Self-adjoint

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

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

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

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

Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian.^[1] Because of that the notations ${\displaystyle {{𝒜}}_h}$, ${\displaystyle {{𝒜}}_H}$ or ${\displaystyle H({𝒜})}$ 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 ${\displaystyle a{a}^{*}}$ 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 $\mathrm{Re}(a)=\frac{1}{2}(a+a^{*})$ and $\mathrm{Im}(a)=\frac{1}{2\mathrm{i}}(a-a^{*})$ are self-adjoint, where ${\displaystyle \mathrm{i}}$ denotes the imaginary unit.^[1]
• If ${\displaystyle a\in {{𝒜}}_N}$ is a normal element of a C*-algebra ${\displaystyle {𝒜}}$, 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 ${\displaystyle {𝒜}}$ be a *-algebra. Then:

• Let ${\displaystyle a\in {𝒜}}$, then ${\displaystyle a^{*}a}$ is self-adjoint, since ${\displaystyle (a^{*}a{)}^{*}=a^{*}(a^{*}{)}^{*}=a^{*}a}$. A similarly calculation yields that ${\displaystyle a{a}^{*}}$ is also self-adjoint.^[6]
• Let ${\displaystyle a=a_1{a}_2}$ be the product of two self-adjoint elements ${\displaystyle a_1,a_2\in {{𝒜}}_{sa}}$. 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 ${\displaystyle {𝒜}}$ is a C*-algebra, then a normal element ${\displaystyle a\in {{𝒜}}_N}$ is self-adjoint if and only if its spectrum is real, i.e. ${\displaystyle \sigma (a)\subseteq \mathbb{ℝ}}$.^[5]

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

• Each element ${\displaystyle a\in {𝒜}}$ can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements ${\displaystyle a_1,a_2\in {{𝒜}}_{sa}}$, so that ${\displaystyle a=a_1+\mathrm{i}{a}_2}$ holds. Where $a_1=\frac{1}{2}(a+a^{*})$ and $a_2=\frac{1}{2\mathrm{i}}(a-a^{*})$.^[1]
• The set of self-adjoint elements ${\displaystyle {{𝒜}}_{sa}}$ is a real linear subspace of ${\displaystyle {𝒜}}$. From the previous property, it follows that ${\displaystyle {𝒜}}$ is the direct sum of two real linear subspaces, i.e. ${\displaystyle {𝒜}={{𝒜}}_{sa}\oplus \mathrm{i}{{𝒜}}_{sa}}$.^[7]
• If ${\displaystyle a\in {{𝒜}}_{sa}}$ is self-adjoint, then ${\displaystyle a}$ is normal.^[1]
• The *-algebra ${\displaystyle {𝒜}}$ is called a hermitian *-algebra if every self-adjoint element ${\displaystyle a\in {{𝒜}}_{sa}}$ has a real spectrum ${\displaystyle \sigma (a)\subseteq \mathbb{ℝ}}$.^[8]

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

• For the spectrum ${\displaystyle \Vert a\Vert \in \sigma (a)}$ or ${\displaystyle -\Vert a\Vert \in \sigma (a)}$ holds, since ${\displaystyle \sigma (a)}$ is real and ${\displaystyle r(a)=\Vert a\Vert }$ holds for the spectral radius, because ${\displaystyle a}$ is normal.^[9]
• According to the continuous functional calculus, there exist uniquely determined positive elements ${\displaystyle a_{+},a_{-}\in {{𝒜}}_{+}}$, such that ${\displaystyle a=a_{+}-a_{-}}$ with ${\displaystyle a_{+}{a}_{-}=a_{-}{a}_{+}=0}$. For the norm, ${\displaystyle \Vert a\Vert =\max (\Vert a_{+}\Vert ,\Vert a_{-}\Vert )}$ 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 $|a|=(a^{*}a{)}^{\frac{1}{2}}$.^[11]
• For every ${\displaystyle a\in {{𝒜}}_{+}}$ and odd ${\displaystyle n\in \mathbb{\mathbb{N}}}$, there exists a uniquely determined ${\displaystyle b\in {{𝒜}}_{+}}$ that satisfies ${\displaystyle b^n=a}$, i.e. a unique ${\displaystyle n}$-th root, as can be shown with the continuous functional calculus.^[12]

• Self-adjoint matrix
• Self-adjoint operator

• 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. 

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