Positive_element

Positive element

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In mathematics, an element of a *-algebra is called positive if it is the sum of elements of the form $a^{*}a$ .^[1]

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 ${\textstyle a=\sum _{k=1}^{n}a_{k}^{*}a_{k}}$ holds.^[1] This is also denoted by $a\geq 0$ .^[2]

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

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.^[1]

In case ${\mathcal {A}}$ is a C*-algebra, the following holds:

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 $f(a)$ .^[3]
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.^[3]

Criteria

Let ${\mathcal {A}}$ be a C*-algebra and $a\in {\mathcal {A}}$ . Then the following are equivalent:^[4]

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:^[5]

$\left\|te-a\right\|\leq t$ 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.

Properties

In *-algebras

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

If $a\in {\mathcal {A}}_{+}$ is a positive element, then $a$ is self-adjoint.^[6]
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 $\alpha \in [0,\infty )$ .^[6]
If $a\in {\mathcal {A}}_{+}$ is a positive element, then $b^{*}ab$ is also positive for every element $b\in {\mathcal {A}}$ .^[7]
For the linear span of ${\mathcal {A}}_{+}$ the following holds: $\langle {\mathcal {A}}_{+}\rangle ={\mathcal {A}}^{2}$ and ${\mathcal {A}}_{+}-{\mathcal {A}}_{+}={\mathcal {A}}_{sa}\cap {\mathcal {A}}^{2}$ .^[8]

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: ${\textstyle |b|=(b^{*}b)^{\frac {1}{2}}}$ .^[9]
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 $a$ .^[7]
Products of commutative positive elements are also positive. So if $ab=ba$ holds for positive $a,b\in {\mathcal {A}}_{+}$ , then $ab\in {\mathcal {A}}_{+}$ .^[5]
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.^[10] For it holds that ${\mathcal {A}}_{sa}={\mathcal {A}}_{+}-{\mathcal {A}}_{+}$ , since ${\mathcal {A}}^{2}={\mathcal {A}}$ .^[8]
If both $a$ and $-a$ are positive $a=0$ holds.^[5]
If ${\mathcal {B}}$ is a C*-subalgebra of ${\mathcal {A}}$ , then ${\mathcal {B}}_{+}={\mathcal {B}}\cap {\mathcal {A}}_{+}$ .^[5]
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.^[11]
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 $a\bot b$ .^[12]

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 $b\geq a$ .^[13]

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 $t\in [0,\infty )$ .^[8]

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.^[14]
If $-b\leq a\leq b$ holds, then $\left\|a\right\|\leq \left\|b\right\|$ .^[15]
If $0\leq a\leq b$ holds, then ${\textstyle a^{\alpha }\leq b^{\alpha }}$ holds for all real numbers $0<\alpha \leq 1$ .^[16]
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.^[15]

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[FOOTNOTEBlackadar200663-2] [2]
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[FOOTNOTEKadisonRingrose1983250-15] [15]
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