Russo–Dye theorem

The set of Russo - Dye, named after Bernard Russo ( born 1939 ) and Henry Abel Dye, is a set of the mathematical theory of C *-algebras from the year 1966. Evidence could be greatly simplified by Laurence Gardner Terrell. In a C * - algebra with identity, there are many unitary elements, in so many, that the conclusion of their convex hull already fills the entire unit sphere.

Wording of the sentence

In a C * - algebra with unit element the unit sphere is equal to the norm completion of the convex hull of the unitary elements.

Comments

A unitary element of a C * - algebra with unit element 1 is an element with. From the defining C * follows property so that unitary elements lie on the boundary of the unit sphere. Therefore, the standard completion of the convex hull of the unitary elements is safely contained in the unit sphere; the reverse inclusion is the non-trivial part of the theorem of Russo - Dye. The given below in the textbook by KR Davidson reproduced proof even shows that the convex hull of unitary elements (no degree education) includes the interior of the unit sphere, which is a stronger statement.

In any complex Banach algebra with unit element 1 and this one element is an extreme point of the unit sphere. Since multiplication by a unitary element is an isometric isomorphism, all elements are unitary extremum of the unit sphere. Therefore, it immediately follows the weaker statement

It is true the statement of the theorem of Krein - Milman, but note that the unit sphere is non-standard compact in an infinite-dimensional C * - algebra, that is, the set of Russo - Dye is not a consequence of the Krein - Milman.

In addition to the unitary elements, there are other extreme points. One can show that the extreme points of a C *-algebra with unit element 1 if the partial isometries are with. Thus the statement of the theorem of Russo - Dye is even stronger than that of the set of Krein - Milman, because you do not need all the extreme points, the unitary elements are sufficient. It even comes with even less as TW Palmer has shown in 1968:

The continuous functional calculus shows that the elements of the mold are unitary, and there are examples of unitary elements which are not of this form.