INVARIANT STATES OF VON NEUMANN ALGEBRAS 255

be an infinite orthogonal sequence of projections in 0t, Then o^{F^ -> 0. Let e>0 be given and choose (5>0 as above. Choose n^ such that if u^Uq then co(P^) < Ô, But then a){g{P^)) < s for n^nQ,so that o){g(Pn)) -> 0 uniformly in G. By the lemma there is a normal ö-invariant state q of ^ such that QI ^^ = CO \ Si^, As pointed out in the proof of the lemma the support of q majorizes that of oi. Since со is faithful, the support of q is the identity, hence q is faithful.

Theorem . Let 3i be a von Neumann algebra and G a group of "^- morphisms of ^. Then M is G-finite if and only if G is relatively compact in the relative weak-operator topology on jSf^(^).

Proof . Suppose âl is ö-finite. Let {g^} be a net of automorphisms in G. We have to show that this net has a subnet which converges in the weak-operator topology to a map in JSf^(^). Since G is contained in the unit ball in JSf(^), which is weak-operator compact [7], there is a subnet {g^} which converges to a positive linear map (p in ^(ß) such that (p{I) — I, see [7]. In order to show (p e JSf^(^) it suffices to show that if CO is a normal state of Si then so is cooç?. Let

К = [œog : geG) .

Then C0099 is in the tü*-closure of K, Since ^ is G-finite, there is a ful normal ö-invariant projection map Ф oiM onto ^^, and every normal G-invariant state of M is of the form q = q\^^o0 (see [8] or [3]). In particular, с = со|^^оф satisfies the condition in the lemma, so by the lemma and the theorem of Akemann [1] referred to there the set К is weakly relatively compact in ^^. Therefore the ti;*-closure of К is tained in ^^, hence со о 99 is normal. Therefore cp e »Sf^(^), and G is tively compact.

Conversely , assume G is relatively compact in the relative weak- operator topology on c5f^(^). Let со be a normal state of ^. Then K=[(oog : ^ G 0} is weakly relatively compact in M^. Indeed, if {оуод^ is a net in К converging to a state q in the î^;*-topology, we can choose a subnet {g^} of [g^ converging in the weak-operator topology to (p E ^^(âê). Then coo^^-> C00Ç?, hence ^ = cooç? is a normal state, and К is weakly relatively compact. From the lemma and its proof ^ is G-finite.

Corollary . Let âl be a von Neumann algebra. Let G be a group of "^-automorhisms of ^ leaving the center of ^ element-wise fixed and con-