polar decomposition in von Neumann algebras

- Let be a von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath acting on a Hilbert spaceMathworldPlanetmath H and T. If T=VR is the polar decompositionMathworldPlanetmath for T with KerV=KerR, then both V and R belong to .

Proof :

  • As is a C*-algebraPlanetmathPlanetmath (http://planetmath.org/CAlgebra), it is known that R=T*T belongs to . (proof will be added later)

  • To see that V also belongs to , by the double commutant theorem, it suffices to show that V belongs to ′′ (the double commutant of ).

    Suppose S. We intend to prove that V commutes with S.

    For xH we have that




    So SV and VS agree on RanR¯.

    As R is self-adjointPlanetmathPlanetmath, RanR¯=KerR, and so it remains to show that SV and VS agree on KerR. Recall that, by hypothesis, KerR=KerV.

    Let xKerR. We have that RSx=SRx=0 and therefore


    and so we can conclude that VS is identically zero in KerR.

    Clearly SV is also identically zero on KerR=KerV.

    Thus VS and SV agree on KerR. Therefore SV=VS and so V′′=

Title polar decomposition in von Neumann algebras
Canonical name PolarDecompositionInVonNeumannAlgebras
Date of creation 2013-03-22 17:28:54
Last modified on 2013-03-22 17:28:54
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Result
Classification msc 47A05
Classification msc 46L10