GeezerVison

GeezerVison is showing en.wikipedia.org — Self-adjoint element - Wikipedia

(Redirected from Self adjoint)

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

The set of self-adjoint elements is referred to as $𝒜_{s a}$ .

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

A special case of particular importance is the case where $𝒜$ is a complete normed *-algebra, that satisfies the C*-identity ( $‖ a^{*} a ‖ = {‖ a ‖}^{2} \forall a \in 𝒜$ ), 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 $𝒜_{h}$ , $𝒜_{H}$ 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 $a a^{*}$ and $a^{*} a$ are self-adjoint, since * is an involutive antiautomorphism.^[4]
For each element $a$ of a *-algebra, the real and imaginary parts $Re (a) = \frac{1}{2} (a + a^{*})$ and $Im (a) = \frac{1}{2 i} (a - a^{*})$ are self-adjoint, where $i$ denotes the imaginary unit.^[1]
If $a \in 𝒜_{N}$ 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 $a \in 𝒜$ , then $a^{*} a$ is self-adjoint, since $(a^{*} a)^{*} = a^{*} (a^{*})^{*} = a^{*} a$ . A similarly calculation yields that $a a^{*}$ is also self-adjoint.^[6]
Let $a = a_{1} a_{2}$ be the product of two self-adjoint elements $a_{1}, a_{2} \in 𝒜_{s a}$ . 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 $𝒜$ is a C*-algebra, then a normal element $a \in 𝒜_{N}$ is self-adjoint if and only if its spectrum is real, i.e. $σ (a) \subseteq ℝ$ .^[5]

Properties

In *-algebras

Let $𝒜$ be a *-algebra. Then:

Each element $a \in 𝒜$ can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements $a_{1}, a_{2} \in 𝒜_{s a}$ , so that $a = a_{1} + i a_{2}$ holds. Where $a_{1} = \frac{1}{2} (a + a^{*})$ and $a_{2} = \frac{1}{2 i} (a - a^{*})$ .^[1]
The set of self-adjoint elements $𝒜_{s a}$ 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. $𝒜 = 𝒜_{s a} \oplus i 𝒜_{s a}$ .^[7]
If $a \in 𝒜_{s a}$ is self-adjoint, then $a$ is normal.^[1]
The *-algebra $𝒜$ is called a hermitian *-algebra if every self-adjoint element $a \in 𝒜_{s a}$ has a real spectrum $σ (a) \subseteq ℝ$ .^[8]

In C*-algebras

Let $𝒜$ be a C*-algebra and $a \in 𝒜_{s a}$ . Then:

For the spectrum $‖ a ‖ \in σ (a)$ or $- ‖ a ‖ \in σ (a)$ 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 $a_{+}, a_{-} \in 𝒜_{+}$ , 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 $| a | = (a^{*} a)^{\frac{1}{2}}$ .^[11]
For every $a \in 𝒜_{+}$ and odd $n \in ℕ$ , there exists a uniquely determined $b \in 𝒜_{+}$ that satisfies $b^{n} = a$ , i.e. a unique $n$ -th root, as can be shown with the continuous functional calculus.^[12]

Notes

1 2 3 4 5 6 Dixmier 1977, p. 4.
↑ Dixmier 1977, p. 3.
↑ Palmer 2001, p. 800.
↑ Dixmier 1977, pp. 3–4.
1 2 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.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Self-adjoint_element&oldid=1355094475"

Categories:

Hidden categories: