Positive element

In mathematics, an element of a *-algebra is called positive if it is the sum of elements of the form

a

An element

is called positive if there are finitely many elements

holds.

[1] This is also denoted by

[2] The set of positive elements is denoted by

A special case from particular importance is the case where

is a complete normed *-algebra, that satisfies the C*-identity (

), which is called a C*-algebra.

In case

is a C*-algebra, the following holds: Let

Then the following are equivalent:[4] If

is a unital *-algebra with unit element

, then in addition the following statements are equivalent:[5] Let

The property of being a positive element defines a translation invariant partial order on the set of self-adjoint elements

{\displaystyle {\mathcal {A}}_{sa}}

holds for

, one writes

[13] This partial order fulfills the properties

{\displaystyle ta\leq tb}

{\displaystyle a,b,c\in {\mathcal {A}}_{sa}}

is a C*-algebra, the partial order also has the following properties for