Self-adjoint

Let $\backslash mathcal\{A\}$ be a *-algebra. An element $a\; \backslash in\; \backslash mathcal\{A\}$ is called self-adjoint if {{nowrap|$a\; =\; a^*$.{{sfn|Dixmier|1977|p=4}}}}

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

A subset $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash mathcal\{A\}$ that is closed under the involution *, i.e. $\backslash mathcal\{B\}\; =\; \backslash mathcal\{B\}^*$, is called {{nowrap|self-adjoint.{{sfn|Dixmier|1977|p=3}}}}

A special case of particular importance is the case where $\backslash mathcal\{A\}$ is a complete normed *-algebra, that satisfies the C*-identity ($\backslash left\backslash |\; a^*a\; \backslash right\backslash |\; =\; \backslash left\backslash |\; a\; \backslash right\backslash |^2\; \backslash \; \backslash forall\; a\; \backslash in\; \backslash mathcal\{A\}$), 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 $\backslash mathcal\{A\}\_h$, $\backslash mathcal\{A\}\_H$ or $H(\backslash mathcal\{A\})$ for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

• Each positive element of a C*-algebra is {{nowrap|self-adjoint.{{sfn|Palmer|2001|p=800}}}}
• For each element $a$ of a *-algebra, the elements $aa^*$ and $a^*a$ are self-adjoint, since * is an {{nowrap|involutive antiautomorphism.{{sfn|Dixmier|1977|pages=3-4}}}}
• For each element $a$ of a *-algebra, the real and imaginary parts $\backslash operatorname\{Re\}(a)\; =\; \backslash frac\{1\}\{2\}\; (a+a^*)$ and $\backslash operatorname\{Im\}(a)\; =\; \backslash frac\{1\}\{2\; \backslash mathrm\{i\}\; \}\; (a-a^*)$ are self-adjoint, where $\backslash mathrm\{i\}$ denotes the {{nowrap|imaginary unit.{{sfn|Dixmier|1977|p=4}}}}
• If $a\; \backslash in\; \backslash mathcal\{A\}\_N$ is a normal element of a C*-algebra $\backslash mathcal\{A\}$, then for every real-valued function $f$, which is continuous on the spectrum of $a$, the continuous functional calculus defines a self-adjoint element {{nowrap|$f(a)$.{{sfn|Kadison|Ringrose|1983|p=271}}}}

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

• Let $a\; \backslash in\; \backslash mathcal\{A\}$, then $a^*a$ is self-adjoint, since $(a^*a)^*\; =\; a^*(a^*)^*\; =\; a^*a$. A similarly calculation yields that $aa^*$ is also {{nowrap|self-adjoint.{{sfn|Palmer|2001|pages=798-800}}}}
• Let $a\; =\; a\_1\; a\_2$ be the product of two self-adjoint elements {{nowrap|$a\_1,a\_2\; \backslash in\; \backslash mathcal\{A\}\_\{sa\}$.}} 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 {{nowrap|holds.{{sfn|Dixmier|1977|p=4}}}}
• If $\backslash mathcal\{A\}$ is a C*-algebra, then a normal element $a\; \backslash in\; \backslash mathcal\{A\}\_N$ is self-adjoint if and only if its spectrum is real, i.e. {{nowrap|$\backslash sigma(a)\; \backslash subseteq\; \backslash R$.{{sfn|Kadison|Ringrose|1983|p=271}}}}

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

• Each element $a\; \backslash in\; \backslash mathcal\{A\}$ can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements $a\_1,a\_2\; \backslash in\; \backslash mathcal\{A\}\_\{sa\}$, so that $a\; =\; a\_1\; +\; \backslash mathrm\{i\}\; a\_2$ holds. Where $a\_1\; =\; \backslash frac\{1\}\{2\}\; (a\; +\; a^*)$ and {{nowrap|$a\_2\; =\; \backslash frac\{1\}\{2\; \backslash mathrm\{i\}\}\; (a\; -\; a^*)$.{{sfn|Dixmier|1977|p=4}}}}
• The set of self-adjoint elements $\backslash mathcal\{A\}\_\{sa\}$ is a real linear subspace of {{nowrap|$\backslash mathcal\{A\}$.}} From the previous property, it follows that $\backslash mathcal\{A\}$ is the direct sum of two real linear subspaces, i.e. {{nowrap|$\backslash mathcal\{A\}\; =\; \backslash mathcal\{A\}\_\{sa\}\; \backslash oplus\; \backslash mathrm\{i\}\; \backslash mathcal\{A\}\_\{sa\}$.{{sfn|Palmer|2001|p=798}}}}
• If $a\; \backslash in\; \backslash mathcal\{A\}\_\{sa\}$ is self-adjoint, then $a$ is {{nowrap|normal.{{sfn|Dixmier|1977|p=4}}}}
• The *-algebra $\backslash mathcal\{A\}$ is called a hermitian *-algebra if every self-adjoint element $a\; \backslash in\; \backslash mathcal\{A\}\_\{sa\}$ has a real spectrum {{nowrap|$\backslash sigma(a)\; \backslash subseteq\; \backslash R$.{{sfn|Palmer|2001|p=1008}}}}

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

• For the spectrum $\backslash left\backslash |\; a\; \backslash right\backslash |\; \backslash in\; \backslash sigma(a)$ or $-\backslash left\backslash |\; a\; \backslash right\backslash |\; \backslash in\; \backslash sigma(a)$ holds, since $\backslash sigma(a)$ is real and $r(a)\; =\; \backslash left\backslash |\; a\; \backslash right\backslash |$ holds for the spectral radius, because $a$ is {{nowrap|normal.{{sfn|Kadison|Ringrose|1983|p=238}}}}
• According to the continuous functional calculus, there exist uniquely determined positive elements $a\_+,a\_-\; \backslash in\; \backslash mathcal\{A\}\_+$, such that $a\; =\; a\_+\; -\; a\_-$ with {{nowrap|$a\_+\; a\_-\; =\; a\_-\; a\_+\; =\; 0$.}} For the norm, $\backslash left\backslash |\; a\; \backslash right\backslash |\; =\; \backslash max(\backslash left\backslash |a\_+\backslash right\backslash |,\backslash left\backslash |a\_-\backslash right\backslash |)$ holds.{{sfn|Kadison|Ringrose|1983|p=246}} 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 {{nowrap|$|a|\; =\; (a^*\; a)^\backslash frac\{1\}\{2\}$.{{sfn|Dixmier|1977|p=15}}}}
• For every $a\; \backslash in\; \backslash mathcal\{A\}\_+$ and odd $n\; \backslash in\; \backslash mathbb\{N\}$, there exists a uniquely determined $b\; \backslash in\; \backslash mathcal\{A\}\_+$ that satisfies $b^n\; =\; a$, i.e. a unique $n$-th root, as can be shown with the continuous functional {{nowrap|calculus.{{sfn|Blackadar|2006|p=63}}}}

• Self-adjoint matrix
• Self-adjoint operator

{{reflist}}

• {{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 |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