Positive element

In mathematics, especially functional analysis, a hermitian element "A" of a C*-algebra is a positive element if its spectrum consists of nonnegative real numbers. It is possible to prove that an element "x" of a C*-algebra "A" is positive if and only if there is some "b" in "A" such that "x" = "b"^*"b".

If "A" is a bounded linear operator on a Hilbert space "H", then this notion coincides with the condition that "A" is self-adjoint and$langle\; Ax,x\; angle$ is positive for every vector "x" in "H". Note that $langle\; Ax,x\; angle$ is real for every "x" in "H" since "A" is self-adjoint.

Partial ordering using positivity

Hermitian elements are also called self-adjoint.By introducing the convention:$A\; leq\; B\; Longleftrightarrow\; B-A\; extrm\{\; is\; \}\; extrm\{\; positive\; \}$for self-adjoint elements in a C*-algebra $mathcal\; A$, one obtains a partial ordering of $mathcal\; A$.

This partial ordering is analoguous to the ordering of real numbers, but only to some extent. For example, it respects multiplication by positive reals and addition with positive elements, but $ABgeq\; CD$ need not hold for positive elements $A,B,C,Din\; mathcal\; A$ with $Ageq\; C$ and $Bgeq\; D$.