Self-adjoint

Examples

• Each positive element of a C*-algebra is self-adjoint.
• For each element ${\displaystyle a}$ of a *-algebra, the elements ${\displaystyle a{a}^{\ast }}$ and ${\displaystyle a^{\ast }a}$ are self-adjoint, since * is an involutive antiautomorphism.
• For each element ${\displaystyle a}$ of a *-algebra, the real and imaginary parts $\mathrm{Re}(a)=\frac{1}{2}(a+a^{\ast })$ and $\mathrm{Im}(a)=\frac{1}{2\mathrm{i}}(a-a^{\ast })$ are self-adjoint, where ${\displaystyle \mathrm{i}}$ denotes the imaginary unit.
• If ${\displaystyle a\in {\mathcal{A}}_N}$ is a normal element of a C*-algebra ${\displaystyle \mathcal{A}}$, then for every real-valued function ${\displaystyle f}$, which is continuous on the spectrum of ${\displaystyle a}$, the continuous functional calculus defines a self-adjoint element ${\displaystyle f(a)}$.

Criteria

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

• Let ${\displaystyle a\in \mathcal{A}}$, then ${\displaystyle a^{\ast }a}$ is self-adjoint, since ${\displaystyle (a^{\ast }a{)}^{\ast }=a^{\ast }(a^{\ast }{)}^{\ast }=a^{\ast }a}$. A similarly calculation yields that ${\displaystyle a{a}^{\ast }}$ is also self-adjoint.
• Let ${\displaystyle a=a_1{a}_2}$ be the product of two self-adjoint elements ${\displaystyle a_1,a_2\in {\mathcal{A}}_{sa}}$. Then ${\displaystyle a}$ is self-adjoint if ${\displaystyle a_1}$ and ${\displaystyle a_2}$ commutate, since ${\displaystyle (a_1{a}_2{)}^{\ast }=a_2^{\ast }{a}_1^{\ast }=a_2{a}_1}$ always holds.
• If ${\displaystyle \mathcal{A}}$ is a C*-algebra, then a normal element ${\displaystyle a\in {\mathcal{A}}_N}$ is self-adjoint if and only if its spectrum is real, i.e. ${\displaystyle \sigma (a)\subseteq \mathbb{R}}$.

Properties

In *-algebras

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

• Each element ${\displaystyle a\in \mathcal{A}}$ can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements ${\displaystyle a_1,a_2\in {\mathcal{A}}_{sa}}$, so that ${\displaystyle a=a_1+\mathrm{i}{a}_2}$ holds. Where $a_1=\frac{1}{2}(a+a^{\ast })$ and $a_2=\frac{1}{2\mathrm{i}}(a-a^{\ast })$.
• The set of self-adjoint elements ${\displaystyle {\mathcal{A}}_{sa}}$ is a real linear subspace of ${\displaystyle \mathcal{A}}$. From the previous property, it follows that ${\displaystyle \mathcal{A}}$ is the direct sum of two real linear subspaces, i.e. ${\displaystyle \mathcal{A}={\mathcal{A}}_{sa}\oplus \mathrm{i}{\mathcal{A}}_{sa}}$.
• If ${\displaystyle a\in {\mathcal{A}}_{sa}}$ is self-adjoint, then ${\displaystyle a}$ is normal.
• The *-algebra ${\displaystyle \mathcal{A}}$ is called a hermitian *-algebra if every self-adjoint element ${\displaystyle a\in {\mathcal{A}}_{sa}}$ has a real spectrum ${\displaystyle \sigma (a)\subseteq \mathbb{R}}$.

In C*-algebras

Let ${\displaystyle \mathcal{A}}$ be a C*-algebra and ${\displaystyle a\in {\mathcal{A}}_{sa}}$. Then:

• For the spectrum ${\displaystyle \Vert a\Vert \in \sigma (a)}$ or ${\displaystyle -\Vert a\Vert \in \sigma (a)}$ holds, since ${\displaystyle \sigma (a)}$ is real and ${\displaystyle r(a)=\Vert a\Vert }$ holds for the spectral radius, because ${\displaystyle a}$ is normal.
• According to the continuous functional calculus, there exist uniquely determined positive elements ${\displaystyle a_{+},a_{-}\in {\mathcal{A}}_{+}}$, such that ${\displaystyle a=a_{+}-a_{-}}$ with ${\displaystyle a_{+}{a}_{-}=a_{-}{a}_{+}=0}$. For the norm, ${\displaystyle \Vert a\Vert =max(\Vert a_{+}\Vert ,\Vert a_{-}\Vert )}$ holds. The elements ${\displaystyle a_{+}}$ and ${\displaystyle a_{-}}$ are also referred to as the positive and negative parts. In addition, ${\displaystyle |a|=a_{+}+a_{-}}$ holds for the absolute value defined for every element $|a|=(a^{\ast }a{)}^{\frac{1}{2}}$.
• For every ${\displaystyle a\in {\mathcal{A}}_{+}}$ and odd ${\displaystyle n\in \mathbb{N}}$, there exists a uniquely determined ${\displaystyle b\in {\mathcal{A}}_{+}}$ that satisfies ${\displaystyle b^n=a}$, i.e. a unique ${\displaystyle n}$-th root, as can be shown with the continuous functional calculus.