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: each unitary U in M should leave invariant the graph of A defined by . the projection onto G(A) should lie in M2(M). each unitary U in M''' should carry D(A), the domain of A, onto itself and satisfy UAU* = A there. each unitary U in M should commute with both operators in the polar decomposition of A. 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: each spectral projection of |A| and the partial isometry in the polar decomposition of A lies in M. In general the operators affiliated with a von Neumann algebra M need not necessarily be well-behaved under either addition or composition. However in the presence of a faithful semi-finite normal trace τ and the standard Gelfand–Naimark–Segal action of M on H = L2(M, τ), Edward Nelson proved that the measurable affiliated operators do form a *-algebra with nice properties: these are operators such that τ(I − E([0,N])) < ∞ for N sufficiently large.