AUTOMORPHISMS AND INVARIANT STATES OF OPERATOR ALGEBRAS

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AUTOMORPHISMS AND INVARIANT STATES OF OPERATOR ALGEBRAS

B Y

E R L I N G STORMER

Mathematical Institute, University of Oslo, Oslo, Norway

1. Introduction

L e t 9~ be a y o n N e u m a n n algebra and G a group of *-automorphisms of 9~ with fixed point algebra N in ~. I f 9X is semi-finite and N contains the center of ~l the normal G-in- v a r i a n t states of 9X were analysed in [3], [12], [13]. I n the present paper we shall extend these studies to the general situation, in which the center is not necessarily left fixed b y G.

The main result, from which the rest follows, states t h a t if 9~ is semi-finite and co a faithful normal G-invariant state of 9~, and if G acts ergodicly on the center of ~, then there exists a faithful normal G-invariant semi-finite trace T of 9~ which is unique up to a scalar multiple, and a positive self-adjoint operator

B ELI(~, ~)

affiliated with B such t h a t

co(A) =T(BA)

for all A Eg~. F o r example, if G is ergodic on 9~ then co is a trace, hence ~I is finite. As an application to C*-algebras we show t h a t if A is an asymptotically abelian C*-algebra (more specifically G-abelian) and @ is an extremal G-invariant state, t h e n either the weak closure of its representation, viz

zo(A)",

is of t y p e I I I , or the cyclic vector xr such t h a t

@(A)=(~e(A)xo,

xe),

A E A ,

is a trace vector for the c o m m u t a n t of gQ(A). This has previously been shown for invariant factor states [12].

The basic technical tool used in this paper is the theory of Tomita [15] and Takesaki [14] on the modular automorphisms associated with faithful normal states of y o n l~eumann algebras. I t will, however, mainly be applied to semi-finite algebras. We recall from [14] t h a t if 9~ is a yon N e u m a n n algebra with a separating and cyclic vector x 0 t h e n the *-operation S:

Axo~Ax o*

is a pre-closed conjugate linear operator with polar decomposition

S=JA 89

where J is a conjugation of the underlying tIilbert space, and A is a positive self-adjoint o p e r a t o r - - t h e modular operator defined b y x 0. The modular automorphism

a t

of 9~ associated with x 0 (or rather the state coxo) is given b y

at(A ) =AtAA -~t.*

Furthermore, J satisfies the relation

J91J=9.I'.

F o r details and further results from this

1 -- 712906 Acta mathematica 127. I m p r i m 6 lo 28 M a i 1971

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2 ERLII~G STORMER

theory we refer the reader to the notes of Takesaki [14]. F o r other references on von N e u m a n n algebras the reader is referred to the book of Dixmier [1].

I n most of the discussion we shall s t u d y faithful normal G-invariant states of 91.

I f a normal G-invariant state w is not faithful then its support E belongs to B, hence we can restrict attention to the y o n N e u m a n n algebra E91E and the automorphisms E A E - ~ E g ( A ) E, gEG, of this yon N e u m a r m algebra, and then a p p l y the results for faithful states.

2. Automorphisms of yon Neumann algebras

I n this section we prove the main results concerning invariant states of yon N e u m a n n algebras.

L E M M A 1. Let 91 be a von N e u m a n n algebra and let G be a group o / u n i t a r y operators such that U91U -1 =91 /or U EG. Suppose x o is a separating and cyclic vector /or 91 such that U x o = x o /or UEG, and let A be its modular operator. Suppose A~t=F(t)F'(t), where F(t) (resp. F'(t)) is a strongly continuous one-parameter unitary group in 91 (resp. 91'). I / UF(t) U - I = F ( t ) and U F ' ( t ) U - I = F ' ( t ) /or all t and U E G , then 91 has a /aith/ul normal G-invariant semi-/inite trace.

This l e m m a follows from the proof of [14, Theorem 14.1J, because the trace constructed in t h a t proof will clearly be G-invariant.

n E M M A 2.

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Let 91 be a von N e u m a n n algebra acting on a Hilbert space ~ . Suppose x o is a separating and cyclic vector/or 91, and let A be its modular operator. Suppose U is a uni- tary operator on ~ such that U91U-l=91 and Uxo=Xo. T h e n U A = A U and U J = J U .

