Affiliated operator

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.This gives another equivalent condition: In general the operators affiliated with a von Neumann algebra M need not necessarily be well-behaved under either addition or composition.This algebra of unbounded operators is complete for a natural topology, generalising the notion of convergence in measure.Indeed in this case, thanks to the Tomita–Takesaki theory, it is known that the non-commutative Lp spaces are no longer realised by operators affiliated with the von Neumann algebra.
mathematicsMurrayvon Neumannvon Neumann algebrasunbounded operatorsAtiyahSingerindex theoremselliptic operatorsclosed manifoldsfundamental groupL2 cohomologyanalysisgeometryvon Neumann algebraHilbert spaceclosedunitary operatorcommutantdomainpolar decompositionpartial isometryself-adjointspectral theoremGelfand–Naimark–SegalEdward Nelson*-algebraconvergence in measureHilbert–Schmidt operatorsSchattenvon Neumann regular ringTomita–Takesaki theoryConnes