Affiliated operator
In mathematics, affiliated operators were introduced by Murray and von Neumann in the theory of von Neumann algebras as a technique for using unbounded operators to study modules generated by a single vector. Later Atiyah and Singer showed that index theorems for elliptic operators on closed manifolds with infinite fundamental group could naturally be phrased in terms of unbounded operators affiliated with the von Neumann algebra of the group. Algebraic properties of affiliated operators have proved important in L2 cohomology, an area between analysis and geometry that evolved from the study of such index theorems.
Let M be a von Neumann algebra acting on a Hilbert space H. A closed and densely defined operator A is said to be affiliated with M if A commutes with every unitary operator U in the commutant of M. Equivalent conditions are that:
The last condition follows by uniqueness of the polar decomposition. If A has a polar decomposition
it says that the partial isometry V should lie in M and that the positive self-adjoint operator |A| should be affiliated with M. However, by the spectral theorem, a positive self-adjoint operator commutes with a unitary operator if and only if each of its spectral projections does. This gives another equivalent condition:
![{\displaystyle E([0,N])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/649b3b2625d1615f57fe0dcef82938e94d02034e)
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