Proo/. As in the proof of [14, Theorem 12.1] 91 is made into a generalized Hilbert algebra via the representation A - > x o ( A ) = A x o with multiplication x o ( A ) x o ( B ) = x o ( A B ) and involution xo(A) ~ =x0(A*), A E91. The u n i t a r y operator U defines an isometric *-auto- morphism of the generalized Hilbert algebra 91 b y Uxo(A ) = x o ( U A U-l), which extends to an isometry of the domain ~ of A t onto itself, cf. [14, Theorem 7.1]. Now for A E91 we have

JA~xo(A ) = xo(A ) ~ = A * x o = U-l( U A U-1)*x0 = U - 1 J A 89 U A U-lxo

= ( U - 1 J U) (U-1A 89 U) xo(A ).

Since the generalized ttilbert algebra 91 is dense in the Hilbert space ~ [14, L e m m a 3.4] we have t h a t JA 89 = ( U - 1 J U ) ( U - 1 A 8 9 for all xE ~ . ttence from the uniqueness of polar decomposition we have J = U - 1 J U and A89 U-1A 89 hence A = U - 1 A U .

(1) A p a r t i a l r e s u l t in this d i r e c t i o n h a s b e e n o b t a i n e d b y W i n n i n k [17, L e m m a I V . 5].

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A U T O M O R P H I S M S A~TD I N V A R I A N T S T A T E S OF O P E R A T O R A L G E B R A S

W e n e x t show our m a i n result. I n t h e t h e o r e m we assume t h a t t h e g r o u p G of auto- m o r p h i s m s of 9~ acts ergodicly on t h e center C of 9~, i.e. B fl C = C, where B is t h e fixed points of G in ~ . This a s s u m p t i o n is m a d e m a i n l y for convenience a n d is analogous to t h a t of s t u d y i n g factors r a t h e r t h a n general y o n N e u m a n n algebras.

T ~ E O R E ~ 1. Let 9~ be a semi-/inite von Neumann algebra and G a group o/ *-automor- phisms o] ~ acting ergodicly on the center o / ~ . Suppose o) is a ]aith/ul normal G-invariant state ol ~. Then there exists up to a scalar multiple a unique faithful normal G-invariant semi-finite trace T o/ 9X, and there is a positive self-adjoint operator B ELI(9~, 7) a//iliated with the /ixed point algebra B of G in 9~ such that co(A)=~(BA) for all A EOJ.

Proof. Uniqueness. Suppose ~ is a n o t h e r n o r m a l G-invariant semi-finite trace of ~ . T h e n it is an easy consequence of t h e R a d o n - N i k o d y m t h e o r e m for n o r m a l traces [1, Ch. I I I , w 4] t h a t its R a d o n - N i k o d y m derivative with respect to ~ will be affiliated with b o t h B a n d t h e center of 9~, so it is a scalar b y hypothesis. T h u s ~v =#~, with #>~0.

Existence. W e first m a k e a digression. Since G is ergodic on t h e center C of 9~ it follows t h a t ~ is either of t y p e I, II1, or IIoo. I n t h e t y p e I a n d I I 1 cases it is easy to show t h e existence of t h e i n v a r i a n t trace 7, a n d we m a y even w e a k e n t h e assumptions a n d only assume t h a t eo is a n o r m a l G-invariant state of C. (I a m indebted to G. Elliott a n d R.

K a d i s o n for valuable c o m m e n t s on these eases.) Indeed, since G is ergodic on C, w is faithful on C. Suppose first 9J is of t y p e I. L e t E be an abclian projection in 9X with central carrier 1. L e t ~p be a faithful n o r m a l center v a l u e d trace of ~ such t h a t ~o(E) = I [1, Ch. I I I , w 4]. I f g is a *-automorphism of 9~ t h e n g(E) is an abelian projection in 9~ with central carrier I , hence g(E) is equivalent to E [1, Ch. I I I , w 3]. Thus l=~v(E)=y~(g(E))=

g-l(yj(g(E))). N o w g-ly~g is a faithful n o r m a l center valued trace on 9X which coincides with y~ on E. Therefore t h e y are equal, hence ~v is G-invariant. T h e n coo~ is a faithful n o r m a l G-invariant semi-finite trace of 9J. N o t e t h a t if 9X is finite there exists a unique faithful n o r m a l center v a l u e d trace ~p of 9~ such t h a t ~ ( I ) = I . B y uniqueness ~v is G-invariant, a n d t h e proof is completed as in t h e t y p e I case. T h u s all t h a t remains is t h e I I ~ case.

