Inmathematics, anelement of a*-algebra is calledself-adjoint if it is the same as itsadjoint (i.e.${\displaystyle a=a^{\ast }}$).

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

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

Asubset${\displaystyle \mathcal{B}\subseteq \mathcal{A}}$ that isclosed under theinvolution *, i.e.${\displaystyle \mathcal{B}={\mathcal{B}}^{\ast }}$, is calledself-adjoint.^[2]

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

Especially in the older literature on *-algebras and C*-algebras, such elements are often calledhermitian.^[1] Because of that the notations${\displaystyle {\mathcal{A}}_h}$,${\displaystyle {\mathcal{A}}_H}$ or${\displaystyle H(\mathcal{A})}$ for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

• Eachpositive element of a C*-algebra isself-adjoint.^[3]
• For each element${\displaystyle a}$ of a *-algebra, the elements${\displaystyle a{a}^{\ast }}$ and${\displaystyle a^{\ast }a}$ are self-adjoint, since * is aninvolutive antiautomorphism.^[4]
• For each element${\displaystyle a}$ of a *-algebra, thereal 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${\displaystyle \mathrm{i}}$ denotes theimaginary unit.^[1]
• If${\displaystyle a\in {\mathcal{A}}_N}$ is anormal element of a C*-algebra${\displaystyle \mathcal{A}}$, then for everyreal-valued function${\displaystyle f}$, which iscontinuous on thespectrum of${\displaystyle a}$, thecontinuous functional calculus defines a self-adjoint element${\displaystyle f(a)}$.^[5]

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

• Let${\displaystyle a\in \mathcal{A}}$, then${\displaystyle a^{\ast }a}$ is self-adjoint, since${\displaystyle (a^{\ast }a{)}^{\ast }=a^{\ast }(a^{\ast }{)}^{\ast }=a^{\ast }a}$. A similarly calculation yields that${\displaystyle a{a}^{\ast }}$ is alsoself-adjoint.^[6]
• Let${\displaystyle a=a_1{a}_2}$ be theproduct of two self-adjoint elements${\displaystyle a_1,a_2\in {\mathcal{A}}_{sa}}$. Then${\displaystyle a}$ is self-adjoint if${\displaystyle a_1}$ and${\displaystyle a_2}$commutate, since${\displaystyle (a_1{a}_2{)}^{\ast }=a_2^{\ast }{a}_1^{\ast }=a_2{a}_1}$ alwaysholds.^[1]
• If${\displaystyle \mathcal{A}}$ is a C*-algebra, then a normal element${\displaystyle a\in {\mathcal{A}}_N}$ is self-adjoint if and only if its spectrum is real, i.e.${\displaystyle \sigma (a)\subseteq \mathbb{R}}$.^[5]

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

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

Let${\displaystyle \mathcal{A}}$ be a C*-algebra and${\displaystyle a\in {\mathcal{A}}_{sa}}$. 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 thespectral radius, because${\displaystyle a}$ isnormal.^[9]
• According to the continuous functional calculus, there exist uniquely determined positive elements${\displaystyle a_{+},a_{-}\in {\mathcal{A}}_{+}}$, 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 thepositive and negative parts. In addition,${\displaystyle |a|=a_{+}+a_{-}}$ holds for the absolute value defined for every element$|a|=(a^{\ast }a{)}^{\frac{1}{2}}$.^[11]
• For every${\displaystyle a\in {\mathcal{A}}_{+}}$ and odd${\displaystyle n\in \mathbb{N}}$, there exists a uniquely determined${\displaystyle b\in {\mathcal{A}}_{+}}$ that satisfies${\displaystyle b^n=a}$, i.e. a unique${\displaystyle n}$-th root, as can be shown with the continuous functionalcalculus.^[12]

• Self-adjoint matrix
• Self-adjoint operator

• 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 ofLes 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.

• Abstract algebra
• C*-algebras

• Articles with short description
• Short description matches Wikidata
• CS1 French-language sources (fr)