Partial Ordering Using Positivity
By introducing the convention
for self-adjoint elements in a C*-algebra, one obtains a partial order on the set of self-adjoint elements in . Note that according to this convention, we have if and only if is positive, which is convenient.
This partial order is analoguous to the natural order on the real numbers, but only to some extent. For example, it respects multiplication by positive reals and addition of self-adjoint elements, but need not hold for positive elements with and .
Read more about this topic: Positive Element
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