Self-adjoint

In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. ).

Definition

Let be a *-algebra. An element is called self-adjoint if

The set of self-adjoint elements is referred to as

A subset that is closed under the involution *, i.e. , is called

A special case of particular importance is the case where is a complete normed *-algebra, that satisfies the C*-identity (), 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 , or 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 of a *-algebra, the elements and are self-adjoint, since * is an
For each element of a *-algebra, the real and imaginary parts and are self-adjoint, where denotes the
If is a normal element of a C*-algebra , then for every real-valued function , which is continuous on the spectrum of , the continuous functional calculus defines a self-adjoint element

Criteria

Let be a *-algebra. Then:

Let , then is self-adjoint, since . A similarly calculation yields that is also
Let be the product of two self-adjoint elements Then is self-adjoint if and commutate, since always
If is a C*-algebra, then a normal element is self-adjoint if and only if its spectrum is real, i.e.

Properties

In *-algebras

Let be a *-algebra. Then:

Each element can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements , so that holds. Where and
The set of self-adjoint elements is a real linear subspace of From the previous property, it follows that is the direct sum of two real linear subspaces, i.e.
If is self-adjoint, then is
The *-algebra is called a hermitian *-algebra if every self-adjoint element has a real spectrum

In C*-algebras

Let be a C*-algebra and . Then:

For the spectrum or holds, since is real and holds for the spectral radius, because is
According to the continuous functional calculus, there exist uniquely determined positive elements , such that with For the norm, holds. The elements and are also referred to as the positive and negative parts. In addition, holds for the absolute value defined for every element
For every and odd , there exists a uniquely determined that satisfies , i.e. a unique -th root, as can be shown with the continuous functional

Notes

References

English translation of