Conditional Expectations on Jordan Algebras

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

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