lemma 7.2 required in depth

Let M be a closed strongly semi-positive and rational symplectic manifold. For an element Ham(M) write for brevity c()=c([M],) and, as above, define as a homogenization of c([M],) : ():=vol(M)limk+c(k)/k. It is easy to see that is not a quasi-morphism already when M is the 2torus - see the cliscussion following Question 8.7 in Section 8. Moreover, a similar argument actually shows that for any (strongly semi-positive, rational) symplectic direct product MT2n the homogenization of any spectral number c(a,) cannot be a quasi-morphism. In spite of this, has a number of nice properties which will enable us to show that the functional given by (4) is a partial symplectic quasi-state. We shall neel the following definition. Given a displaceable open set UM, each Ham(M) can be represented as a product of elements of the form 1 with Ham(U). This follows from Banyaga's fragmentation lemma [9]. Denote by U the minimal number of factors in such a product. Theorem 7.1. Suppose M is strongly semi-positive and rational. The functional :H um (M)R, given by (5), is well defined and has the following properties: (FH)=k1((FH)k)=k1(k(F)+k(H)+rk), where rk