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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 {\displaystyle a^{*}a} .cite-ref-footnotepalmer2001798-1-0[1]

Contents

• Definition

• Examples

• Criteria

• Properties

• In *-algebras

• In C*-algebras

• Partial order

• See also

• Citations

• References

• Bibliography

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.cite-ref-footnotepalmer2001798-1-1[1] This is also denoted by a ≥ 0 {\displaystyle a\geq 0} .cite-ref-footnoteblackadar200663-2-0[2]

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.cite-ref-footnotepalmer2001798-1-2[1]

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)} .cite-ref-footnotekadisonringrose1983271-3-0[3]

• 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.cite-ref-footnotekadisonringrose1983271-3-1[3]

Criteria

Let A {\displaystyle {\mathcal {A}}} be a C*-algebra and a ∈ A {\displaystyle a\in {\mathcal {A}}} . Then the following are equivalent:cite-ref-footnotekadisonringrose1983247-248-4-0[4]

• 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:cite-ref-footnotekadisonringrose1983245-5-0[5]

• ‖ t e − a ‖ ≤ t {\displaystyle \left\|te-a\right\|\leq t} for every t ≥ ‖ a ‖ {\displaystyle t\geq \left\|a\right\|} and a {\displaystyle a} is a self-adjoint element.

• ‖ t e − a ‖ ≤ t {\displaystyle \left\|te-a\right\|\leq t} for some t ≥ ‖ a ‖ {\displaystyle t\geq \left\|a\right\|} and a {\displaystyle a} is a self-adjoint element.

Properties

In *-algebras

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

• If a ∈ A + {\displaystyle a\in {\mathcal {A}}_{+}} is a positive element, then a {\displaystyle a} is self-adjoint.cite-ref-footnotepalmer2001800-6-0[6]

• The set of positive elements A + {\displaystyle {\mathcal {A}}_{+}} is a convex cone in the real vector space of the self-adjoint elements A s a {\displaystyle {\mathcal {A}}_{sa}} . This means that α a , a + b ∈ A + {\displaystyle \alpha a,a+b\in {\mathcal {A}}_{+}} holds for all a , b ∈ A {\displaystyle a,b\in {\mathcal {A}}} and α ∈ [ 0 , ∞ ) {\displaystyle \alpha \in [0,\infty )} .cite-ref-footnotepalmer2001800-6-1[6]

• If a ∈ A + {\displaystyle a\in {\mathcal {A}}_{+}} is a positive element, then b ∗ a b {\displaystyle b^{*}ab} is also positive for every element b ∈ A {\displaystyle b\in {\mathcal {A}}} .cite-ref-footnoteblackadar200664-7-0[7]

• For the linear span of A + {\displaystyle {\mathcal {A}}_{+}} the following holds: ⟨ A + ⟩ = A 2 {\displaystyle \langle {\mathcal {A}}_{+}\rangle ={\mathcal {A}}^{2}} and A + − A + = A s a ∩ A 2 {\displaystyle {\mathcal {A}}_{+}-{\mathcal {A}}_{+}={\mathcal {A}}_{sa}\cap {\mathcal {A}}^{2}} .cite-ref-footnotepalmer2001802-8-0[8]

In C*-algebras

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

• Using the continuous functional calculus, for every a ∈ A + {\displaystyle a\in {\mathcal {A}}_{+}} and n ∈ N {\displaystyle n\in \mathbb {N} } there is a uniquely determined b ∈ A + {\displaystyle b\in {\mathcal {A}}_{+}} that satisfies b n = a {\displaystyle b^{n}=a} , i.e. a unique n {\displaystyle n} -th root. In particular, a square root exists for every positive element. Since for every b ∈ A {\displaystyle b\in {\mathcal {A}}} the element b ∗ b {\displaystyle b^{*}b} is positive, this allows the definition of a unique absolute value: | b | = ( b ∗ b ) 1 2 {\textstyle |b|=(b^{*}b)^{\frac {1}{2}}} .cite-ref-footnoteblackadar200663-65-9-0[9]

• For every real number α ≥ 0 {\displaystyle \alpha \geq 0} there is a positive element a α ∈ A + {\displaystyle a^{\alpha }\in {\mathcal {A}}_{+}} for which a α a β = a α + β {\displaystyle a^{\alpha }a^{\beta }=a^{\alpha +\beta }} holds for all β ∈ [ 0 , ∞ ) {\displaystyle \beta \in [0,\infty )} . The mapping α ↦ a α {\displaystyle \alpha \mapsto a^{\alpha }} is continuous. Negative values for α {\displaystyle \alpha } are also possible for invertible elements a {\displaystyle a} .cite-ref-footnoteblackadar200664-7-1[7]

