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Let G be a countable group of automorphisms on a W* algebra M and let 0o be a w*G6 point of the set of G invariant states on M which belong to w* cl Co E, where E is a set of (possibly pure) states on M. If G is amenable, then the cyclic representation 7ro, corresponding to 4o is contained in {ff lr,; k E E}. This property characterizes amenable groups. Related results are obtained. Introduction. Let M be an infinite dimensional W* algebra, M* its Banach space dual, G a group of automorphisms g: M -* M, and E a set of states on M such that G*E c E. Denote by SG the set of states q in w* cl Co E (the w* closure of the convex hull of E) which are G-invariant, i.e. q(gx) = q(x) for all g in G and x in M. We call q0 in SG a w*Gs point of SG if there is a sequence J = {xn} in M such that {qo} = SG n Jo, where JO = {q E M*; (q, J) = 0}. This just means that the w* topology restricted to SG is first countable at qo. Denote by w*G6 (Si) the set of all such points. We note that 0q being in w*Gs (SG) depends only on the w* topology restricted to SG and does not depend a priori, on what happens in w* cl Co E. Furthermore w*Gs(SG) may be void. If for some countable J c M the set (SG n Jo, w*) is separable metric or even if it has the RNP (even the WRNP is enough in some cases; see the sequel for notations), then w*Gs (Si) G 0. If 4 E M* is positive let 7ro be the GNS representation determined by q (Pedersen [12, 3.3.3]). The main result of this paper is (a refinement of) the following THEOREM. Let q$o E w*G6(SG). Then 7rok0 is contained (not only weakly contained a la Fell) in the direct sum {( 7r,; q$ E E} provided G is countable and amenable. If G is any nonamenable group then there is even an abelian M = L??(X,u) for some nonatomic probability space (X, B, ,i) on which G acts ergodically and measure preservingly such that if E is the set of all pure states on M then w* G (SG) = {,u} = SG, yet 7r1 is not contained (but only weakly contained) in {( 7ro; q E E}. Received by the editors November 4, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L30, 46L40, 46L55; Secondary 46L10. (?)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

A strong containment property for discrete amenable groups of automorphisms on 𝑊* algebras