Fundamental Aspects of Quantum Theory pp 75-79 | Cite as

# Conditional Expectations on Jordan Algebras

Chapter

## Abstract

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$$ \begin{gathered} a\;o\;b = \frac{1}{{2\,}}\,(ab + ba). \hfill \\ \hfill \\ \end{gathered} $$

## Keywords

Conditional Expectation Jordan Algebra Order Interval Isometric Isomorphism Cosine Family
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## References

- 1.P. Jordan, J. Von Neumann and E. Wigner. On an algebraic generalization of the quantum mechanical formalism, Ann. Math., 35:307 (1934).CrossRefGoogle Scholar
- 2.H. Hanche-Olsen and E. Stormer, “Jordan Operator Algebras,” Pitman, Boston, London, Melbourne (1984).MATHGoogle Scholar
- 3.S. Sakai, “C*-algebras and W*-algebras,” Springer, Berlin, Heidelberg, New York (1971).Google Scholar
- 4.U. Haagerup and H. Hanche-Olsen, Tomita-Takesaki theory for Jordan algebras, preprint (1982).Google Scholar
- 5.M. Takesaki, Conditional Expectations in Von Neumann algebras, J. Funct. Anal. 9:306 (1972).MathSciNetMATHCrossRefGoogle Scholar

## Copyright information

© Plenum Press, New York 1986