Self-adjoint

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Element of algebra where x* equals x

title: "Self-adjoint" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["abstract-algebra", "c-algebras"] description: "Element of algebra where x equals x" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Self-adjoint" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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^*).

Definition

Let \mathcal{A} be a *-algebra. An element a \in \mathcal{A} is called self-adjoint if

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

A subset \mathcal{B} \subseteq \mathcal{A} that is closed under the involution , i.e. \mathcal{B} = \mathcal{B}^, is called

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.

Especially in the older literature on -algebras and C-algebras, such elements are often called Because of that the notations \mathcal{A}_h, \mathcal{A}_H or 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
For each element a of a -algebra, the elements aa^ and a^*a are self-adjoint, since * is an
For each element a of a -algebra, the real and imaginary parts \operatorname{Re}(a) = \frac{1}{2} (a+a^) and \operatorname{Im}(a) = \frac{1}{2 \mathrm{i} } (a-a^*) are self-adjoint, where \mathrm{i} denotes the
If a \in \mathcal{A}_N is a normal element of a C*-algebra \mathcal{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

Criteria

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

Let a \in \mathcal{A}, then a^a is self-adjoint, since (a^a)^ = a^(a^)^ = a^a. A similarly calculation yields that aa^ is also
Let a = a_1 a_2 be the product of two self-adjoint elements a_1,a_2 \in \mathcal{A}_{sa}. 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
If \mathcal{A} is a C*-algebra, then a normal element a \in \mathcal{A}_N is self-adjoint if and only if its spectrum is real, i.e.

Properties

In *-algebras

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

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

In C*-algebras

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

For the spectrum \left| a \right| \in \sigma(a) or -\left| a \right| \in \sigma(a) holds, since \sigma(a) is real and r(a) = \left| a \right| holds for the spectral radius, because a is
According to the continuous functional calculus, there exist uniquely determined positive elements a_+,a_- \in \mathcal{A}+, such that a = a+ - a_- with For the norm, \left| a \right| = \max(\left|a_+\right|,\left|a_-\right|) holds. 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
For every a \in \mathcal{A}+ and odd n \in \mathbb{N}, there exists a uniquely determined b \in \mathcal{A}+ that satisfies b^n = a, i.e. a unique n-th root, as can be shown with the continuous functional

Notes

References

English translation of

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