Self-adjoint

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

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

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

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

Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian. Because of that the notations ${\mathcal {A}}_{h}$, ${\mathcal {A}}_{H}$ or $H({\mathcal {A}})$ 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 self-adjoint.
• For each element $a$ of a *-algebra, the elements $aa^{*}$ and $a^{*}a$ are self-adjoint, since * is an involutive antiautomorphism.
• For each element $a$ 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 $\mathrm {i}$ denotes the imaginary unit.
• If $a\in {\mathcal {A}}_{N}$ is a normal element of a C*-algebra ${\mathcal {A}}$, then for every real-valued function $f$, which is continuous on the spectrum of $a$, the continuous functional calculus defines a self-adjoint element $f(a)$.

Criteria

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

• Let $a\in {\mathcal {A}}$, then $a^{*}a$ is self-adjoint, since $(a^{*}a)^{*}=a^{*}(a^{*})^{*}=a^{*}a$. A similarly calculation yields that $aa^{*}$ is also self-adjoint.
• Let $a=a_{1}a_{2}$ be the product of two self-adjoint elements $a_{1},a_{2}\in {\mathcal {A}}_{sa}$. 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 holds.
• If ${\mathcal {A}}$ is a C*-algebra, then a normal element $a\in {\mathcal {A}}_{N}$ is self-adjoint if and only if its spectrum is real, i.e. $\sigma (a)\subseteq \mathbb {R}$.

Properties

In *-algebras

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

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

In C*-algebras

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

• For the spectrum $\left\|a\right\|\in \sigma (a)$ or $-\left\|a\right\|\in \sigma (a)$ holds, since $\sigma (a)$ is real and $r(a)=\left\|a\right\|$ holds for the spectral radius, because $a$ is normal.
• According to the continuous functional calculus, there exist uniquely determined positive elements $a_{+},a_{-}\in {\mathcal {A}}_{+}$, such that $a=a_{+}-a_{-}$ with $a_{+}a_{-}=a_{-}a_{+}=0$. For the norm, $\left\|a\right\|=\max(\left\|a_{+}\right\|,\left\|a_{-}\right\|)$ holds. 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 ${\textstyle |a|=(a^{*}a)^{\frac {1}{2}}}$.
• For every $a\in {\mathcal {A}}_{+}$ and odd $n\in \mathbb {N}$, there exists a uniquely determined $b\in {\mathcal {A}}_{+}$ that satisfies $b^{n}=a$, i.e. a unique $n$-th root, as can be shown with the continuous functional calculus.

See also

• Self-adjoint matrix
• Self-adjoint operator

References

• Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. p. 63. ISBN 3-540-28486-9.
• Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
• Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3.
• Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0.