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Definition

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^{*}} .¹

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.²

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.³ 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.

Examples

• Each positive element of a C*-algebra is self-adjoint.⁴
• For each element a {\displaystyle a} of a *-algebra, the elements a a ∗ {\displaystyle aa^{*}} and a ∗ a {\displaystyle a^{*}a} are self-adjoint, since * is an involutive antiautomorphism.⁵
• For each element a {\displaystyle a} of a *-algebra, the real and imaginary parts Re ⁡ ( a ) = 1 2 ( a + a ∗ ) {\textstyle \operatorname {Re} (a)={\frac {1}{2}}(a+a^{*})} and Im ⁡ ( a ) = 1 2 i ( a − a ∗ ) {\textstyle \operatorname {Im} (a)={\frac {1}{2\mathrm {i} }}(a-a^{*})} are self-adjoint, where i {\displaystyle \mathrm {i} } denotes the imaginary unit.⁶
• If a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} is a normal element of a C*-algebra A {\displaystyle {\mathcal {A}}} , then for every real-valued function f {\displaystyle f} , which is continuous on the spectrum of a {\displaystyle a} , the continuous functional calculus defines a self-adjoint element f ( a ) {\displaystyle f(a)} .⁷

Criteria

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

• Let a ∈ A {\displaystyle a\in {\mathcal {A}}} , then a ∗ a {\displaystyle a^{*}a} is self-adjoint, since ( a ∗ a ) ∗ = a ∗ ( a ∗ ) ∗ = a ∗ a {\displaystyle (a^{*}a)^{*}=a^{*}(a^{*})^{*}=a^{*}a} . A similarly calculation yields that a a ∗ {\displaystyle aa^{*}} is also self-adjoint.⁸
• Let a = a 1 a 2 {\displaystyle a=a_{1}a_{2}} be the product of two self-adjoint elements a 1 , a 2 ∈ A s a {\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}} . Then a {\displaystyle a} is self-adjoint if a 1 {\displaystyle a_{1}} and a 2 {\displaystyle a_{2}} commutate, since ( a 1 a 2 ) ∗ = a 2 ∗ a 1 ∗ = a 2 a 1 {\displaystyle (a_{1}a_{2})^{*}=a_{2}^{*}a_{1}^{*}=a_{2}a_{1}} always holds.⁹
• If A {\displaystyle {\mathcal {A}}} is a C*-algebra, then a normal element a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} is self-adjoint if and only if its spectrum is real, i.e. σ ( a ) ⊆ R {\displaystyle \sigma (a)\subseteq \mathbb {R} } .¹⁰

Properties

In *-algebras

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

• Each element a ∈ A {\displaystyle a\in {\mathcal {A}}} can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements a 1 , a 2 ∈ A s a {\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}} , so that a = a 1 + i a 2 {\displaystyle a=a_{1}+\mathrm {i} a_{2}} holds. Where a 1 = 1 2 ( a + a ∗ ) {\textstyle a_{1}={\frac {1}{2}}(a+a^{*})} and a 2 = 1 2 i ( a − a ∗ ) {\textstyle a_{2}={\frac {1}{2\mathrm {i} }}(a-a^{*})} .¹¹
• The set of self-adjoint elements A s a {\displaystyle {\mathcal {A}}_{sa}} is a real linear subspace of A {\displaystyle {\mathcal {A}}} . From the previous property, it follows that A {\displaystyle {\mathcal {A}}} is the direct sum of two real linear subspaces, i.e. A = A s a ⊕ i A s a {\displaystyle {\mathcal {A}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}} .¹²
• If a ∈ A s a {\displaystyle a\in {\mathcal {A}}_{sa}} is self-adjoint, then a {\displaystyle a} is normal.¹³
• The *-algebra A {\displaystyle {\mathcal {A}}} is called a hermitian *-algebra if every self-adjoint element a ∈ A s a {\displaystyle a\in {\mathcal {A}}_{sa}} has a real spectrum σ ( a ) ⊆ R {\displaystyle \sigma (a)\subseteq \mathbb {R} } .¹⁴

In C*-algebras

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.¹⁵
• 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.¹⁶ 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}}} .¹⁷
• 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.¹⁸

See also

• Self-adjoint matrix
• Self-adjoint operator

Notes

• 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.

References

1. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

2. Dixmier 1977, p. 3. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

3. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

4. Palmer 2001, p. 800. - Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0. ↩

5. Dixmier 1977, pp. 3–4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

6. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

7. Kadison & Ringrose 1983, p. 271. - 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. ↩

8. Palmer 2001, pp. 798–800. - Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0. ↩

9. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

10. Kadison & Ringrose 1983, p. 271. - 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. ↩

11. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

12. Palmer 2001, p. 798. - Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0. ↩

13. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

14. Palmer 2001, p. 1008. - Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0. ↩

15. Kadison & Ringrose 1983, p. 238. - 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. ↩

16. Kadison & Ringrose 1983, p. 246. - 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. ↩

17. Dixmier 1977, p. 15. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

18. Blackadar 2006, p. 63. - Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. p. 63. ISBN 3-540-28486-9. ↩