Self-adjoint element

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

Definition

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

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. $ℬ={ℬ}^{\ast }$, is called self-adjoint.

A special case of particular importance is the case where $𝒜$ is a complete normed *-algebra, that satisfies the C*-identity ($\Vert a^{\ast }a\Vert ={\Vert a\Vert }^2\mathrm{\forall }a\in 𝒜$), 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 $a{a}^{\ast }$ and $a^{\ast }a$ are self-adjoint, since * is an involutive antiautomorphism.
• For each element $a$ of a *-algebra, the real and imaginary parts $\mathrm{Re}(a)=\frac{1}{2}(a+a^{\ast })$ and $\mathrm{Im}(a)=\frac{1}{2\mathrm{i}}(a-a^{\ast })$ 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 $a^{\ast }a$ is self-adjoint, since $(a^{\ast }a{)}^{\ast }=a^{\ast }(a^{\ast }{)}^{\ast }=a^{\ast }a$. A similarly calculation yields that $a{a}^{\ast }$ 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 $(a_1{a}_2{)}^{\ast }=a_2^{\ast }{a}_1^{\ast }=a_2{a}_1$ 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. $\sigma (a)\subseteq ℝ$.

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 $a_1=\frac{1}{2}(a+a^{\ast })$ and $a_2=\frac{1}{2\mathrm{i}}(a-a^{\ast })$.
• 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 $\sigma (a)\subseteq ℝ$.

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 $|a|=(a^{\ast }a{)}^{\frac{1}{2}}$.
• 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 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 français). 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.