In mathematics, an element of a *-algebra is called positive if it is the sum of elements of the form a ∗ a {\displaystyle a^{*}a} .
Definition Let A {\displaystyle {\mathcal {A}}} be a *-algebra. An element a ∈ A {\displaystyle a\in {\mathcal {A}}} is called positive if there are finitely many elements a k ∈ A ( k = 1 , 2 , … , n ) {\displaystyle a_{k}\in {\mathcal {A}}\;(k=1,2,\ldots ,n)} , so that a = ∑ k = 1 n a k ∗ a k {\textstyle a=\sum _{k=1}^{n}a_{k}^{*}a_{k}} holds. This is also denoted by a ≥ 0 {\displaystyle a\geq 0} . The set of positive elements is denoted by A + {\displaystyle {\mathcal {A}}_{+}} . A special case from particular importance is the case where A {\displaystyle {\mathcal {A}}} is a complete normed *-algebra, that satisfies the C*-identity ( ‖ a ∗ a ‖ = ‖ a ‖ 2 ∀ a ∈ A {\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}} ), which is called a C*-algebra.
Examples The unit element e {\displaystyle e} of an unital *-algebra is positive. For each element a ∈ A {\displaystyle a\in {\mathcal {A}}} , the elements a ∗ a {\displaystyle a^{*}a} and a a ∗ {\displaystyle aa^{*}} are positive by definition. In case A {\displaystyle {\mathcal {A}}} is a C*-algebra, the following holds:
Let a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} be a normal element, then for every positive function f ≥ 0 {\displaystyle f\geq 0} which is continuous on the spectrum of a {\displaystyle a} the continuous functional calculus defines a positive element f ( a ) {\displaystyle f(a)} . Every projection, i.e. every element a ∈ A {\displaystyle a\in {\mathcal {A}}} for which a = a ∗ = a 2 {\displaystyle a=a^{*}=a^{2}} holds, is positive. For the spectrum σ ( a ) {\displaystyle \sigma (a)} of such an idempotent element, σ ( a ) ⊆ { 0 , 1 } {\displaystyle \sigma (a)\subseteq \{0,1\}} holds, as can be seen from the continuous functional calculus.
Criteria Let A {\displaystyle {\mathcal {A}}} be a C*-algebra and a ∈ A {\displaystyle a\in {\mathcal {A}}} . Then the following are equivalent:
For the spectrum σ ( a ) ⊆ [ 0 , ∞ ) {\displaystyle \sigma (a)\subseteq [0,\infty )} holds and a {\displaystyle a} is a normal element. There exists an element b ∈ A {\displaystyle b\in {\mathcal {A}}} , such that a = b b ∗ {\displaystyle a=bb^{*}} . There exists a (unique) self-adjoint element c ∈ A s a {\displaystyle c\in {\mathcal {A}}_{sa}} such that a = c 2 {\displaystyle a=c^{2}} . If A {\displaystyle {\mathcal {A}}} is a unital *-algebra with unit element e {\displaystyle e} , then in addition the following statements are equivalent:
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