Self-adjoint

Template:Short description 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 Template:Nowrap

The set of self-adjoint elements is referred to as Template:Nowrap

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 Template:Nowrap

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 Template:Nowrap 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 Template:Nowrap
• 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 Template:Nowrap
• 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 Template:Nowrap
• 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 Template:Nowrap

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 Template:Nowrap
• Let <math>a = a_1 a_2</math> be the product of two self-adjoint elements Template:Nowrap 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 Template:Nowrap
• 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. Template:Nowrap

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 Template:Nowrap (a - a^*)</math>.Template:Sfn}}
• The set of self-adjoint elements <math>\mathcal{A}_{sa}</math> is a real linear subspace of Template:Nowrap From the previous property, it follows that <math>\mathcal{A}</math> is the direct sum of two real linear subspaces, i.e. Template:Nowrap
• If <math>a \in \mathcal{A}_{sa}</math> is self-adjoint, then <math>a</math> is Template:Nowrap
• 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 Template:Nowrap

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 Template:Nowrap
• 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 Template:Nowrap For the norm, <math>\left\| a \right\| = \max(\left\|a_+\right\|,\left\|a_-\right\|)</math> holds.Template:Sfn 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 Template:Nowrap
• 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 Template:Nowrap

See also

• Self-adjoint matrix
• Self-adjoint operator

References

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