  We discuss new approach to mathematical foundations of quantum theory, which
is completely independent of Hilbert spaces and measure spaces. New kinematics
is defined by non-linear geometry of spaces of integrals on abstract
non-commutative algebras. New dynamics is defined by constrained maximisation
of quantum relative entropy. We recover Hilbert space based approach (including
unitary evolution and the von Neumann--L\"{u}ders rule) and measure theoretic
approach to probability theory (including Bayes' rule) as special cases of our
approach.