• Products of commutative positive elements are also positive. So if a b = b a {\displaystyle ab=ba} holds for positive a , b ∈ A + {\displaystyle a,b\in {\mathcal {A}}_{+}} , then a b ∈ A + {\displaystyle ab\in {\mathcal {A}}_{+}} .cite-ref-footnotekadisonringrose1983245-5-1[5]

• Each element a ∈ A {\displaystyle a\in {\mathcal {A}}} can be uniquely represented as a linear combination of four positive elements. To do this, 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.cite-ref-footnotekadisonringrose1983247-10-0[10] For it holds that A s a = A + − A + {\displaystyle {\mathcal {A}}_{sa}={\mathcal {A}}_{+}-{\mathcal {A}}_{+}} , since A 2 = A {\displaystyle {\mathcal {A}}^{2}={\mathcal {A}}} .cite-ref-footnotepalmer2001802-8-1[8]

• If both a {\displaystyle a} and − a {\displaystyle -a} are positive a = 0 {\displaystyle a=0} holds.cite-ref-footnotekadisonringrose1983245-5-2[5]

• If B {\displaystyle {\mathcal {B}}} is a C*-subalgebra of A {\displaystyle {\mathcal {A}}} , then B + = B ∩ A + {\displaystyle {\mathcal {B}}_{+}={\mathcal {B}}\cap {\mathcal {A}}_{+}} .cite-ref-footnotekadisonringrose1983245-5-3[5]

• If B {\displaystyle {\mathcal {B}}} is another C*-algebra and Φ {\displaystyle \Phi } is a *-homomorphism from A {\displaystyle {\mathcal {A}}} to B {\displaystyle {\mathcal {B}}} , then Φ ( A + ) = Φ ( A ) ∩ B + {\displaystyle \Phi ({\mathcal {A}}_{+})=\Phi ({\mathcal {A}})\cap {\mathcal {B}}_{+}} holds.cite-ref-footnotedixmier197718-11-0[11]

• If a , b ∈ A + {\displaystyle a,b\in {\mathcal {A}}_{+}} are positive elements for which a b = 0 {\displaystyle ab=0} , they commutate and ‖ a + b ‖ = max ( ‖ a ‖ , ‖ b ‖ ) {\displaystyle \left\|a+b\right\|=\max(\left\|a\right\|,\left\|b\right\|)} holds. Such elements are called orthogonal and one writes a ⊥ b {\displaystyle a\bot b} .cite-ref-footnoteblackadar200667-12-0[12]

Partial order

Let 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 A s a {\displaystyle {\mathcal {A}}_{sa}} . If b − a ∈ A + {\displaystyle b-a\in {\mathcal {A}}_{+}} holds for a , b ∈ A {\displaystyle a,b\in {\mathcal {A}}} , one writes a ≤ b {\displaystyle a\leq b} or b ≥ a {\displaystyle b\geq a} .cite-ref-footnotepalmer2001799-13-0[13]

This partial order fulfills the properties t a ≤ t b {\displaystyle ta\leq tb} and a + c ≤ b + c {\displaystyle a+c\leq b+c} for all a , b , c ∈ A s a {\displaystyle a,b,c\in {\mathcal {A}}_{sa}} with a ≤ b {\displaystyle a\leq b} and t ∈ [ 0 , ∞ ) {\displaystyle t\in [0,\infty )} .cite-ref-footnotepalmer2001802-8-2[8]

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

• If a ≤ b {\displaystyle a\leq b} holds, then c ∗ a c ≤ c ∗ b c {\displaystyle c^{*}ac\leq c^{*}bc} is true for every c ∈ A {\displaystyle c\in {\mathcal {A}}} . For every c ∈ A + {\displaystyle c\in {\mathcal {A}}_{+}} that commutates with a {\displaystyle a} and b {\displaystyle b} even a c ≤ b c {\displaystyle ac\leq bc} holds.cite-ref-footnotekadisonringrose1983249-14-0[14]

• If − b ≤ a ≤ b {\displaystyle -b\leq a\leq b} holds, then ‖ a ‖ ≤ ‖ b ‖ {\displaystyle \left\|a\right\|\leq \left\|b\right\|} .cite-ref-footnotekadisonringrose1983250-15-0[15]

• If 0 ≤ a ≤ b {\displaystyle 0\leq a\leq b} holds, then a α ≤ b α {\textstyle a^{\alpha }\leq b^{\alpha }} holds for all real numbers 0 < α ≤ 1 {\displaystyle 0<\alpha \leq 1} .cite-ref-footnoteblackadar200666-16-0[16]

• If a {\displaystyle a} is invertible and 0 ≤ a ≤ b {\displaystyle 0\leq a\leq b} holds, then b {\displaystyle b} is invertible and for the inverses b − 1 ≤ a − 1 {\displaystyle b^{-1}\leq a^{-1}} holds.cite-ref-footnotekadisonringrose1983250-15-1[15]