  In this paper we give a decomposition of a state on a $C^*$-algebra into a
family of pure states and a decomposition of a representation into a family of
irreducible representation. Then, we use it to solve the following three
problems and/or conjectures.. (1) The noncommutative Stone-Weierstrass problem,
(2) The extension problem (asked by Arveson) of a pure state on a nonseparable
operator system to a boundary state on the generated $C^*$-algebra, and (3) The
hyperrigidity problem of an operator system under the hypothesis that pure
states have the unique extension property, conjectured by Arveson.