Since t h e t y p e I a n d II~ cases come u n d e r t h e a r g u m e n t we shall give, we only assume is semi-finite.

Considering t h e G e l f a n d - N a i m a r k - S e g a l construction for o~ we m a y assume eo = O~x~

with x 0 a separating a n d cyclic unit v e c t o r for 9~ in t h e underlying Hilbert space :H, a n d t h a t there is a u n i t a r y representation g-> Ug of G on ~ such t h a t Ugx o = x 0 a n d UgA U g 1 =

g(A) for all gEG, A Eg.I.

L e t E 0 be t h e orthogonal projection on t h e subspace of ~ consisting of all vectors y E

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4 ]~RLING STORMER

such t h a t

Ugy=y

for all

gEG.

Then

Eoxo=Xo,

so E 0 4 0 . From the ergodic theorem [11, w 144] there exists a net {~2~ Ua~}~eK in cony (Ug:

gEG)

which converges strongly to E 0.

B y [7, Theorem 2] there exists a unique faithful normal G-invariant projection map (I) of 9~ onto B, and b y [2, Corollary 1] we have

O(A) = strong lim ~ 2~

VgTA V ~

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for all A E9~.

Let Tr be a faithful normal semi-finite trace of 9~ [1, p. 99], and let H be a positive self-adjoint operator in L1(9~, Tr) such t h a t cox0(A)

=Tr(HA)

for all A E9~ [1, p. 107]. L e t A be the modular operator and J the unitary involution defined b y x 0. B y Lemma 2 UgA*t=A~tUg and

UgJ=JUg

for all

gEG.

B y [14, Corollary 14.1 and end of w 14]

Ait=H~tJHiJ*

so t h a t

Ht=JH-iJAit

(recall t h a t

J92(J=~').

Thus for

gEG

we have

UgH~U~ 1 = JUgH-~tU[1JAit.*

Therefore we have from (1) t h a t

O ( H ~t) =

JCP(H-~)*

J A ~*.

L e t

B,~(~(H~).*

Then B, E B, and furthermore

B , =

JB JJH~JH ~,

so t h a t

B,H-~=JB~HiJE9.I*

N 9 ' = C, where C is the center of 9/. Therefore

B,=C,H ~*

with

CtE C.

Let F, be the range projection of B,. Then FiE B. But F , is also the range projection of

C,,

hence belongs to C, so t h a t F, E ~ A C, which equals the scalar operators b y assump- tion. Thus either F t

=0

o r _F t = / r . Since 9 is strongly continuous on bounded sets and

H~--->I

strongly as

t->O, B,=~P(H~)--->I*

strongly as t-+0. Therefore there is a neighborhood of 0 in tt such t h a t Ft = I for t E ~/. Let Bt = V, I Bt ] and C, = U, ] Ct [ be the polar decom- positions of B, and Ct. Then Vt and Ut are unitary operators in B and C respectively for tE~/. Since B , =

Vt[Btl =CtH ~= UtH**]Ct[*

it follows from the uniqueness of polar decomposition of an operator t h a t Vt=

U,H ~*

and

I B,[ = I Ct[

for all t. Therefore there is a number 2t ~>0 such t h a t B, =2t Vt =2t

UtH ~,*

and 21 > 0 for t E ~/.

The map t-+ V, is strongly continuous for tE ~/. Indeed, t->Bt is strongly continuous, and so is t-> B_, = B~. Since I] Bt[] ~< l, t-->2, = I Bt I = (B* Bt)89 is strongly continuous [6].

Therefore

t---> Vt=27~Bt

is strongly continuous for tE ~/.

