  Practically and intrinsically, inclusions of operator algebras are of
fundamental interest. The subject of this paper is intermediate operator
algebras of inclusions. There are two previously known theorems which naturally
and completely describe all intermediate operator algebras: the Galois
Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the
third, new complete description theorem which gives a canonical bijective
correspondence between intermediate operator algebras and intermediate
extensions of dynamical systems. One can also regard this theorem as a crossed
product splitting theorem, analogous to the Tensor Splitting Theorem. We then
give concrete applications, particularly to maximal amenability problem and a
new realization result of intermediate operator algebra lattice.
