Self-adjoint

Short description: Element of algebra where x* equals x

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 $a=a^{*}$.

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

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

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.

Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian. 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 self-adjoint.
• For each element $a$ of a *-algebra, the elements $a{a}^{*}$ and $a^{*}a$ are self-adjoint, since * is an involutive antiautomorphism.
• 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 imaginary unit.
• 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 $f(a)$.

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 self-adjoint.
• Let $a=a_1{a}_2$ be the product of two self-adjoint elements $a_1,a_2\in {{𝒜}}_{sa}$. 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 ${𝒜}$ 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. $\sigma (a)\subseteq \mathbb{ℝ}$.

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 $a_2=\frac{1}{2\mathrm{i}}(a-a^{*})$.
• The set of self-adjoint elements ${{𝒜}}_{sa}$ is a real linear subspace of ${𝒜}$. From the previous property, it follows that ${𝒜}$ is the direct sum of two real linear subspaces, i.e. ${𝒜}={{𝒜}}_{sa}\oplus \mathrm{i}{{𝒜}}_{sa}$.
• If $a\in {{𝒜}}_{sa}$ is self-adjoint, then $a$ is normal.
• The *-algebra ${𝒜}$ is called a hermitian *-algebra if every self-adjoint element $a\in {{𝒜}}_{sa}$ has a real spectrum $\sigma (a)\subseteq \mathbb{ℝ}$.

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 normal.
• According to the continuous functional calculus, there exist uniquely determined positive elements $a_{+},a_{-}\in {{𝒜}}_{+}$, such that $a=a_{+}-a_{-}$ with $a_{+}{a}_{-}=a_{-}{a}_{+}=0$. For the norm, $\Vert a\Vert =\max (\Vert a_{+}\Vert ,\Vert a_{-}\Vert )$ 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 $|a|=(a^{*}a{)}^{\frac{1}{2}}$.
• For every $a\in {{𝒜}}_{+}$ and odd $n\in \mathbb{\mathbb{N}}$, 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 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. pp. 63. ISBN 3-540-28486-9. 
• Dixmier, Jacques (1977). C*-algebras. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1.  English translation of Dixmier, Jacques (1969) (in fr). Les C*-algèbres et leurs représentations. Gauthier-Villars. 
• 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. (1994). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras.. Cambridge university press. ISBN 0-521-36638-0.