Positive Element
In mathematics, especially functional analysis, a self-adjoint (or hermitian) element of a C*-algebra is called positive if its spectrum consists of non-negative real numbers. Moreover, an element of a C*-algebra is positive if and only if there is some in such that . A positive element is self-adjoint and thus normal.
If is a bounded linear operator on a Hilbert space, then this notion coincides with the condition that is non-negative for every vector in . Note that is real for every in if and only if ' is self-adjoint. Hence, a positive operator on a Hilbert space is always self-adjoint (and a self-adjoint everywhere defined operator on a Hilbert space is always bounded because of the Hellinger-Toeplitz theorem).
The set of positive elements of a C*-algebra forms a convex cone.
Read more about Positive Element: Positive and Positive Definite Operators, Examples, Partial Ordering Using Positivity
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