Self-adjoint

Short description: Element of algebra where x* equals x

In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. [math]\displaystyle{ a = a^* }[/math]).

## Contents

* 1 Definition
* 2 Examples
* 3 Criteria
* 4 Properties
* 4.1 In *-algebras
* 4.2 In C*-algebras
* 5 See also
* 6 Notes
* 7 References

## Definition

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

The set of self-adjoint elements is referred to as [math]\displaystyle{ \mathcal{A}_{sa} }[/math].

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

A special case from particular importance is the case where [math]\displaystyle{ \mathcal{A} }[/math] is a complete normed *-algebra, that satisfies the C*-identity ([math]\displaystyle{ \left\| a^*a \right\| = \left\| a \right\|^2 \ \forall a \in \mathcal{A} }[/math]), 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 [math]\displaystyle{ \mathcal{A}_h }[/math], [math]\displaystyle{ \mathcal{A}_H }[/math] or [math]\displaystyle{ H(\mathcal{A}) }[/math] 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 [math]\displaystyle{ a }[/math] of a *-algebra, the elements [math]\displaystyle{ aa^* }[/math] and [math]\displaystyle{ a^*a }[/math] are self-adjoint, since * is an involutive antiautomorphism.[4]
* For each element [math]\displaystyle{ a }[/math] of a *-algebra, the real and imaginary parts [math]\displaystyle{ \operatorname{Re}(a) = \frac{1}{2} (a+a^*) }[/math] and [math]\displaystyle{ \operatorname{Im}(a) = \frac{1}{2 \mathrm{i} } (a-a^*) }[/math] are self-adjoint, where [math]\displaystyle{ \mathrm{i} }[/math] denotes the imaginary unit.[1]
* If [math]\displaystyle{ a \in \mathcal{A}_N }[/math] is a normal element of a C*-algebra [math]\displaystyle{ \mathcal{A} }[/math], then for every real-valued function [math]\displaystyle{ f }[/math], which is continuous on the spectrum of [math]\displaystyle{ a }[/math], the continuous functional calculus defines a self-adjoint element [math]\displaystyle{ f(a) }[/math].[5]

## Criteria

Let [math]\displaystyle{ \mathcal{A} }[/math] be a *-algebra. Then:

* Let [math]\displaystyle{ a \in \mathcal{A} }[/math], then [math]\displaystyle{ a^*a }[/math] is self-adjoint, since [math]\displaystyle{ (a^*a)^* = a^*(a^*)^* = a^*a }[/math]. A similarly calculation yields that [math]\displaystyle{ aa^* }[/math] is also self-adjoint.[6]
* Let [math]\displaystyle{ a = a_1 a_2 }[/math] be the product of two self-adjoint elements [math]\displaystyle{ a_1,a_2 \in \mathcal{A}_{sa} }[/math]. Then [math]\displaystyle{ a }[/math] is self-adjoint if [math]\displaystyle{ a_1 }[/math] and [math]\displaystyle{ a_2 }[/math] commutate, since [math]\displaystyle{ (a_1 a_2)^* = a_2^* a_1^* = a_2 a_1 }[/math] always holds.[1]
* If [math]\displaystyle{ \mathcal{A} }[/math] is a C*-algebra, then a normal element [math]\displaystyle{ a \in \mathcal{A}_N }[/math] is self-adjoint if and only if its spectrum is real, i.e. [math]\displaystyle{ \sigma(a) \subseteq \R }[/math].[5]

## Properties

### In *-algebras

Let [math]\displaystyle{ \mathcal{A} }[/math] be a *-algebra. Then:

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

### In C*-algebras

Let [math]\displaystyle{ \mathcal{A} }[/math] be a C*-algebra and [math]\displaystyle{ a \in \mathcal{A}_{sa} }[/math]. Then:

* For the spectrum [math]\displaystyle{ \left\| a \right\| \in \sigma(a) }[/math] or [math]\displaystyle{ -\left\| a \right\| \in \sigma(a) }[/math] holds, since [math]\displaystyle{ \sigma(a) }[/math] is real and [math]\displaystyle{ r(a) = \left\| a \right\| }[/math] holds for the spectral radius, because [math]\displaystyle{ a }[/math] is normal.[9]
* According to the continuous functional calculus, there exist uniquely determined positive elements [math]\displaystyle{ a_+,a_- \in \mathcal{A}_+ }[/math], such that [math]\displaystyle{ a = a_+ - a_- }[/math] with [math]\displaystyle{ a_+ a_- = a_- a_+ = 0 }[/math]. For the norm, [math]\displaystyle{ \left\| a \right\| = \max(\left\|a_+\right\|,\left\|a_-\right\|) }[/math] holds.[10] The elements [math]\displaystyle{ a_+ }[/math] and [math]\displaystyle{ a_- }[/math] are also referred to as the positive and negative parts. In addition, [math]\displaystyle{ |a| = a_+ + a_- }[/math] holds for the absolute value defined for every element [math]\displaystyle{ |a| = (a^* a)^\frac{1}{2} }[/math].[11]
* For every [math]\displaystyle{ a \in \mathcal{A}_+ }[/math] and odd [math]\displaystyle{ n \in \mathbb{N} }[/math], there exists a uniquely determined [math]\displaystyle{ b \in \mathcal{A}_+ }[/math] that satisfies [math]\displaystyle{ b^n = a }[/math], i.e. a unique [math]\displaystyle{ n }[/math]-th root, as can be shown with the continuous functional calculus.[12]

## See also

* Self-adjoint matrix
* Self-adjoint operator

## Notes

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Dixmier 1977, p. 4.
2. ↑ Dixmier 1977, p. 3.
3. ↑ Palmer 1977, p. 800.
4. ↑ Dixmier 1977, pp. 3-4.
5. ↑ 5.0 5.1 Kadison 1983, p. 271.
6. ↑ Palmer 1977, pp. 798-800.
7. ↑ Palmer 1977, p. 798.
8. ↑ Palmer 1977, p. 1008.
9. ↑ Kadison 1983, p. 238.
10. ↑ Kadison 1983, p. 246.
11. ↑ Dixmier 1977, p. 15.
12. ↑ Blackadar 2006, p. 63.

## References

* 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.

* 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 of a spectrum|

* (Continuous
* Point
* Residual)
* Approximate point
* Compression
* Discrete
* Spectral abscissa

Spectral Theorem|

* Borel functional calculus
* Min-max theorem
* 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
* Uniform algebra

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
* Limiting absorption principle
* 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

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