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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 $a=a^{*}$ .^[1]

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.^[2]

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.^[1] 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.^[3]
For each element $a$ of a *-algebra, the elements $aa^{*}$ and $a^{*}a$ are self-adjoint, since * is an involutive antiautomorphism.^[4]
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.^[1]
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 $f(a)$ .^[5]

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.^[6]
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.^[1]
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. $\sigma (a)\subseteq \mathbb {R}$ .^[5]

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 ${\textstyle a_{2}={\frac {1}{2\mathrm {i} }}(a-a^{*})}$ .^[1]
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. ${\mathcal {A}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}$ .^[7]
If $a\in {\mathcal {A}}_{sa}$ is self-adjoint, then $a$ is normal.^[1]
The *-algebra ${\mathcal {A}}$ is called a hermitian *-algebra if every self-adjoint element $a\in {\mathcal {A}}_{sa}$ has a real spectrum $\sigma (a)\subseteq \mathbb {R}$ .^[8]

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.^[9]
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.^[10] 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 ${\textstyle |a|=(a^{*}a)^{\frac {1}{2}}}$ .^[11]
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.^[12]

Notes

^ ^a ^b ^c ^d ^e ^f Dixmier 1977, p. 4.
^ Dixmier 1977, p. 3.
^ Palmer 2001, p. 800.
^ Dixmier 1977, pp. 3–4.
^ ^a ^b Kadison & Ringrose 1983, p. 271.
^ Palmer 2001, pp. 798–800.
^ Palmer 2001, p. 798.
^ Palmer 2001, p. 1008.
^ Kadison & Ringrose 1983, p. 238.
^ Kadison & Ringrose 1983, p. 246.
^ Dixmier 1977, p. 15.
^ Blackadar 2006, p. 63.

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.

[FOOTNOTEDixmier19774-1] ^ ^a ^b ^c ^d ^e ^f Dixmier 1977, p. 4.

[FOOTNOTEDixmier19773-2] Dixmier 1977, p. 3.

[FOOTNOTEPalmer2001800-3] Palmer 2001, p. 800.

[FOOTNOTEDixmier19773–4-4] Dixmier 1977, pp. 3–4.

[FOOTNOTEKadisonRingrose1983271-5] Kadison & Ringrose 1983, p. 271.

[FOOTNOTEPalmer2001798–800-6] Palmer 2001, pp. 798–800.

[FOOTNOTEPalmer2001798-7] Palmer 2001, p. 798.

[FOOTNOTEPalmer20011008-8] Palmer 2001, p. 1008.

[FOOTNOTEKadisonRingrose1983238-9] Kadison & Ringrose 1983, p. 238.

[FOOTNOTEKadisonRingrose1983246-10] Kadison & Ringrose 1983, p. 246.

[FOOTNOTEDixmier197715-11] Dixmier 1977, p. 15.

[FOOTNOTEBlackadar200663-12] Blackadar 2006, p. 63.

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Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

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