  We provide a framework for Bell inequalities which is based on multilinear
contractions. The derivation of the inequalities allows for an intuitive
geometric depiction and their violation within quantum mechanics can be seen as
a direct consequence of non-vanishing commutators. The approach is motivated by
generalizing recent work on non-linear inequalities which was based on the
moduli of complex numbers, quaternions and octonions. We extend results on
Peres conjecture about the validity of Bell inequalities for quantum states
with positive partial transposes. Moreover, we show the possibility of
obtaining unbounded quantum violations albeit we also prove that quantum
mechanics can only violate the derived inequalities if three or more parties
are involved.
