Unitary element

In mathematics, an element of a *-algebra is called unitary if it is invertible and its inverse element is the same as its adjoint element.

Definition

Let be a *-algebra with unit An element is called unitary if In other words, if is invertible and holds, then is unitary.

The set of unitary elements is denoted by or

A special case from particular importance is the case where is a complete normed *-algebra. This algebra satisfies the C*-identity () and is called a C*-algebra.

Criteria

Let be a unital C*-algebra and a normal element. Then, is unitary if the spectrum consists only of elements of the circle group , i.e.

Examples

The unit is unitary.

Let be a unital C*-algebra, then:

Every projection, i.e. every element with , is unitary. For the spectrum of a projection consists of at most and , as follows from the
If is a normal element of a C*-algebra , then for every continuous function on the spectrum the continuous functional calculus defines an unitary element , if

Properties

Let be a unital *-algebra and Then:

The element is unitary, since In particular, forms a
The element is normal.
The adjoint element is also unitary, since holds for the involution
If is a C*-algebra, has norm 1, i.e.

Notes

References

English translation of