Self-adjoint explained

In mathematics, an element of a

• -algebra

is called self-adjoint if it is the same as its adjoint (i.e.

a = a ^{*} ).

Definition

Let

l{A} be a *-algebra. An element a \in l{A} is called self-adjoint if

The set of self-adjoint elements is referred to as

l{B} \subseteq l{A} that is closed under the involution *, i.e. l{B} = l{B} ^{*} , is called

A special case of particular importance is the case where

l{A} is a complete normed *-algebra, that satisfies the C*-identity (\left\| a ^{*a} \right\| = \left\| a \right\| ^{2} \forall a \in l{A} ), which is called a C*-algebra.

Especially in the older literature on *-algebras and C*-algebras, such elements are often called Because of that the notations

l{A} _{h} , l{A} _{H} or H( l{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
• For each element

a of a *-algebra, the elements aa ^{*} and a ^{*a} are self-adjoint, since * is an

• For each element

a of a *-algebra, the real and imaginary parts $\backslash operatorname(a)\; =\; \backslash frac\; (a+a^*)$ and $\backslash operatorname(a)\; =\; \backslash frac\; (a-a^*)$ are self-adjoint, where i denotes the

• If

a \in l{A} _{N} is a normal element of a C*-algebra l{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

Criteria

Let

l{A} be a *-algebra. Then:

• Let

a \in l{A} , then a ^{*a} is self-adjoint, since (a ^{*a)} ^{*} = a ^{*(a} ^{*)} ^{*} = a ^{*a} . A similarly calculation yields that aa ^{*} is also

• Let

a = a _{1} a _{2} be the product of two self-adjoint elements 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

• If

l{A} is a C*-algebra, then a normal element a \in l{A} _{N} is self-adjoint if and only if its spectrum is real, i.e.

Properties

In *-algebras

Let

l{A} be a *-algebra. Then:

• Each element

a \in l{A} can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements a _{1,a} _{2} \in l{A} _{sa} , so that a = a _{1} + i a _{2} holds. Where $a\_1\; =\; \backslash frac\; (a\; +\; a^*)$ and

• The set of self-adjoint elements

l{A} _{sa} is a real linear subspace of From the previous property, it follows that l{A} is the direct sum of two real linear subspaces, i.e.

• If

a \in l{A} _{sa} is self-adjoint, then a is

• The *-algebra

l{A} is called a hermitian *-algebra if every self-adjoint element a \in l{A} _{sa} has a real spectrum

In C*-algebras

Let

l{A} be a C*-algebra and a \in l{A} _{sa} . Then:

• For the spectrum

\left\| a \right\| \in \sigma(a) or - \left\| a \right\| \in \sigma(a) holds, since \sigma(a) is real and r(a) = \left\| a \right\| holds for the spectral radius, because a is

• According to the continuous functional calculus, there exist uniquely determined positive elements

a _{+,a} _{-} \in l{A} _{+} , such that a = a _{+} - a _{-} with For the norm, \left\| a \right\| = max (\left\|a _{+\right\|,\left\|a} _{-\right\|)} holds. 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

• For every

a \in l{A} _{+} and odd n \in N , there exists a uniquely determined b \in l{A} _{+} that satisfies b ^{n} = a , i.e. a unique n -th root, as can be shown with the continuous functional

See also

• Self-adjoint matrix
• Self-adjoint operator

References

• Book: Blackadar, Bruce . Operator Algebras. Theory of C*-Algebras and von Neumann Algebras . Springer . Berlin/Heidelberg . 2006 . 3-540-28486-9 . 63 .
• Book: Dixmier, Jacques . C*-algebras . North-Holland . Amsterdam/New York/Oxford . 1977 . 0-7204-0762-1 . Jellett . Francis . English translation of Book: Dixmier, Jacques . 0 . Les C*-algèbres et leurs représentations . fr . Gauthier-Villars . 1969 .
• Book: Kadison . Richard V. . Ringrose . John R. . Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. . Academic Press . New York/London . 1983 . 0-12-393301-3.
• Book: Palmer, Theodore W. . Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. . Cambridge university press . 2001 . 0-521-36638-0 .