Normal element

In mathematics, an element of a *-algebra is called normal if it commutates with its adjoint.^[1]

Definition

Let ${\mathcal {A}}$ be a *-Algebra. An element $a\in {\mathcal {A}}$ is called normal if it commutes with $a^{*}$ , i.e. it satisfies the equation $aa^{*}=a^{*}a$ .^[1]

The set of normal elements is denoted by ${\mathcal {A}}_{N}$ or $N({\mathcal {A}})$ .

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.

Examples

Every self-adjoint element of a a *-algebra is normal.^[1]
Every unitary element of a a *-algebra is normal.^[2]
If ${\mathcal {A}}$ is a C*-Algebra and $a\in {\mathcal {A}}_{N}$ a normal element, then for every continuous function $f$ on the spectrum of $a$ the continuous functional calculus defines another normal element $f(a)$ .^[3]

Criteria

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

An element $a\in {\mathcal {A}}$ is normal if and only if the *-subalgebra generated by $a$ , meaning the smallest *-algebra containing $a$ , is commutative.^[2]
Every element $a\in {\mathcal {A}}$ can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements $a_{1},a_{2}\in {\mathcal {A}}_{sa}$ , such that $a=a_{1}+\mathrm {i} a_{2}$ , where $\mathrm {i}$ denotes the imaginary unit. Exactly then $a$ is normal if $a_{1}a_{2}=a_{2}a_{1}$ , i.e. real and imaginary part commutate.^[1]

Properties

In *-algebras

Let $a\in {\mathcal {A}}_{N}$ be a normal element of a *-algebra ${\mathcal {A}}$ . Then:

The adjoint element $a^{*}$ is also normal, since $a=(a^{*})^{*}$ holds for the involution *.^[4]

In C*-algebras

Let $a\in {\mathcal {A}}_{N}$ be a normal element of a C*-algebra ${\mathcal {A}}$ . Then:

It is $\left\|a^{2}\right\|=\left\|a\right\|^{2}$ , since for normal elements using the C*-identity $\left\|a^{2}\right\|^{2}=\left\|(a^{2})(a^{2})^{*}\right\|=\left\|(a^{*}a)^{*}(a^{*}a)\right\|=\left\|a^{*}a\right\|^{2}=\left(\left\|a\right\|^{2}\right)^{2}$ holds.^[5]
Every normal element is a normaloid element, i.e. the spectral radius $r(a)$ equals the norm of $a$ , i.e. $r(a)=\left\|a\right\|$ .^[6] This follows from the spectral radius formula by repeated application of the previous property.^[7]
A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of $a$ to $a$ .^[3]

Notes

^ ^a ^b ^c ^d Dixmier 1977, p. 4.
^ ^a ^b Dixmier 1977, p. 5.
^ ^a ^b Dixmier 1977, p. 13.
^ Dixmier 1977, pp. 3–4.
^ Werner 2018, p. 518.
^ Heuser 1982, p. 390.
^ Werner 2018, pp. 284–285, 518.

References

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.
Heuser, Harro (1982). Functional analysis. Translated by Horvath, John. John Wiley & Sons Ltd. ISBN 0-471-10069-2.
Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer. ISBN 978-3-662-55407-4.

[FOOTNOTEDixmier19774-1] Dixmier 1977, p. 4.

[FOOTNOTEDixmier19775-2] Dixmier 1977, p. 5.

[FOOTNOTEDixmier197713-3] Dixmier 1977, p. 13.

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

[FOOTNOTEWerner2018518-5] Werner 2018, p. 518.

[FOOTNOTEHeuser1982390-6] Heuser 1982, p. 390.

[FOOTNOTEWerner2018284–285,_518-7] Werner 2018, pp. 284–285, 518.

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v t e 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