{{Short description|Element of *-algebra where x* equals x}} In [[mathematics]], an [[Element (mathematics)|element]] of a [[*-algebra]] is called '''self-adjoint''' if it is the same as its [[Adjoint_operator|adjoint]] (i.e. <math>a = a^*</math>).
== Definition ==
Let <math>\mathcal{A}</math> be a *-algebra. An element <math>a \in \mathcal{A}</math> is called self-adjoint if {{nowrap|<math>a = a^*</math>.{{sfn|Dixmier|1977|p=4}}}}
The [[Set (mathematics)|set]] of self-adjoint elements is referred to as {{nowrap|<math>\mathcal{A}_{sa}</math>.}}
A [[subset]] <math>\mathcal{B} \subseteq \mathcal{A}</math> that is [[Closed set|closed]] under the [[Involution (mathematics)|involution]] *, i.e. <math>\mathcal{B} = \mathcal{B}^*</math>, is called {{nowrap|self-adjoint.{{sfn|Dixmier|1977|p=3}}}}
A special case of particular importance is the case where <math>\mathcal{A}</math> is a [[Banach algebra#Banach *-algebras|complete normed *-algebra]], that satisfies the C*-identity (<math>\left\| a^*a \right\| = \left\| a \right\|^2 \ \forall a \in \mathcal{A}</math>), which is called a [[C*-algebra]].
Especially in the older literature on *-algebras and C*-algebras, such elements are often called {{nowrap|hermitian.{{sfn|Dixmier|1977|p=4}}}} Because of that the notations <math>\mathcal{A}_h</math>, <math>\mathcal{A}_H</math> or <math>H(\mathcal{A})</math> 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 {{nowrap|self-adjoint.{{sfn|Palmer|2001|p=800}}}} * For each element <math>a</math> of a *-algebra, the elements <math>aa^*</math> and <math>a^*a</math> are self-adjoint, since * is an {{nowrap|[[Antihomomorphism#Involutions|involutive antiautomorphism]].{{sfn|Dixmier|1977|pages=3-4}}}} * For each element <math>a</math> of a *-algebra, the [[real and imaginary parts]] <math display="inline">\operatorname{Re}(a) = \frac{1}{2} (a+a^*)</math> and <math display="inline">\operatorname{Im}(a) = \frac{1}{2 \mathrm{i} } (a-a^*)</math> are self-adjoint, where <math>\mathrm{i}</math> denotes the {{nowrap|[[imaginary unit]].{{sfn|Dixmier|1977|p=4}}}} * If <math>a \in \mathcal{A}_N</math> is a [[normal element]] of a C*-algebra <math>\mathcal{A}</math>, then for every [[real-valued function]] <math>f</math>, which is [[Continuous function|continuous]] on the [[Banach algebra#Spectral theory|spectrum]] of <math>a</math>, the [[continuous functional calculus]] defines a self-adjoint element {{nowrap|<math>f(a)</math>.{{sfn|Kadison|Ringrose|1983|p=271}}}}
== Criteria ==
Let <math>\mathcal{A}</math> be a *-algebra. Then:
* Let <math>a \in \mathcal{A}</math>, then <math>a^*a</math> is self-adjoint, since <math>(a^*a)^* = a^*(a^*)^* = a^*a</math>. A similarly calculation yields that <math>aa^*</math> is also {{nowrap|self-adjoint.{{sfn|Palmer|2001|pages=798-800}}}} * Let <math>a = a_1 a_2</math> be the [[Product (mathematics)|product]] of two self-adjoint elements {{nowrap|<math>a_1,a_2 \in \mathcal{A}_{sa}</math>.}} Then <math>a</math> is self-adjoint if <math>a_1</math> and <math>a_2</math> [[Commutative property|commutate]], since <math>(a_1 a_2)^* = a_2^* a_1^* = a_2 a_1</math> always {{nowrap|holds.{{sfn|Dixmier|1977|p=4}}}} * If <math>\mathcal{A}</math> is a C*-algebra, then a normal element <math>a \in \mathcal{A}_N</math> is self-adjoint if and only if its spectrum is real, i.e. {{nowrap|<math>\sigma(a) \subseteq \R</math>.{{sfn|Kadison|Ringrose|1983|p=271}}}}
== Properties == === In *-algebras ===
Let <math>\mathcal{A}</math> be a *-algebra. Then:
* Each element <math>a \in \mathcal{A}</math> can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements <math>a_1,a_2 \in \mathcal{A}_{sa}</math>, so that <math>a = a_1 + \mathrm{i} a_2</math> holds. Where <math display="inline">a_1 = \frac{1}{2} (a + a^*)</math> and {{nowrap|<math display="inline">a_2 = \frac{1}{2 \mathrm{i}} (a - a^*)</math>.{{sfn|Dixmier|1977|p=4}}}} * The set of self-adjoint elements <math>\mathcal{A}_{sa}</math> is a [[Real number|real]] [[linear subspace]] of {{nowrap|<math>\mathcal{A}</math>.}} From the previous property, it follows that <math>\mathcal{A}</math> is the [[direct sum]] of two real linear subspaces, i.e. {{nowrap|<math>\mathcal{A} = \mathcal{A}_{sa} \oplus \mathrm{i} \mathcal{A}_{sa}</math>.{{sfn|Palmer|2001|p=798}}}} * If <math>a \in \mathcal{A}_{sa}</math> is self-adjoint, then <math>a</math> is {{nowrap|normal.{{sfn|Dixmier|1977|p=4}}}} * The *-algebra <math>\mathcal{A}</math> is called a hermitian *-algebra if every self-adjoint element <math>a \in \mathcal{A}_{sa}</math> has a real spectrum {{nowrap|<math>\sigma(a) \subseteq \R</math>.{{sfn|Palmer|2001|p=1008}}}}
=== In C*-algebras ===
Let <math>\mathcal{A}</math> be a C*-algebra and <math>a \in \mathcal{A}_{sa}</math>. Then:
* For the spectrum <math>\left\| a \right\| \in \sigma(a)</math> or <math>-\left\| a \right\| \in \sigma(a)</math> holds, since <math>\sigma(a)</math> is real and <math>r(a) = \left\| a \right\|</math> holds for the [[Banach algebra#Spectral theory|spectral radius]], because <math>a</math> is {{nowrap|normal.{{sfn|Kadison|Ringrose|1983|p=238}}}} * According to the continuous functional calculus, there exist uniquely determined positive elements <math>a_+,a_- \in \mathcal{A}_+</math>, such that <math>a = a_+ - a_-</math> with {{nowrap|<math>a_+ a_- = a_- a_+ = 0</math>.}} For the norm, <math>\left\| a \right\| = \max(\left\|a_+\right\|,\left\|a_-\right\|)</math> holds.{{sfn|Kadison|Ringrose|1983|p=246}} The elements <math>a_+</math> and <math>a_-</math> are also referred to as the [[positive and negative parts]]. In addition, <math>|a| = a_+ + a_-</math> holds for the absolute value defined for every element {{nowrap|<math display="inline">|a| = (a^* a)^\frac{1}{2}</math>.{{sfn|Dixmier|1977|p=15}}}} * For every <math>a \in \mathcal{A}_+</math> and odd <math>n \in \mathbb{N}</math>, there exists a uniquely determined <math>b \in \mathcal{A}_+</math> that satisfies <math>b^n = a</math>, i.e. a unique <math>n</math>[[Nth root|-th root]], as can be shown with the continuous functional {{nowrap|calculus.{{sfn|Blackadar|2006|p=63}}}}
== See also==
* [[Hermitian matrix|Self-adjoint matrix]] * [[Self-adjoint operator]]
== Notes == {{reflist}}
== References == * {{cite book |last=Blackadar|first=Bruce |title=Operator Algebras. Theory of C*-Algebras and von Neumann Algebras |publisher=Springer |location=Berlin/Heidelberg |year=2006 |isbn=3-540-28486-9 |pages=63 }} * {{cite book |last=Dixmier |first=Jacques |author-link=Jacques Dixmier |title=C*-algebras |publisher=North-Holland |location=Amsterdam/New York/Oxford |year=1977 |isbn=0-7204-0762-1 |translator-last=Jellett |translator-first=Francis }} English translation of {{cite book |display-authors=0 |last=Dixmier |first=Jacques |title=Les C*-algèbres et leurs représentations |language=fr |publisher=Gauthier-Villars |year=1969 }} * {{cite book |last1=Kadison |first1=Richard V. |last2=Ringrose |first2=John R. |title=Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. |publisher=Academic Press |location=New York/London |year=1983 |isbn=0-12-393301-3}} * {{cite book |last=Palmer|first=Theodore W. |title=Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. |publisher=Cambridge university press |year=2001 |isbn=0-521-36638-0 }}
{{SpectralTheory}}
[[Category:Abstract algebra]] [[Category:C*-algebras]]