Positive element

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

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

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

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

• The unit element ${\displaystyle e}$ of an unital *-algebra is positive.
• For each element ${\displaystyle a\in {\mathcal {A}}}$, the elements ${\displaystyle a^{*}a}$ and ${\displaystyle aa^{*}}$ are positive by definition.¹

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

• Let ${\displaystyle a\in {\mathcal {A}}_{N}}$ be a normal element, then for every positive function ${\displaystyle f\geq 0}$ which is continuous on the spectrum of ${\displaystyle a}$ the continuous functional calculus defines a positive element ${\displaystyle f(a)}$.³
• Every projection, i.e. every element ${\displaystyle a\in {\mathcal {A}}}$ for which ${\displaystyle a=a^{*}=a^{2}}$ holds, is positive. For the spectrum ${\displaystyle \sigma (a)}$ of such an idempotent element, ${\displaystyle \sigma (a)\subseteq \{0,1\}}$ holds, as can be seen from the continuous functional calculus.³

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

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

If ${\displaystyle {\mathcal {A}}}$ is a unital *-algebra with unit element ${\displaystyle e}$, then in addition the following statements are equivalent:⁵

• ${\displaystyle \left\|te-a\right\|\leq t}$ for every ${\displaystyle t\geq \left\|a\right\|}$ and ${\displaystyle a}$ is a self-adjoint element.
• ${\displaystyle \left\|te-a\right\|\leq t}$ for some ${\displaystyle t\geq \left\|a\right\|}$ and ${\displaystyle a}$ is a self-adjoint element.

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

• If ${\displaystyle a\in {\mathcal {A}}_{+}}$ is a positive element, then ${\displaystyle a}$ is self-adjoint.⁶
• The set of positive elements ${\displaystyle {\mathcal {A}}_{+}}$ is a convex cone in the real vector space of the self-adjoint elements ${\displaystyle {\mathcal {A}}_{sa}}$. This means that ${\displaystyle \alpha a,a+b\in {\mathcal {A}}_{+}}$ holds for all ${\displaystyle a,b\in {\mathcal {A}}}$ and ${\displaystyle \alpha \in [0,\infty )}$.⁶
• If ${\displaystyle a\in {\mathcal {A}}_{+}}$ is a positive element, then ${\displaystyle b^{*}ab}$ is also positive for every element ${\displaystyle b\in {\mathcal {A}}}$.⁷
• For the linear span of ${\displaystyle {\mathcal {A}}_{+}}$ the following holds: ${\displaystyle \langle {\mathcal {A}}_{+}\rangle ={\mathcal {A}}^{2}}$ and ${\displaystyle {\mathcal {A}}_{+}-{\mathcal {A}}_{+}={\mathcal {A}}_{sa}\cap {\mathcal {A}}^{2}}$.⁸

