  The spectral metric, defined by Schwarz and Oh using Floer-theoretical
method, is a bi-invariant metric on the Hamiltonian diffeomorphism group. We
show in this note that for certain symplectic manifolds, this metric can not be
extended to a bi-invariant metric on the full group of symplectomorphisms. We
also study the bounded isometry conjecture of Lalonde and Polterovich in the
context of the spectral metric. In particular, we show that the conjecture
holds for the torus with all linear symplectic forms.