We next want to define V, for those t for which Bt =0. Let 2 E ~/, 2 40, and let N = [ - 2 , 2]. Consider V, as only defined for t EN. If s CN with s > 0 let t be the largest number in 2V such that

s=tn

with n a positive integer. Let

Vs=(Vt) ~.

If s < 0 let

V~=V~.*

We show t h a t s-+ V~ is strongly continuous for s 4 n 2 and continuous from below (resp. above)

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AUTOMORPHISMS AND INVARIhNT STATES OF OPERATOR ALGEBRAS

if

s=n2,

n > 0 (rcsp. n < 0 ) . Indeed, it suffices to show this for s > 0 . L e t

s = n t

with t t h e largest n u m b e r in 2V which divides s in an integer. Since t h e function

t-+nt

is open a n d continuous there exists a n e i g h b o r h o o d ~ of s such t h a t if

s'E :U~

t h e n

s' =nt'

with t' in a n e i g h b o r h o o d of t. Assume first t=4=2. L e t

s'E'~s,

so

s'=nt', t'EN.

I f

s,=(n+ic)t 1

with t 1 E N, /c a positive integer, t h e n t I < t. I f s ' = ( n - 1)t

2, t~ E N,

t h e n if

s'

is sufficiently close t o s it follows f r o m t h e above a r g u m e n t t h a t s = ( n - 1 ) t a with taE2V. B u t t h e n t a > t contradicting t h e m a x i m a l i t y of t. Therefore s' is n o t of t h e f o r m ( n - 1 ) t e with t~EN.

If s ' = ( n - l c ) t 2

with t~E~V, n - k > l , t h e n also

s ' = ( n - 1 ) t 1

with

t l = ( n - l c ) ( n - 1 ) - l t 2 < t 2 ,

so

t l E N ,

a case which is ruled out. Therefore there is a n e i g h b o r h o o d ~ of s such t h a t if s' E ~ t h e n s' =

nt'

with t' in a n e i g h b o r h o o d of t, a n d ~' is t h e largest n u m b e r in N which divides s' in an integer. If s = n 2 t h e n t h e same holds for s ' E ~ = {s'E ~8: s' ~<s}. N o w let x~ .... , Xr be r vectors in ~/ a n d e > 0 . Since

t-~Vt

is strongly continuous for

tEN,

so is t-+ V~. Therefore, if 1~ is a sufficiently small n e i g h b o r h o o d of s contained in ~s (or in ~Ws

if s=n2)

t h e n

II(V~-V~,)x~II=II(V~-V~.)x~II<e

for s ' E : ~ . T h u s

s--->V,

is strongly continuous for s =~n2 a n d strongly continuous f r o m below for

s=n)~,

as asserted.

~zn_ Unrips

H e n c e if A E ~ L e t

s=nt, t ~ N .

T h e n r t ~ U t H ~t with

UtEC,

a n d V ~ = ~ t - t .

we h a v e

V~AVs~=H~SAH -~s.

N o t e t h a t

VsV~,AV2,1V2 I=Hi(s+8')AH-(~+~')* V,+~,AV -1

8 + 8 " "

N o w

Vs Vs, V2+~s , =y(s, s') I

with

y(s, s')

in t h e circle g r o u p T1, because

V~ Vs, V2+1~, E

N n C = C . One can easily show t h a t y: R x R - + T 1 is a Borel map. F u r t h e r m o r e , since

Vt=HisUr~

all t h e V~ c o m m u t e with each other. Therefore it is trivial to show t h a t

7(82, 83)7(81 -/-82, 83)--1~(81, 82 @83)~](81, 82) -1 = 1

for all sl, s2, saER. T h u s y is a 2-cocycle as a cochain on R with coefficients in T1 (with trivial action on T1) in t h e usual c o h o m o l o g y t h e o r y of groups cf. [10]. Since H~(l~, T1) = 0 [10, T h e o r e m 11.5] y is a 2 - c o b o u n d a r y , so there is a function ~(s) on R with values in T 1 such t h a t ~'(S, S ' ) = ~(8)-1~(8')-1~(8-~-8'), a n d as pointed o u t b y K a d i s o n [5, p.

