Positive element

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

Definition

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

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

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

Examples

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

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

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

Criteria

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

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

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

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

Properties

In *-algebras

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

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

In C*-algebras

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

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

Partial order

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

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

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

• If $a\leq b$ holds, then $c^{*}ac\leq c^{*}bc$ is true for every $c\in {\mathcal {A}}$. For every $c\in {\mathcal {A}}_{+}$ that commutates with $a$ and $b$ even $ac\leq bc$ holds.
• If $-b\leq a\leq b$ holds, then $\left\|a\right\|\leq \left\|b\right\|$.
• If $0\leq a\leq b$ holds, then ${\textstyle a^{\alpha }\leq b^{\alpha }}$ holds for all real numbers $0<\alpha \leq 1$.
• If $a$ is invertible and $0\leq a\leq b$ holds, then $b$ is invertible and for the inverses $b^{-1}\leq a^{-1}$ 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. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
• 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.