Editing C*-algebra (section)

=== Self-adjoint elements ===

Self-adjoint elements are those of the form <math> x = x^* </math>. The set of elements of a C*-algebra ''A'' of the form <math> x^*x </math> forms a closed [[convex cone]].  This cone is identical to the elements of the form <math> xx^* </math>. Elements of this cone are called ''non-negative'' (or sometimes ''positive'', even though this terminology conflicts with its use for elements of <math>\mathbb{R}</math>)

The set of self-adjoint elements of a C*-algebra ''A'' naturally has the structure of a [[partial order|partially ordered]] [[vector space]]; the ordering is usually denoted <math> \geq </math>. In this ordering, a self-adjoint element <math> x \in A </math> satisfies <math> x \geq 0 </math> if and only if the [[Spectrum (functional analysis)|spectrum]] of <math> x </math>  is non-negative, if and only if <math> x = s^*s </math> for some <math> s \in A</math>. Two self-adjoint elements <math>x</math> and <math> y </math> of ''A'' satisfy <math> x \geq y </math> if <math> x - y \geq 0 </math>.

This partially ordered subspace allows the definition of a [[positive linear functional]] on a C*-algebra, which in turn is used to define the [[State (functional analysis)|states]] of a C*-algebra, which in turn can be used to construct the [[spectrum of a C*-algebra]] using the [[GNS construction]].