Positive element

In mathematics, an element of a *-algebra is called positive if it is the sum of elements of the form Template:Nowrap

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 $a={\sum }_{k=1}^n{a}_k^{*}{a}_k$ Template:Nowrap This is also denoted by Template:Nowrap

The set of positive elements is denoted by Template:Nowrap

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 Template:Nowrap

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 Template:Nowrap
• 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 Template:Nowrap

Criteria

Let $𝒜$ be a C*-algebra and Template:Nowrap Then the following are equivalent:Template:Sfn

• 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 Template:Nowrap
• There exists a (unique) self-adjoint element $c\in {𝒜}_{sa}$ such that Template:Nowrap

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

• $\Vert te-a\Vert \le t$ 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.Template:Sfn
• The set of positive elements ${𝒜}_{+}$ is a convex cone in the real vector space of the self-adjoint elements Template:Nowrap This means that $\alpha a,a+b\in {𝒜}_{+}$ holds for all $a,b\in 𝒜$ and Template:Nowrap
• If $a\in {𝒜}_{+}$ is a positive element, then $b^{*}ab$ is also positive for every element Template:Nowrap
• For the linear span of ${𝒜}_{+}$ the following holds: $⟨{𝒜}_{+}⟩={𝒜}^2$ and Template:Nowrap

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: Template:Nowrap
• 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 Template:Nowrap The mapping $\alpha \mapsto a^{\alpha }$ is continuous. Negative values for $\alpha$ are also possible for invertible elements Template:Nowrap
• Products of commutative positive elements are also positive. So if $ab=ba$ holds for positive $a,b\in {𝒜}_{+}$, then Template:Nowrap
• 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 Template:Nowrap For it holds that ${𝒜}_{sa}={𝒜}_{+}-{𝒜}_{+}$, since Template:Nowrap
• If both $a$ and $-a$ are positive $a=0$ Template:Nowrap
• If $ℬ$ is a C*-subalgebra of $𝒜$, then Template:Nowrap
• If $ℬ$ is another C*-algebra and $\mathrm{\Phi }$ is a *-homomorphism from $𝒜$ to $ℬ$, then $\mathrm{\Phi }({𝒜}_{+})=\mathrm{\Phi }(𝒜)\cap {ℬ}_{+}$ Template:Nowrap
• 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 Template:Nowrap

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 Template:Nowrap If $b-a\in {𝒜}_{+}$ holds for $a,b\in 𝒜$, one writes $a\le b$ or Template:Nowrap

This partial order fulfills the properties $ta\le tb$ and $a+c\le b+c$ for all $a,b,c\in {𝒜}_{sa}$ with Template:NowrapTemplate:Sfn

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

See also

• Nonnegative matrix
• Positive operator (Hilbert space)

Citations

Bibliography

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