Set of Russo-Dye

The Russo-Dye theorem , named after Bernard Russo (* 1939) and Henry Abel Dye , is a theorem from the mathematical theory of C * -algebras from 1966. The proof was considerably simplified by Laurence Terrell Gardner . In a C * -algebra with one element there are very many unitary elements , so many that the end of their convex hull already fills the whole unit sphere .

Formulation of the sentence

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

Remarks

A unitary element of a C * -algebra with one element 1 is an element with . From the defining C * property it follows that unitary elements lie on the edge of the unit sphere. Therefore, the norm closure of the convex hull of the unitary elements is surely contained in the unit sphere; the reverse inclusion is the non-trivial part of Russo-Dye's theorem. The proof given in the textbook by KR Davidson given below even shows that the convex hull of the unitary elements (without formation of a closure) encompasses the interior of the unit sphere, which is a stronger statement. ${\displaystyle A}$ ${\displaystyle u\in A}$ ${\displaystyle u^{*}u=uu^{*}=1}$ ${\displaystyle \|u\|^{2}=\|u^{*}u\|=\|1\|=1}$

In every complex Banach algebra with unit 1 and this unit is an extreme point of the unit sphere. Since the multiplication by a unitary element is an isometric isomorphism , all unitary elements are also extremal points of the unit sphere. Therefore the weaker statement immediately follows ${\displaystyle \|1\|=1}$

In a C * -algebra with a unit element, the unit sphere is equal to the standard closure of the convex hull of the extremal points.

So the statement of the Kerin-Milman theorem applies , but note that the unit sphere in an infinite-dimensional C * -algebra is not norm-compact, i.e. the Russo-Dye theorem is not a consequence of the Kerin-Milman theorem.

In addition to the unitary elements, there are other extreme points. One can show that the extreme points of a C * -algebra with unit 1 are exactly the partial isometries with . Thus the statement of Russo-Dye's theorem is even stronger than that of Kerin-Milman's theorem, because one does not need all extreme points, the unitary elements are sufficient. You can get by with even less, as TW Palmer showed in 1968: ${\displaystyle A}$ ${\displaystyle v\in A}$ ${\displaystyle (1-v^{*}v)A(1-vv^{*})=0}$

In a C * -algebra with one element, the unit sphere is equal to the standard closure of the convex hull of the elements of the form , whereby all self-adjoint elements pass through. ${\displaystyle \exp(ia)}$ ${\displaystyle a}$

The continuous functional calculus shows that the elements of form are unitary, and there are examples of unitary elements that are not of this form. ${\displaystyle \exp(ia),a^{*}=a}$

Individual evidence

1. ^ B. Russo .; HA Dye, A Note on Unitary Operators in C * Algebras , Duke Mathematical Journal Volume 33.2 (1966), pages 413-416
2. ^ LT Gardner: An elementary proof of the Russo-Dye theorem , Proc. Amer. Math. Soc. Volume 90.1 (1984), page 171
3. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Theorem I.8.4
4. ^ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , Cambridge University Press (1971), ISBN 0-521-07988-8 , Chapter 1, §4, Theorem 5
5. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 7.3.1
6. ↑ TW Palmer: Characterizations of C * -algebras , Bull. Amer. Math. Soc. 74: 538-540 (1968)