Self-adjoint

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

Definition

Let $𝒜$ be a *-algebra. An element $a\in 𝒜$ is called self-adjoint if Template:Nowrap

The set of self-adjoint elements is referred to as Template:Nowrap

A subset $ℬ\subseteq 𝒜$ that is closed under the involution *, i.e. $ℬ={ℬ}^{*}$, is called Template:Nowrap

A special case of 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.

Especially in the older literature on *-algebras and C*-algebras, such elements are often called Template:Nowrap Because of that the notations ${𝒜}_h$, ${𝒜}_H$ or $H(𝒜)$ for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

Examples

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

Criteria

Let $𝒜$ be a *-algebra. Then:

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

Properties

In *-algebras

Let $𝒜$ be a *-algebra. Then:

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

In C*-algebras

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

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

See also

• Self-adjoint matrix
• Self-adjoint operator

References

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