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 $𝒜$ be a *-algebra. An element $a\in 𝒜$ is called positive if there are finitely many elements $a_k\in 𝒜(k=1,2,\dots ,n)$, so that $a={\sum }_{k=1}^n{a}_k^{*}{a}_k$ holds. This is also denoted by $a\ge 0$.

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

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

Examples

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

In case $𝒜$ is a C*-algebra, the following holds:

• Let $a\in {𝒜}_N$ be a normal element, then for every positive function $f\ge 0$ which is continuous on the spectrum of $a$ the continuous functional calculus defines a positive element $f(a)$.
• Every projection, i.e. every element $a\in 𝒜$ 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 $𝒜$ be a C*-algebra and $a\in 𝒜$. Then the following are equivalent:

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

If $𝒜$ is a unital *-algebra with unit element $e$, then in addition the following statements are equivalent:

• $\Vert te-a\Vert \le t$ for every $t\ge \Vert a\Vert$ and $a$ is a self-adjoint element.
• $\Vert te-a\Vert \le t$ for some $t\ge \Vert a\Vert$ and $a$ is a self-adjoint element.

Properties

In *-algebras

Let $𝒜$ be a *-algebra. Then:

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

In C*-algebras

Let $𝒜$ be a C*-algebra. Then:

• Using the continuous functional calculus, for every $a\in {𝒜}_{+}$ and $n\in \mathbb{N}$ there is a uniquely determined $b\in {𝒜}_{+}$ 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 $b\in 𝒜$ the element $b^{*}b$ is positive, this allows the definition of a unique absolute value: $|b|=(b^{*}b{)}^{\frac{1}{2}}$.
• For every real number $\alpha \ge 0$ there is a positive element $a^{\alpha }\in {𝒜}_{+}$ for which $a^{\alpha }{a}^{\beta }=a^{\alpha +\beta }$ holds for all $\beta \in [0,\mathrm{\infty })$. 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 $a,b\in {𝒜}_{+}$, then $ab\in {𝒜}_{+}$.
• Each element $a\in 𝒜$ 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 ${𝒜}_{sa}={𝒜}_{+}-{𝒜}_{+}$, since ${𝒜}^2=𝒜$.
• If both $a$ and $-a$ are positive $a=0$ holds.
• If $ℬ$ is a C*-subalgebra of $𝒜$, then ${ℬ}_{+}=ℬ\cap {𝒜}_{+}$.
• If $ℬ$ is another C*-algebra and $\Phi$ is a *-homomorphism from $𝒜$ to $ℬ$, then $\Phi ({𝒜}_{+})=\Phi (𝒜)\cap {ℬ}_{+}$ holds.
• If $a,b\in {𝒜}_{+}$ are positive elements for which $ab=0$, they commutate and $\Vert a+b\Vert =\max (\Vert a\Vert ,\Vert b\Vert )$ holds. Such elements are called orthogonal and one writes $a\mathrm{\perp }b$.

Partial order

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

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

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

• If $a\le b$ holds, then $c^{*}ac\le c^{*}bc$ is true for every $c\in 𝒜$. For every $c\in {𝒜}_{+}$ that commutates with $a$ and $b$ even $ac\le bc$ holds.
• If $-b\le a\le b$ holds, then $\Vert a\Vert \le \Vert b\Vert$.
• If $0\le a\le b$ holds, then $a^{\alpha }\le b^{\alpha }$ holds for all real numbers $0<\alpha \le 1$.
• If $a$ is invertible and $0\le a\le b$ holds, then $b$ is invertible and for the inverses $b^{-1}\le a^{-1}$ holds.

See also

• Nonnegative matrix
• Positive operator (Hilbert space)

Citations

Bibliography

• English translation of