Self-Adjoint

In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. [math]\displaystyle{ a = a^* }[/math]).

Definition

Let [math]\displaystyle{ \mathcal{A} }[/math] be a *-algebra. An element [math]\displaystyle{ a \in \mathcal{A} }[/math] is called self-adjoint if [math]\displaystyle{ a = a^* }[/math].^[1]

The set of self-adjoint elements is referred to as [math]\displaystyle{ \mathcal{A}_{sa} }[/math].

A subset [math]\displaystyle{ \mathcal{B} \subseteq \mathcal{A} }[/math] that is closed under the involution *, i.e. [math]\displaystyle{ \mathcal{B} = \mathcal{B}^* }[/math], is called self-adjoint.^[2]

A special case from particular importance is the case where [math]\displaystyle{ \mathcal{A} }[/math] is a complete normed *-algebra, that satisfies the C*-identity ([math]\displaystyle{ \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 hermitian.^[1] Because of that the notations [math]\displaystyle{ \mathcal{A}_h }[/math], [math]\displaystyle{ \mathcal{A}_H }[/math] or [math]\displaystyle{ 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 self-adjoint.^[3]
• For each element [math]\displaystyle{ a }[/math] of a *-algebra, the elements [math]\displaystyle{ aa^* }[/math] and [math]\displaystyle{ a^*a }[/math] are self-adjoint, since * is an involutive antiautomorphism.^[4]
• For each element [math]\displaystyle{ a }[/math] of a *-algebra, the real and imaginary parts [math]\displaystyle{ \operatorname{Re}(a) = \frac{1}{2} (a+a^*) }[/math] and [math]\displaystyle{ \operatorname{Im}(a) = \frac{1}{2 \mathrm{i} } (a-a^*) }[/math] are self-adjoint, where [math]\displaystyle{ \mathrm{i} }[/math] denotes the imaginary unit.^[1]
• If [math]\displaystyle{ a \in \mathcal{A}_N }[/math] is a normal element of a C*-algebra [math]\displaystyle{ \mathcal{A} }[/math], then for every real-valued function [math]\displaystyle{ f }[/math], which is continuous on the spectrum of [math]\displaystyle{ a }[/math], the continuous functional calculus defines a self-adjoint element [math]\displaystyle{ f(a) }[/math].^[5]

Criteria

Let [math]\displaystyle{ \mathcal{A} }[/math] be a *-algebra. Then:

• Let [math]\displaystyle{ a \in \mathcal{A} }[/math], then [math]\displaystyle{ a^*a }[/math] is self-adjoint, since [math]\displaystyle{ (a^*a)^* = a^*(a^*)^* = a^*a }[/math]. A similarly calculation yields that [math]\displaystyle{ aa^* }[/math] is also self-adjoint.^[6]
• Let [math]\displaystyle{ a = a_1 a_2 }[/math] be the product of two self-adjoint elements [math]\displaystyle{ a_1,a_2 \in \mathcal{A}_{sa} }[/math]. Then [math]\displaystyle{ a }[/math] is self-adjoint if [math]\displaystyle{ a_1 }[/math] and [math]\displaystyle{ a_2 }[/math] commutate, since [math]\displaystyle{ (a_1 a_2)^* = a_2^* a_1^* = a_2 a_1 }[/math] always holds.^[1]
• If [math]\displaystyle{ \mathcal{A} }[/math] is a C*-algebra, then a normal element [math]\displaystyle{ a \in \mathcal{A}_N }[/math] is self-adjoint if and only if its spectrum is real, i.e. [math]\displaystyle{ \sigma(a) \subseteq \R }[/math].^[5]

Properties

In *-algebras

Let [math]\displaystyle{ \mathcal{A} }[/math] be a *-algebra. Then:

• Each element [math]\displaystyle{ a \in \mathcal{A} }[/math] can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements [math]\displaystyle{ a_1,a_2 \in \mathcal{A}_{sa} }[/math], so that [math]\displaystyle{ a = a_1 + \mathrm{i} a_2 }[/math] holds. Where [math]\displaystyle{ a_1 = \frac{1}{2} (a + a^*) }[/math] and [math]\displaystyle{ a_2 = \frac{1}{2 \mathrm{i}} (a - a^*) }[/math].^[1]
• The set of self-adjoint elements [math]\displaystyle{ \mathcal{A}_{sa} }[/math] is a real linear subspace of [math]\displaystyle{ \mathcal{A} }[/math]. From the previous property, it follows that [math]\displaystyle{ \mathcal{A} }[/math] is the direct sum of two real linear subspaces, i.e. [math]\displaystyle{ \mathcal{A} = \mathcal{A}_{sa} \oplus \mathrm{i} \mathcal{A}_{sa} }[/math].^[7]
• If [math]\displaystyle{ a \in \mathcal{A}_{sa} }[/math] is self-adjoint, then [math]\displaystyle{ a }[/math] is normal.^[1]
• The *-algebra [math]\displaystyle{ \mathcal{A} }[/math] is called a hermitian *-algebra if every self-adjoint element [math]\displaystyle{ a \in \mathcal{A}_{sa} }[/math] has a real spectrum [math]\displaystyle{ \sigma(a) \subseteq \R }[/math].^[8]

In C*-algebras

Let [math]\displaystyle{ \mathcal{A} }[/math] be a C*-algebra and [math]\displaystyle{ a \in \mathcal{A}_{sa} }[/math]. Then:

• For the spectrum [math]\displaystyle{ \left\| a \right\| \in \sigma(a) }[/math] or [math]\displaystyle{ -\left\| a \right\| \in \sigma(a) }[/math] holds, since [math]\displaystyle{ \sigma(a) }[/math] is real and [math]\displaystyle{ r(a) = \left\| a \right\| }[/math] holds for the spectral radius, because [math]\displaystyle{ a }[/math] is normal.^[9]
• According to the continuous functional calculus, there exist uniquely determined positive elements [math]\displaystyle{ a_+,a_- \in \mathcal{A}_+ }[/math], such that [math]\displaystyle{ a = a_+ - a_- }[/math] with [math]\displaystyle{ a_+ a_- = a_- a_+ = 0 }[/math]. For the norm, [math]\displaystyle{ \left\| a \right\| = \max(\left\|a_+\right\|,\left\|a_-\right\|) }[/math] holds.^[10] The elements [math]\displaystyle{ a_+ }[/math] and [math]\displaystyle{ a_- }[/math] are also referred to as the positive and negative parts. In addition, [math]\displaystyle{ |a| = a_+ + a_- }[/math] holds for the absolute value defined for every element [math]\displaystyle{ |a| = (a^* a)^\frac{1}{2} }[/math].^[11]
• For every [math]\displaystyle{ a \in \mathcal{A}_+ }[/math] and odd [math]\displaystyle{ n \in \mathbb{N} }[/math], there exists a uniquely determined [math]\displaystyle{ b \in \mathcal{A}_+ }[/math] that satisfies [math]\displaystyle{ b^n = a }[/math], i.e. a unique [math]\displaystyle{ n }[/math]-th root, as can be shown with the continuous functional calculus.^[12]

See also

• Self-adjoint matrix
• Self-adjoint operator

Notes

References

• Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. pp. 63. ISBN 3-540-28486-9. 
• Dixmier, Jacques (1977). C*-algebras. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1.  English translation of Dixmier, Jacques (1969) (in fr). Les C*-algèbres et leurs représentations. Gauthier-Villars. 
• 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. (1994). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras.. Cambridge university press. ISBN 0-521-36638-0. 

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