197] it follows f r o m [9, Thdorgme 2] t h a t ~(s) can be chosen as a Borel function. Since

~(s, - s ) = I a n d we m a y normalize ~ so t h a t ~ ( 0 ) = I , we h a v e t h a t ~ ( s ) - ~ = ~ ( - s ) . W e n e x t show t h a t ~(s) is continuous at 0, a n d for this we m o d i f y t h e proof of [4, T h e o r e m 22.18]. L e t W 0 be a s y m m e t r i c n e i g h b o r h o o d of 1 in T1, a n d let W be a s y m m e t r i c n e i g h b o r h o o d of 1 in T 1 such t h a t W a c W 0. Since T 1 is c o m p a c t there is a finite subset

Yl

. . . y r ~ . T 1 such t h a t T I =

[.J~=IWy~.

N o w V is continuous in a n e i g h b o r h o o d of 0 in R x R . L e t A be an open s y m m e t r i c n e i g h b o r h o o d of 0 in R such t h a t if

a, bEA

t h e n

y(a, - b ) E W.

W e h a v e t h a t A = (J r =1 (~-l(Wyn)

N A).

Since ~(s) is Borel b y t h e preceding

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6 ]~RLING STORMF.R

paragraph, we have at least one value of n for which ~-l(Wyn)N A is Borel measurable and has positive Lebesgue measure. B y [4, Corollary 20.17] there is a neighborhood E of 0 in R such t h a t

V c (~-l(Wy~) N A ) - ( ~ - l ( W y ~ ) N A ) .

L e t sE V. L e t a, b E ~ - l ( W y n ) N A be such t h a t s = a - b . Then ~(a)=wlyn, ~(b)=w2y n with Wl, w~ E W. Thus we have

~(s) = ~ ( a - b) = 7(a, - b) ~(a) ~(b) -1 = ~(a, - b) w l w ; 1 E W a ~ W o.

Thus ~ is continuous at 0 as asserted.

L e t F(s)=~(s) Vs. Then

P(s + s ' ) = ~(s + s ' ) Vs+s, = ~(s +s')~(s, s')-~ Vs Vs,

= 4(8 +8t)~(S)~(St)~(8 ~ - 8 ' ) - 1 V s V s, = [ a ( 8 ) l ~ ( s t ) ,

so t h a t s ~ F ( s ) is a one-parameter u n i t a r y representation in B, which is strongly continuous a t 0, hence strongly continuous everywhere. Furthermore, if A E 2 t h e n

F(s) AF( - s) = Vs A V21 = H i S A H -~ = A ~ A A - %

L e t F ' ( s ) = F ( - s ) A% Then s-+ F'(s) is a strongly continuous one p a r a m e t e r u n i t a r y group in 9~', and A ~ = F(s)F'(s) for all s E R. Therefore the assumptions in L e m m a 1 are satisfied, so 2 has a faithful normal G-invariant semi-finite trace 3. L e t B be the positive self-adjoint operator in L1(9~, T) such t h a t c0(A)=v(BA) for A E2. Then if g E G we have

~:( U a B U ; ~ A ) = ~( BUg~ A Us) = co( Ual A Ua) = co(A) = ~( B A ).

B y the uniqueness of B, B = UoBUg 1 for all gE G, hence B is affiliated with ]~. This completes the proof of the theorem.

We note t h a t the converse of the theorem is a triviality.

COROLLARY 1.(1) Let assumptions and notation be as in Theorem 1. T h e n ~ is semi- finite.

P r o @ B y Theorem 1 co(A)=~(BA) for A E~, with B affiliated with B. Thus the modular automorphism at of a) is a t ( A ) = B * t A B -i~. Since B is affiliated with ~, at is also the modular automorphism of co restricted to ~. Since at l 73 is inner, ~ is semi-finite b y [14, Theorem 14.1].

The n e x t two corollaries are direct generalizations of theorems of Hugenholtz [3] and the author [12], see also [1, p. 101, Thdor~me 7].

COROLL-~Ru 2. Let 9.1 be a semi-finite von N e u m a n n algebra and G an ergodic group o/

*-automorphisms o/92[. Suppose co is a ]aith/ul normal G-invariant state o/9.i. T h e n 9.I is finite and co is a trace.

(1) This corollary also follows from [16].

