Self-adjoint element

In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. ${\displaystyle a=a^{*}}$).

Let ${\displaystyle {\mathcal {A}}}$ be a *-algebra. An element ${\displaystyle a\in {\mathcal {A}}}$ is called self-adjoint if ${\displaystyle a=a^{*}}$.¹

The set of self-adjoint elements is referred to as ${\displaystyle {\mathcal {A}}_{sa}}$.

A subset ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {A}}}$ that is closed under the involution *, i.e. ${\displaystyle {\mathcal {B}}={\mathcal {B}}^{*}}$, is called self-adjoint.²

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

Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian.¹ Because of that the notations ${\displaystyle {\mathcal {A}}_{h}}$, ${\displaystyle {\mathcal {A}}_{H}}$ or ${\displaystyle H({\mathcal {A}})}$ for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

• Each positive element of a C*-algebra is self-adjoint.³
• For each element ${\displaystyle a}$ of a *-algebra, the elements ${\displaystyle aa^{*}}$ and ${\displaystyle a^{*}a}$ are self-adjoint, since * is an involutive antiautomorphism.⁴
• For each element ${\displaystyle a}$ of a *-algebra, the real and imaginary parts ${\textstyle \operatorname {Re} (a)={\frac {1}{2}}(a+a^{*})}$ and ${\textstyle \operatorname {Im} (a)={\frac {1}{2\mathrm {i} }}(a-a^{*})}$ are self-adjoint, where ${\displaystyle \mathrm {i} }$ denotes the imaginary unit.¹
• 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 ${\displaystyle f(a)}$.⁵

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

• Let ${\displaystyle a\in {\mathcal {A}}}$, then ${\displaystyle a^{*}a}$ is self-adjoint, since ${\displaystyle (a^{*}a)^{*}=a^{*}(a^{*})^{*}=a^{*}a}$. A similarly calculation yields that ${\displaystyle aa^{*}}$ is also self-adjoint.⁶
• 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 ${\displaystyle (a_{1}a_{2})^{*}=a_{2}^{*}a_{1}^{*}=a_{2}a_{1}}$ always holds.¹
• 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. ${\displaystyle \sigma (a)\subseteq \mathbb {R} }$.⁵

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 ${\textstyle a_{1}={\frac {1}{2}}(a+a^{*})}$ and ${\textstyle a_{2}={\frac {1}{2\mathrm {i} }}(a-a^{*})}$.¹
• 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. ${\displaystyle {\mathcal {A}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}}$.⁷
• If ${\displaystyle a\in {\mathcal {A}}_{sa}}$ is self-adjoint, then ${\displaystyle a}$ is normal.¹
• 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 ${\displaystyle \sigma (a)\subseteq \mathbb {R} }$.⁸

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

• For the spectrum ${\displaystyle \left\|a\right\|\in \sigma (a)}$ or ${\displaystyle -\left\|a\right\|\in \sigma (a)}$ holds, since ${\displaystyle \sigma (a)}$ is real and ${\displaystyle r(a)=\left\|a\right\|}$ holds for the spectral radius, because ${\displaystyle a}$ is normal.⁹
• 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, ${\displaystyle \left\|a\right\|=\max(\left\|a_{+}\right\|,\left\|a_{-}\right\|)}$ holds.¹⁰ 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 ${\textstyle |a|=(a^{*}a)^{\frac {1}{2}}}$.¹¹
• 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.¹²