Positive element

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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\rangle is positive for every vector x in H. Note that \langle Ax,x\rangle is real for every x in H since A is self-adjoint.

[edit] Partial ordering using positivity

Hermitian elements are also called self-adjoint. By introducing the convention

A \leq B \Longleftrightarrow B-A \ \textrm{ is }\ \textrm{ 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 AB\geq CD need not hold for positive elements A,B,C,D\in \mathcal A with A\geq C and B\geq D.