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A U T O M O R P H I S M S A N D I N V A R I A N T S T A T E S O F O P E R A T O R A L G E B R A S 7 Proo/. L e t ~ a n d B be as in T h e o r e m 1. Since B = CI, B is a scalar h i , 2 > 0 . T h u s co(A) =~T(A) is a finite trace of 9~. I n particular 9~ is finite.

A m o r e direct proof of this corollary can be obtained if we notice t h a t if B~=C~H u as in t h e proof of T h e o r e m 1, t h e n Bt is a scalar, hence H ~ A H - ~ = A for t in a n e i g h b o r h o o d of 0 for all A ~ . T h u s H is affiliated with t h e center of 9~, so o~(A) = T r (HA) is a trace on 9~.

COROLLARY 3. Let ~ be a von N e u m a n n algebra acting on a Hilbert space ~ . Let G be a group o / u n i t a r y operators on ~ such that Ug~U -1 =9~ /or U E G. Suppose there exists a unit vector x o E ~ such that

(i) x o is cyclic /or 9~,

(if) Cx o is the set o/vectors in ~ invariant under G.

Then 9~ is o/type I I I i / a n d only i / x o is not a trace vector/or 9~'.

Proo/. L e t F = [9~'x0]. T h e n F is t h e s u p p o r t of t h e G-invariant state cox~ so F E B - - t h e fixed p o i n t algebra of G in 9~, Since x 0 is cyclic for 2 , it is separating for 0/', hence i~I' ~ i~I'F.

T h u s cox~ ]9~' is a trace if a n d o n l y if cox~ [~ ' ~ is a trace. I f 9~ is of t y p e I I I t h e n so is 9~', hence wx, ]9/' is n o t a trace. Conversely, assume cox~ is n o t a trace, hence o~x, ]9~'F is n o t a trace. W e show t h a t u n d e r this a s s u m p t i o n FO/F is of t y p e I I I , hence 9/'2' is of t y p e I I I , so t h a t 9~' is of t y p e I I I , a n d therefore 9~ is of t y p e I I I . W e m a y therefore assume F = I , i.e. we assume x 0 is separating a n d cyclic for 9~. L e t E 0 be t h e one dimensional projection on (~x 0. B y (if) a n d t h e ergodic t h e o r e m [11, w 144] E o E c o n v (U: UEG)-, so E o E B ' . T h u s c%~ is a faithful h o m o m o r p h i s m of B o n t o (~, so B = CI. Since t h e central projections in 9~

on t h e different t y p e portions of ~ are i n v a r i a n t u n d e r t h e automorphisms, t h e y are in B = CI. Therefore 9~ is either semi-finite or of t y p e I I I . I f 9~ is semi-finite t h e n b y Corollary 2 9~ is finite a n d wx. is a trace. Since x 0 is separating a n d cyclic for 9~, wx~ [ 9~' is also a trace, contradicting our hypothesis. Therefore 9~ is of t y p e I I I .

Remark. I f t h e y o n N e u m a n n algebra 9/is n o t semi-finite we can o b t a i n an analogue of T h e o r e m 1 as follows. Suppose eo a n d ~ are n o r m a l G-invariant states of 9~ with eo faithful. T h e n there exists a positive self-adjoint operator H affiliated with B such t h a t o(A) = eo(HAH) for all A E~. I n d e e d b y [7], see also [2], there exists a u n i q u e faithful n o r m a l G-invariant projection (I) of 9~ onto B such t h a t ~ = ( ~ ] B)o(I). B y t h e R a d o n - N i k o d y m T h e o r e m for y o n N e u m a n n algebras [14, T h e o r e m 15.1] there exists a positive self-adjoint operator H affiliated with B such t h a t o ( B ) = w ( H B H ) for B E B, hence o ( A ) = ~ ( r eo(H~P(A)H) for AE9/. B u t t h e state A ~ o ~ ( H A H ) is n o r m a l a n d G-invariant. H e n c e

~(A) = w ( H ( b ( A ) H ) =o~(HAH), A e ~ , as asserted.

