  The relation between Bell-CHSH violation and factorization of Hilbert space
is considered here. That is, a state which is local in the sense of the
Bell-CHSH inequality under a certain factorization of the underlying Hilbert
space can be Bell-CHSH non-local under a different factorization. While this
question has been addressed with respect to separability , the relation of the
factorization with Bell-CHSH violation has remained hitherto unexplored. We
find here, that there is a set containing density matrices which do not exhibit
Bell-CHSH violation under any factorization of the Hilbert space brought about
by global unitary operations. Using the Cartan decomposition of $ SU(4) $,we
characterize the set in terms of a necessary and sufficient criterion based on
the spectrum of density matrices. Sufficient conditions are obtained to
characterize such density matrices based on their bloch representations. For
some classes of density matrices, necessary and sufficient conditions are
derived in terms of bloch parameters. Furthermore, an estimation of the volume
of such density matrices is achieved in terms of purity. The criterion is
applied to some well-known class of states in two qubits.Since, both local
filtering and global unitary operations influence Bell-CHSH violation of a
state, a comparative study is made between the two operations. The
inequivalence of the two operations(in terms of increasing Bell-CHSH violation)
is exemplified through their action on some classes of states.