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

• Using the continuous functional calculus, for every ${\displaystyle a\in {\mathcal {A}}_{+}}$ and ${\displaystyle n\in \mathbb {N} }$ there is a uniquely determined ${\displaystyle b\in {\mathcal {A}}_{+}}$ that satisfies ${\displaystyle b^{n}=a}$, i.e. a unique ${\displaystyle n}$-th root. In particular, a square root exists for every positive element. Since for every ${\displaystyle b\in {\mathcal {A}}}$ the element ${\displaystyle b^{*}b}$ is positive, this allows the definition of a unique absolute value: ${\textstyle |b|=(b^{*}b)^{\frac {1}{2}}}$.⁹
• For every real number ${\displaystyle \alpha \geq 0}$ there is a positive element ${\displaystyle a^{\alpha }\in {\mathcal {A}}_{+}}$ for which ${\displaystyle a^{\alpha }a^{\beta }=a^{\alpha +\beta }}$ holds for all ${\displaystyle \beta \in [0,\infty )}$. The mapping ${\displaystyle \alpha \mapsto a^{\alpha }}$ is continuous. Negative values for ${\displaystyle \alpha }$ are also possible for invertible elements ${\displaystyle a}$.⁷
• Products of positive commutative elements are also positive. So if ${\displaystyle ab=ba}$ holds for positive ${\displaystyle a,b\in {\mathcal {A}}_{+}}$, then ${\displaystyle ab\in {\mathcal {A}}_{+}}$.⁵
• Each element ${\displaystyle a\in {\mathcal {A}}}$ can be uniquely represented as a linear combination of four positive elements. To do this, ${\displaystyle a}$ 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 ${\displaystyle {\mathcal {A}}_{sa}={\mathcal {A}}_{+}-{\mathcal {A}}_{+}}$, since ${\displaystyle {\mathcal {A}}^{2}={\mathcal {A}}}$.⁸
• If both ${\displaystyle a}$ and ${\displaystyle -a}$ are positive ${\displaystyle a=0}$ holds.⁵
• If ${\displaystyle {\mathcal {B}}}$ is a C*-subalgebra of ${\displaystyle {\mathcal {A}}}$, then ${\displaystyle {\mathcal {B}}_{+}={\mathcal {B}}\cap {\mathcal {A}}_{+}}$.⁵
• If ${\displaystyle {\mathcal {B}}}$ is another C*-algebra and ${\displaystyle \Phi }$ is a *-homomorphism from ${\displaystyle {\mathcal {A}}}$ to ${\displaystyle {\mathcal {B}}}$, then ${\displaystyle \Phi ({\mathcal {A}}_{+})=\Phi ({\mathcal {A}})\cap {\mathcal {B}}_{+}}$ holds.¹¹
• If ${\displaystyle a,b\in {\mathcal {A}}_{+}}$ are positive elements for which ${\displaystyle ab=0}$, they commutate and ${\displaystyle \left\|a+b\right\|=\max(\left\|a\right\|,\left\|b\right\|)}$ holds. Such elements are called orthogonal and one writes ${\displaystyle a\bot b}$.¹²

Let ${\displaystyle {\mathcal {A}}}$ be a *-algebra. The property of being a positive element defines a translation invariant partial order on the set of self-adjoint elements ${\displaystyle {\mathcal {A}}_{sa}}$. If ${\displaystyle b-a\in {\mathcal {A}}_{+}}$ holds for ${\displaystyle a,b\in {\mathcal {A}}}$, one writes ${\displaystyle a\leq b}$ or ${\displaystyle b\geq a}$.¹³

This partial order fulfills the properties ${\displaystyle ta\leq tb}$ and ${\displaystyle a+c\leq b+c}$ for all ${\displaystyle a,b,c\in {\mathcal {A}}_{sa}}$ with ${\displaystyle a\leq b}$ and ${\displaystyle t\in [0,\infty )}$.⁸

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

• If ${\displaystyle a\leq b}$ holds, then ${\displaystyle c^{*}ac\leq c^{*}bc}$ is true for every ${\displaystyle c\in {\mathcal {A}}}$. For every ${\displaystyle c\in {\mathcal {A}}_{+}}$ that commutes with ${\displaystyle a}$ and ${\displaystyle b}$ even ${\displaystyle ac\leq bc}$ holds.¹⁴
• If ${\displaystyle -b\leq a\leq b}$ holds, then ${\displaystyle \left\|a\right\|\leq \left\|b\right\|}$.¹⁵
• If ${\displaystyle 0\leq a\leq b}$ holds, then ${\textstyle a^{\alpha }\leq b^{\alpha }}$ holds for all real numbers ${\displaystyle 0<\alpha \leq 1}$.¹⁶
• If ${\displaystyle a}$ is invertible and ${\displaystyle 0\leq a\leq b}$ holds, then ${\displaystyle b}$ is invertible and for the inverses ${\displaystyle b^{-1}\leq a^{-1}}$ holds.¹⁵