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8 ERLI~G STORMER

**3. Asymptotically abelian C*-algebras**

I t was shown in [12] t h a t t h e specialization of Corollary 3 to factors was applicable to describe t h e t y p e s of i n v a r i a n t f a c t o r states of a s y m p t o t i c a l l y abelian C*-algebras. W e can n o w give a criterion valid for all extremal i n v a r i a n t states, a n d this can be done for t h e m o s t general of t h e different notions of a s y m p t o t i c abelianness, n a m e l y t h a t of G-abelian i n t r o d u c e d b y L a n f o r d a n d Ruelle [8]; see [2] for t h e other notions.

L e t .4 be a C*-algebra a n d G a g r o u p of *-automorphisms of .4. W e s a y .4 is G-abelian if for each G-invariant state @ of .4 a n d all self-adjoint operators A, BEOX we h a v e

0 = inf { I@([A', B])]: A ' Econv(g(A): gEG)).

L e t @(A)=(~Q(A)xQ, xQ) be its G e l f a n d - N a i m a r k - S e g a l decomposition, a n d g-+ Ug a uni- t a r y representation of G on t h e Hilbert space :H~ such t h a t UgxQ=x~, a n d ~Q(g(A))=

Ugz~(A) U~ 1, A C .4. T h e n @ is extremal i n v a r i a n t if a n d only if xo is up to a scalar multiple t h e unique v e c t o r y C ~H Q such t h a t Ugy = y for all g C G. W e t h u s h a v e t h e following i m m e d i a t e consequence of Corollary 3.

COROLLARY 4. Let ,-4 be a C*-algebra and G a group o/*-automorphisms o/.4. Suppose .4,4 is G-abelian and that @ is an extremal G-invariant state o/ .4. Then ~Q(.4)" is a von Neumann algebra o/ type I I I i/ and only i/coxq is not a trace when restricted to zQ(.4)'.

References

[1]. DIXMIER, J., Les alg~bres d'opdrateurs dans l'espace Hilbertien. Gauthier-u Paris, 1969.

[2]. DOPLICHER, S., KASTZER, D., & STOlaMER, E., Invariant states and asymptotic abelianness. J. Functional Analysis, 3 (1969), 419-434.

[3]. HUGENHOLTZ, INT., On the factor type of equilibrium states in q u a n t u m statistical mechanics. Comm. Math. Phys., 6 (1967), 189-193.

[4]. I-IEwI~T, E. & ROSS, K. A., Abstract harmonic analysis I. Springer-Verlag, Berlin, 1963.

[5]. KADISON, R. V., Transformations of states in operator theory and dynamics. Topology, 3 (1965), 177-198.

[6]. KArLANSKY, I., A theorem on rings of operators. Paci]ic J. Math., 1 (1951), 227-232.

[7]. KovXcs, I. & Szi2cs, J., Ergodic type theorems in yon Neumann algebras. Acta Sci.

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[8]. LAI~FORD, O. • R U E L L E , D., Integral representations of invariant states o n B*-algebras.

J. Math. Phys., 8 (1967), 1460-1463.

[9]. M A C K E Y , G., Les ensembles bordlicns ctles extensions des groupes. J. Math. Pures A p p L , 36 (1957), 171-178.

[10]. M A c L A ~ E , S., Homology. Springer-Verlag, Berlin, 1963.

[11]. RIESZ, F. & Sz.-NAGv, B., Lemons d'analyse ]onctioneUe. Akaddmiai Kiado, Budapest, 1955.

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Comm. Math. Phys., 6 (1967), 194-204.

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AUTOMORPHISMS AND INVARIANT STATES OF OPERATOR ALGEBRAS

[13]. STORMER, E., States a n d i n v a r i a n t maps of operator algebras. J. Functional Analysis, 5 (1970), 44-65.

[14]. TAKESAKI, M., Tomita's theory of modular Hilbert algebras and its applications. Springer- Verlag, Lecture notes in mathematics, 128 (1970).

[15]. TOMITA, M., S t a n d a r d forms of v o n I~eumann algebras. The Vth functional analysis sym- posium of the Math. Soc. of Japan, Sendal, 1967.

[16]. TOMIYAMA. J , On the projection of n o r m one in W*-algebras, I I I . T6hoku Math. J., 11 (1959), 125-129.

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Received November 28, 1969

AUTOMORPHISMS AND INVARIANT STATES OF OPERATOR ALGEBRAS