Self-adjoint element

{{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. <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 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 under the 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 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|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 on the 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 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> 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 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 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>-th root, as can be shown with the continuous functional {{nowrap|calculus.{{sfn|Blackadar|2006|p=63}}}}

== 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 }}

Category:Abstract algebra Category:C*-algebras