Positive element

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

Definition

Let be a *-algebra. An element is called positive if there are finitely many elements, so that This is also denoted by
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.

Examples

The unit element of an unital *-algebra is positive.
For each element, the elements and are positive by

In case is a C*-algebra, the following holds:

Let be a normal element, then for every positive function which is continuous on the spectrum of the continuous functional calculus defines a positive element
Every projection, i.e. every element for which holds, is positive. For the spectrum of such an idempotent element, holds, as can be seen from the continuous functional

Criteria

Let be a C*-algebra and Then the following are equivalent:

For the spectrum holds and is a normal element.
There exists an element, such that
There exists a self-adjoint element such that

If is a unital *-algebra with unit element, then in addition the following statements are

for every and is a self-adjoint element.
for some and is a self-adjoint element.

Properties

In *-algebras

Let be a *-algebra. Then:

If is a positive element, then is self-adjoint.
The set of positive elements is a convex cone in the real vector space of the self-adjoint elements This means that holds for all and
If is a positive element, then is also positive for every element
For the linear span of the following holds: and

In C*-algebras

Let be a C*-algebra. Then:

Using the continuous functional calculus, for every and there is a uniquely determined that satisfies, i.e. a unique -th root. In particular, a square root exists for every positive element. Since for every the element is positive, this allows the definition of a unique absolute value:
For every real number there is a positive element for which holds for all The mapping is continuous. Negative values for are also possible for invertible elements
Products of positive commutative elements are also positive. So if holds for positive, then
Each element can be uniquely represented as a linear combination of four positive elements. To do this, is first decomposed into the self-adjoint real and imaginary parts and these are then decomposed into positive and negative parts using the continuous functional For it holds that, since
If both and are positive
If is a C*-subalgebra of, then
If is another C*-algebra and is a *-homomorphism from to, then
If are positive elements for which, they commutate and holds. Such elements are called orthogonal and one writes

Partial order

Let be a *-algebra. The property of being a positive element defines a translation invariant partial order on the set of self-adjoint elements If holds for, one writes or
This partial order fulfills the properties and for all with
If is a C*-algebra, the partial order also has the following properties for :

If holds, then is true for every For every that commutes with and even
If holds, then
If holds, then holds for all real numbers
If is invertible and holds, then is invertible and for the inverses