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Crossed ProductsCompact Quantum GroupHomology TheoryAbelian GroupTensor ProductVon Neumann AlgebraDirac OperatorsDifferential OperatorsFundamental GroupHecke AlgebraVector BundleIsomorphismAbelian GroupsNoncommutative GeometryDynamical Systemthe Locally Symmetric SpaceCompact Lie GroupRham CohomologyIwasawa TheoryOperator Algebras