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. a=a*).

## 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 𝒜 be a *-algebra. An element a∈𝒜 is called self-adjoint if a=a*.[1]

The set of self-adjoint elements is referred to as 𝒜sa.

A subset ℬ⊆𝒜 that is closed under the involution *, i.e. ℬ=ℬ*, is called self-adjoint.[2]

A special case from particular importance is the case where 𝒜 is a complete normed *-algebra, that satisfies the C*-identity (‖a*a‖=‖a‖2 ∀a∈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 𝒜h, 𝒜H or H(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 Re⁡(a)=12(a+a*) and Im⁡(a)=12i(a−a*) are self-adjoint, where i denotes the imaginary unit.[1]
* If a∈𝒜N is a normal element of a C*-algebra 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 𝒜 be a *-algebra. Then:

* Let a∈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=a1a2 be the product of two self-adjoint elements a1,a2∈𝒜sa. Then a is self-adjoint if a1 and a2 commutate, since (a1a2)*=a2*a1*=a2a1 always holds.[1]
* If 𝒜 is a C*-algebra, then a normal element a∈𝒜N is self-adjoint if and only if its spectrum is real, i.e. σ(a)⊆ℝ.[5]

## Properties

### In *-algebras

Let 𝒜 be a *-algebra. Then:

* Each element a∈𝒜 can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements a1,a2∈𝒜sa, so that a=a1+ia2 holds. Where a1=12(a+a*) and a2=12i(a−a*).[1]
* The set of self-adjoint elements 𝒜sa is a real linear subspace of 𝒜. From the previous property, it follows that 𝒜 is the direct sum of two real linear subspaces, i.e. 𝒜=𝒜sa⊕i𝒜sa.[7]
* If a∈𝒜sa is self-adjoint, then a is normal.[1]
* The *-algebra 𝒜 is called a hermitian *-algebra if every self-adjoint element a∈𝒜sa has a real spectrum σ(a)⊆ℝ.[8]

### In C*-algebras

Let 𝒜 be a C*-algebra and a∈𝒜sa. Then:

* For the spectrum ‖a‖∈σ(a) or −‖a‖∈σ(a) holds, since σ(a) is real and r(a)=‖a‖ holds for the spectral radius, because a is normal.[9]
* According to the continuous functional calculus, there exist uniquely determined positive elements a+,a−∈𝒜+, such that a=a+−a− with a+a−=a−a+=0. For the norm, ‖a‖=max⁡(‖a+‖,‖a−‖) 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 |a|=(a*a)12.[11]
* For every a∈𝒜+ and odd n∈ℕ, there exists a uniquely determined b∈𝒜+ that satisfies bn=a, i.e. a unique n-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. sfn error: no target: CITEREFPalmer1977 (help)
4. ↑ Dixmier 1977, pp. 3–4.
5. ↑ 5.0 5.1 Kadison 1983, p. 271. sfn error: no target: CITEREFKadison1983 (help)
6. ↑ Palmer 1977, pp. 798–800. sfn error: no target: CITEREFPalmer1977 (help)
7. ↑ Palmer 1977, p. 798. sfn error: no target: CITEREFPalmer1977 (help)
8. ↑ Palmer 1977, p. 1008. sfn error: no target: CITEREFPalmer1977 (help)
9. ↑ Kadison 1983, p. 238. sfn error: no target: CITEREFKadison1983 (help)
10. ↑ Kadison 1983, p. 246. sfn error: no target: CITEREFKadison1983 (help)
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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