Conditional Expectations on Jordan Algebras

  • C. M. Edwards
Part of the NATO ASI Series book series (NSSB, volume 144)


In the late twenties Von Neumann’s model for quantum mechanics was introduced. His proposal was that the bounded observables of a quantum system should be represented by elements of the self-adjoint part L(H)sa of the algebra L(H) of bounded operators on a Hilbert space H. Since L(H)sa is not closed under the formation of products the usual algebraic structure clearly had no immediate physical relevance. However, L(H)sa is closed under the Jordan product defined for elements a and b in L(H)sa by
$$ \begin{gathered} a\;o\;b = \frac{1}{{2\,}}\,(ab + ba). \hfill \\ \hfill \\ \end{gathered} $$


Conditional Expectation Jordan Algebra Order Interval Isometric Isomorphism Cosine Family 
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    M. Takesaki, Conditional Expectations in Von Neumann algebras, J. Funct. Anal. 9:306 (1972).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • C. M. Edwards
    • 1
  1. 1.The Queen’s CollegeOxfordUK

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