
If G is nondiscrete
admits
VN(G) *.
77
272e A(G).
K is
so does
of \u \ in V/V(G)* is an invariant
is the predual
on VN(G).
K such
then
9] VN(K)*
VN(G).
v '
n choose
[5,
subgroup
10 again,
to invariance
ti-*-lim u
[7] applies
that
is a contradiction.
in fact have at least
mean
assume
(see
is discrete
G has a countable
e. For each
point
of 272that
in A(G).
so that a theorem
complete.
9 that
invariant
at
normal
If G/K
by Proposition
\ is norm-convergent
7^*-accumulation
by the uniqueness
VN(G)
basis
3, \u
follows
tially
by Corollary
K of G gener-
and Kodaira
of G and by [2, Theorem
But then
is a unique
^
sets.
mean.
10, we may already
of Kakutani
for its open
subgroup
mean.
there
V . By Proposition
Theorem
basis
assume
that
of e. The subgroup
Proposition
A theorem
nondiscrete
one invariant
more than one invariant
and we can find a compact
has a countable
more than
sets.
generated.
8.7]) now
G/K
„. 7) - (722', 7>| < e.
= (272', 7).
We can now prove
Theorem
(772', 7)1 . e/2 + | «
7)|
then the unique
extreme
m is an invariant
722(7)= 0 for all 7 £ C*p(G).
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invariant
mean
we have
mean on VN(G)
then
1972]
INVARIANT MEANS ON A CLASS OF VON NEUMANN ALGEBRAS
Remark.
The algebra
tion on L2iG).
Proof.
C*ÍG)
If G is amenable
Since
is the C*-completion
of L ,(G)
acting
291
by convolu-
then C*(G) = C*iG).
L (G) is dense
in C*ÍG)
it suffices
to show that
2?2(L,) = 0 for
all / £ L j(G).
Let f £ L ,(G), e > 0. Let U be a neighbourhood of e such that /J /(g)| <7g< e
and choose
zz £ P with
\u \ is »-convergent
spt u C U. Choose
to invariance
{uj
C P such
that
tzv*-lim zza = m. Then
so that
miLf) = lim Jfíg)uaíg)dg = lim J f íg)uíg)uaíg) dg.
But
//W«)«a(«)«te| < f\fig)\ Wig)\
\»aig)\dg
< jjfig^dg
Hence
<«
fotall
a.
722(L,) = 0.
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C. F. Dunkl
and D. E. Ramirez,
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P. Eymard,
L'algèbre
de
Fourier
Math. France 92 (1964), 181-236.
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F. P. Greenleaf,
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#4776.
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Berlin,
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MR 28
#158.
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I. Namioka,
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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CANTERBURY, CHRISTCHURCH, NEW
ZEALAND
Current
address:
Department
of Mathematics,
University
of Toronto,
Canada
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Toronto,
Ontario,