# 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. $\displaystyle{ a = a^* }$).

## Definition

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

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

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

A special case from particular importance is the case where $\displaystyle{ \mathcal{A} }$ is a complete normed *-algebra, that satisfies the C*-identity ($\displaystyle{ \left\| a^*a \right\| = \left\| a \right\|^2 \ \forall a \in \mathcal{A} }$), 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 $\displaystyle{ \mathcal{A}_h }$, $\displaystyle{ \mathcal{A}_H }$ or $\displaystyle{ H(\mathcal{A}) }$ for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

## Examples

• Each positive element of a C*-algebra is self-adjoint.[3]
• For each element $\displaystyle{ a }$ of a *-algebra, the elements $\displaystyle{ aa^* }$ and $\displaystyle{ a^*a }$ are self-adjoint, since * is an
• For each element $\displaystyle{ a }$ of a *-algebra, the real and imaginary parts $\displaystyle{ \operatorname{Re}(a) = \frac{1}{2} (a+a^*) }$ and $\displaystyle{ \operatorname{Im}(a) = \frac{1}{2 \mathrm{i} } (a-a^*) }$ are self-adjoint, where $\displaystyle{ \mathrm{i} }$ denotes the
• If $\displaystyle{ a \in \mathcal{A}_N }$ is a normal element of a C*-algebra $\displaystyle{ \mathcal{A} }$, then for every real-valued function $\displaystyle{ f }$, which is continuous on the spectrum of $\displaystyle{ a }$, the continuous functional calculus defines a self-adjoint element $\displaystyle{ f(a) }$.[5]

## Criteria

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

• Let $\displaystyle{ a \in \mathcal{A} }$, then $\displaystyle{ a^*a }$ is self-adjoint, since $\displaystyle{ (a^*a)^* = a^*(a^*)^* = a^*a }$. A similarly calculation yields that $\displaystyle{ aa^* }$ is also self-adjoint.[6]
• Let $\displaystyle{ a = a_1 a_2 }$ be the product 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 $\displaystyle{ (a_1 a_2)^* = a_2^* a_1^* = a_2 a_1 }$ always holds.[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. $\displaystyle{ \sigma(a) \subseteq \R }$.[5]

## Properties

### In *-algebras

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 $\displaystyle{ a_1 = \frac{1}{2} (a + a^*) }$ and $\displaystyle{ a_2 = \frac{1}{2 \mathrm{i}} (a - a^*) }$.[1]
• The set of self-adjoint elements $\displaystyle{ \mathcal{A}_{sa} }$ is a real linear subspace of $\displaystyle{ \mathcal{A} }$. From the previous property, it follows that $\displaystyle{ \mathcal{A} }$ is the direct 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 $\displaystyle{ a }$ is normal.[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 $\displaystyle{ \sigma(a) \subseteq \R }$.[8]

### In C*-algebras

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

• For the spectrum $\displaystyle{ \left\| a \right\| \in \sigma(a) }$ or $\displaystyle{ -\left\| a \right\| \in \sigma(a) }$ holds, since $\displaystyle{ \sigma(a) }$ is real and $\displaystyle{ r(a) = \left\| a \right\| }$ holds for the spectral radius, because $\displaystyle{ a }$ is normal.[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, $\displaystyle{ \left\| a \right\| = \max(\left\|a_+\right\|,\left\|a_-\right\|) }$ holds.[10] The elements $\displaystyle{ a_+ }$ and $\displaystyle{ a_- }$ are also referred to as the positive and negative parts. In addition, $\displaystyle{ |a| = a_+ + a_- }$ holds for the absolute value defined for every element $\displaystyle{ |a| = (a^* a)^\frac{1}{2} }$.[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 functional calculus.[12]

## Notes

1. Dixmier 1977, p. 4.
2. Dixmier 1977, p. 3.
3. Palmer 1977, p. 800.
4. Dixmier 1977, pp. 3-4.
5. 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.