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

Definition

is called self-adjoint if

The set of self-adjoint elements is referred to as

A subset that is closed under the involution, is called

is a complete normed *-algebra), which is called a C*-algebra.

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

• are self-adjoint, since * is an

• of a *-algebra, the real and imaginary parts denotes the

• is a normal element, then for every real-valued function, which is continuous on the spectrum, the continuous functional calculus defines a self-adjoint element

Criteria

be a *-algebra. Then:

• is also

• be the product of two self-adjoint elements commutate always

• is self-adjoint if and only if its spectrum is real, i.e.

Properties

In *-algebras

be a *-algebra. Then:

• and

• is a real linear subspace of is the direct sum of two real linear subspaces, i.e.

• is

• has a real spectrum

In C*-algebras

. Then:

• holds for the spectral radius is

• with holds. are also referred to as the positive and negative parts holds for the absolute value defined for every element

• -th root, as can be shown with the continuous functional

See also

• Self-adjoint matrix

• Self-adjoint operator

Notes

References

• English translation of