Positive element

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

Definition
Let $$\mathcal{A}$$ be a *-algebra. An element $$a \in \mathcal{A}$$ is called positive if there are finitely many elements $$a_k \in \mathcal{A} \; (k = 1,2,\ldots,n)$$, so that $a = \sum_{k=1}^n a_k^*a_k$ holds. This is also denoted by $a \geq 0$.

The set of positive elements is denoted by $\mathcal{A}_+$.

A special case from particular importance is the case where $$\mathcal{A}$$ is a complete normed *-algebra, that satisfies the C*-identity ($$\left\| a^*a \right\| = \left\| a \right\|^2 \ \forall a \in \mathcal{A}$$), which is called a C*-algebra.

Examples
In case $$\mathcal{A}$$ is a C*-algebra, the following holds:
 * The unit element $$e$$ of an unital *-algebra is positive.
 * For each element $$a \in \mathcal{A}$$, the elements $$a^* a$$ and $$aa^*$$ are positive by definition.
 * Let $$a \in \mathcal{A}_N$$ be a normal element, then for every positive function $$f \geq 0$$ which is continuous on the spectrum of $$a$$ the continuous functional calculus defines a positive element $f(a)$.
 * Every projection, i.e. every element $$a \in \mathcal{A}$$ for which $$a = a^* = a^2$$ holds, is positive. For the spectrum $$\sigma(a)$$ of such an idempotent element, $$\sigma(a) \subseteq \{ 0, 1 \}$$ holds, as can be seen from the continuous functional calculus.

Criteria
Let $$\mathcal{A}$$ be a C*-algebra and $a \in \mathcal{A}$. Then the following are equivalent:


 * For the spectrum $$\sigma(a) \subseteq [0, \infty)$$ holds and $$a$$ is a normal element.
 * There exists an element $$b \in \mathcal{A}$$, such that $a = bb^*$.
 * There exists a (unique) self-adjoint element $$c \in \mathcal{A}_{sa}$$ such that $a = c^2$.

If $$\mathcal{A}$$ is a unital *-algebra with unit element $$e$$, then in addition the following statements are equivalent:
 * $$\left\| te - a \right\| \leq t$$ for every $$t \geq \left\| a \right\|$$ and $$a$$ is a self-adjoint element.
 * $$\left\| te - a \right\| \leq t$$ for some $$t \geq \left\| a \right\|$$ and $$a$$ is a self-adjoint element.

In *-algebras
Let $$\mathcal{A}$$ be a *-algebra. Then:


 * If $$a \in \mathcal{A}_+$$ is a positive element, then $$a$$ is self-adjoint.
 * The set of positive elements $$\mathcal{A}_+$$ is a convex cone in the real vector space of the self-adjoint elements $\mathcal{A}_{sa}$. This means that $$\alpha a, a+b \in \mathcal{A}_+$$ holds for all $$a,b \in \mathcal{A}$$ and $\alpha \in [0, \infty)$.
 * If $$a \in \mathcal{A}_+$$ is a positive element, then $$b^*ab$$ is also positive for every element $b \in \mathcal{A}$.
 * For the linear span of $$\mathcal{A}_+$$ the following holds: $$\langle \mathcal{A}_+ \rangle = \mathcal{A}^2$$ and $\mathcal{A}_+ - \mathcal{A}_+ = \mathcal{A}_{sa} \cap \mathcal{A}^2$.

In C*-algebras
Let $$\mathcal{A}$$ be a C*-algebra. Then:


 * Using the continuous functional calculus, for every $$a \in \mathcal{A}_+$$ and $$n \in \mathbb{N}$$ there is a uniquely determined $$b \in \mathcal{A}_+$$ that satisfies $$b^n = a$$, i.e. a unique $$n$$-th root. In particular, a square root exists for every positive element. Since for every $$b \in \mathcal{A}$$ the element $$b^*b$$ is positive, this allows the definition of a unique absolute value: $
 * For every real number $$\alpha \geq 0$$ there is a positive element $$a^\alpha \in \mathcal{A}_+$$ for which $$a^\alpha a^\beta = a^{\alpha + \beta}$$ holds for all $\beta \in [0, \infty)$. The mapping $$\alpha \mapsto a^\alpha$$ is continuous. Negative values for $$\alpha$$ are also possible for invertible elements $a$.
 * Products of commutative positive elements are also positive. So if $$ab = ba$$ holds for positive $$a,b \in \mathcal{A}_+$$, then $ab \in \mathcal{A}_+$.
 * Each element $$a \in \mathcal{A}$$ can be uniquely represented as a linear combination of four positive elements. To do this, $$a$$ 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 calculus. For it holds that $$\mathcal{A}_{sa} = \mathcal{A}_+ - \mathcal{A}_+$$, since $\mathcal{A}^2 = \mathcal{A}$.
 * If both $$a$$ and $$-a$$ are positive $$a = 0$$ holds.
 * If $$\mathcal{B}$$ is a C*-subalgebra of $$\mathcal{A}$$, then $\mathcal{B}_+ = \mathcal{B} \cap \mathcal{A}_+$.
 * If $$\mathcal{B}$$ is another C*-algebra and $$\Phi$$ is a *-homomorphism from $$\mathcal{A}$$ to $$\mathcal{B}$$, then $$\Phi(\mathcal{A}_+) = \Phi(\mathcal{A}) \cap \mathcal{B}_+$$ holds.
 * If $$a,b \in \mathcal{A}_+$$ are positive elements for which $$ab = 0$$, they commutate and $$\left\| a + b \right\| = \max(\left\| a \right\|, \left\| b \right\|)$$ holds. Such elements are called orthogonal and one writes $a \bot b$.

Partial order
Let $$\mathcal{A}$$ be a *-algebra. The property of being a positive element defines a translation invariant partial order on the set of self-adjoint elements $\mathcal{A}_{sa}$. If $$b - a \in \mathcal{A}_+$$ holds for $$a,b \in \mathcal{A}$$, one writes $$a \leq b$$ or $b \geq a$.

This partial order fulfills the properties $$ta \leq tb$$ and $$a + c \leq b + c$$ for all $$a,b,c \in \mathcal{A}_{sa}$$ with $a \leq b$ and $t \in [0, \infty)$.

If $$\mathcal{A}$$ is a C*-algebra, the partial order also has the following properties for $$a,b \in \mathcal{A}$$:


 * If $$a \leq b$$ holds, then $$c^*ac \leq c^*bc$$ is true for every $c \in \mathcal{A}$. For every $$c \in \mathcal{A}_+$$ that commutates with $$a$$ and $$b$$ even $$ac \leq bc$$ holds.
 * If $$-b \leq a \leq b$$ holds, then $\left\
 * If $$0 \leq a \leq b$$ holds, then $a^\alpha \leq b^\alpha$ holds for all real numbers $0 < \alpha \leq 1$.
 * If $$a$$ is invertible and $$0 \leq a \leq b$$ holds, then $$b$$ is invertible and for the inverses $$b^{-1} \leq a^{-1}$$ holds.