Positive element

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

Definition

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

The set of positive elements is denoted by [math]\displaystyle{ \mathcal{A}_+ }[/math].

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

Examples

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

In case [math]\displaystyle{ \mathcal{A} }[/math] is a C*-algebra, the following holds:

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

Criteria

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

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

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

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

Properties

In *-algebras

Let [math]\displaystyle{ \mathcal{A} }[/math] be a *-algebra. Then:

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

In C*-algebras

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

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

Partial order

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

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

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

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

See also

• Nonnegative matrix
• Positive operator (Hilbert space)

Citations

Bibliography

• Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. ISBN 3-540-28486-9. 
• Dixmier, Jacques (1977). C*-algebras. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1.  English translation of Dixmier, Jacques (1969) (in fr). Les C*-algèbres et leurs représentations. Gauthier-Villars. 
• Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory.. New York/London: Academic Press. ISBN 0-12-393301-3. 
• Palmer, Theodore W. (1994). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras.. Cambridge university press. ISBN 0-521-36638-0.