  Schwarz showed that when a closed symplectic manifold (M,\om) is
symplectically aspherical (i.e. the symplectic form and the first Chern class
vanish on \pi_2(M)) then the spectral invariants, which are initially defined
on the universal cover of the Hamiltonian group, descend to the Hamiltonian
group Ham (M,\om). In this note we describe less stringent conditions on the
Chern class and quantum homology of M under which the (asymptotic) spectral
invariants descend to Ham (M,\om). For example, they descend if the quantum
multiplication of M is undeformed and H_2(M) has rank >1, or if the minimal
Chern number is at least n+1 (where \dim M=2n) and the even cohomology of M is
generated by divisors. The proofs are based on certain calculations of genus
zero Gromov--Witten invariants.
  As an application, we show that the Hamiltonian group of the one point blow
up of T^4 admits a Calabi quasimorphism. Moreover, whenever the (asymptotic)
spectral invariants descend it is easy to see that Ham (M,\om) has infinite
diameter in the Hofer norm. Hence our results establish the infinite diameter
of Ham in many new cases. We also show that the area pseudonorm -- a geometric
version of the Hofer norm -- is nontrivial on the (compactly supported)
Hamiltonian group for all noncompact manifolds as well as for a large class of
closed manifolds.
