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Self-adjoint - Wikipedia
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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 ∗ {\displaystyle a=a^{*}} ).
Let A {\displaystyle {\mathcal {A}}} be a *-algebra. An element a ∈ A {\displaystyle a\in {\mathcal {A}}} is called self-adjoint if a = a ∗ {\displaystyle a=a^{*}} .
The set of self-adjoint elements is referred to as A s a {\displaystyle {\mathcal {A}}_{sa}} .
A subset B ⊆ A {\displaystyle {\mathcal {B}}\subseteq {\mathcal {A}}} that is closed under the involution *, i.e. B = B ∗ {\displaystyle {\mathcal {B}}={\mathcal {B}}^{*}} , is called self-adjoint.
A special case of particular importance is the case where A {\displaystyle {\mathcal {A}}} is a complete normed *-algebra, that satisfies the C*-identity ( ‖ a ∗ a ‖ = ‖ a ‖ 2 ∀ a ∈ A {\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. Because of that the notations A h {\displaystyle {\mathcal {A}}_{h}} , A H {\displaystyle {\mathcal {A}}_{H}} or H ( A ) {\displaystyle H({\mathcal {A}})} for the set of self-adjoint elements are also sometimes used, even in the more recent literature.
Let A {\displaystyle {\mathcal {A}}} be a *-algebra. Then:
Let A {\displaystyle {\mathcal {A}}} be a *-algebra. Then:
Let A {\displaystyle {\mathcal {A}}} be a C*-algebra and a ∈ A s a {\displaystyle a\in {\mathcal {A}}_{sa}} . Then:
- For the spectrum ‖ a ‖ ∈ σ ( a ) {\displaystyle \left\|a\right\|\in \sigma (a)} or − ‖ a ‖ ∈ σ ( a ) {\displaystyle -\left\|a\right\|\in \sigma (a)} holds, since σ ( a ) {\displaystyle \sigma (a)} is real and r ( a ) = ‖ a ‖ {\displaystyle r(a)=\left\|a\right\|} holds for the spectral radius, because a {\displaystyle a} is normal.
- According to the continuous functional calculus, there exist uniquely determined positive elements a + , a − ∈ A + {\displaystyle a_{+},a_{-}\in {\mathcal {A}}_{+}} , such that a = a + − a − {\displaystyle a=a_{+}-a_{-}} with a + a − = a − a + = 0 {\displaystyle a_{+}a_{-}=a_{-}a_{+}=0} . For the norm, ‖ a ‖ = max ( ‖ a + ‖ , ‖ a − ‖ ) {\displaystyle \left\|a\right\|=\max(\left\|a_{+}\right\|,\left\|a_{-}\right\|)} holds. The elements a + {\displaystyle a_{+}} and a − {\displaystyle a_{-}} are also referred to as the positive and negative parts. In addition, | a | = a + + a − {\displaystyle |a|=a_{+}+a_{-}} holds for the absolute value defined for every element | a | = ( a ∗ a ) 1 2 {\textstyle |a|=(a^{*}a)^{\frac {1}{2}}} .
- For every a ∈ A + {\displaystyle a\in {\mathcal {A}}_{+}} and odd n ∈ N {\displaystyle n\in \mathbb {N} } , there exists a uniquely determined b ∈ A + {\displaystyle b\in {\mathcal {A}}_{+}} that satisfies b n = a {\displaystyle b^{n}=a} , i.e. a unique n {\displaystyle n} -th root, as can be shown with the continuous functional calculus.
- 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.
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