  After the mathematical notion of "Micro-Macro Duality" for understanding
mutual relations between microsopic quantum systems (Micro) and their
macroscopic manifestations (Macro) is explained on the basis of the notion of
sectors and order parameters, a general mathematical scheme is proposed for
detecting the state-structure inside of a sector through measurement processes
of a maximal abelian subalgebra of the algebra of observables. For this
purpose, the Kac-Takesaki operators controlling group duality play essential
roles, which naturally leads to the composite system of the observed system and
the measuring system formulated by a crossed product. This construction of
composite systems will be shown to make it possible for the Micro to be
reconstructed from its observational data as Macro in the light of the Takesaki
duality for crossed products.
