Positive element

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Let ${\displaystyle {\mathcal {A}}}${\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 ${\displaystyle {\mathcal {A}}_{sa}}${\displaystyle {\mathcal {A}}_{sa}}. If ${\displaystyle b-a\in {\mathcal {A}}_{+}}${\displaystyle b-a\in {\mathcal {A}}_{+}} holds for ${\displaystyle a,b\in {\mathcal {A}}}${\displaystyle a,b\in {\mathcal {A}}}, one writes ${\displaystyle a\leq b}${\displaystyle a\leq b} or ${\displaystyle b\geq a}${\displaystyle b\geq a}.^[13]

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

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

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

• Nonnegative matrix
• Positive operator (Hilbert space)

• Abstract algebra
• C*-algebras

• CS1 French-language sources (fr)