A REDUCTION THEORY FOR OPERATORS IN TYPE IN VON NEUMANN ALGEBRAS

In this paper, we study the structures of operators in a type I-n von Neumann algebra A. As an analogue of the Jordan canonical form theorem, for an operator A in A, we prove that if {A}' boolean AND A contains a bounded maximal Boolean algebra of idempotents, then the bounded maximal Boolean algebras of idempotents in the relative commutant {A}' boolean AND A are the same up to similarity. Meanwhile we characterize the structures for operators in A whose relative commutants contain bounded maximal Boolean algebras of idempotents. We also classify this class of operators by K-theory for Banach algebras. We use techniques of von Neumann's reduction theory in our proofs.