Russo–Dye theorem explained

In mathematics, the Russo–Dye theorem is a result in the field of functional analysis. It states that in a unital C*-algebra, the closure of the convex hull of the unitary elements is the closed unit ball.^[1] The theorem was published by B. Russo and H. A. Dye in 1966.^[2]

Other formulations and generalizations

Results similar to the Russo–Dye theorem hold in more general contexts. For example, in a unital *-Banach algebra, the closed unit ball is contained in the closed convex hull of the unitary elements.

A more precise result is true for the C*-algebra of all bounded linear operators on a Hilbert space: If T is such an operator and ||T|| < 1 - 2/n for some integer n > 2, then T is the mean of n unitary operators.^[3]

Applications

This example is due to Russo & Dye, Corollary 1: If U(A) denotes the unitary elements of a C*-algebra A, then the norm of a linear mapping f from A to a normed linear space B is

In other words, the norm of an operator can be calculated using only the unitary elements of the algebra.

Further reading

• An especially simple proof of the theorem is given in: Gardner . L. T. . 1984 . An elementary proof of the Russo–Dye theorem . 2044692 . Proceedings of the American Mathematical Society . 90 . 1. 171 . 10.2307/2044692 .

Notes

1. Book: Doran, Robert S. . Victor A. Belfi . Characterizations of C*-Algebras: The Gelfand–Naimark Theorems . Marcel Dekker . New York . 1986 . 0-8247-7569-4 .
2. Russo . B. . H. A. Dye . 1966. A Note on Unitary Operators in C*-Algebras . Duke Mathematical Journal . 33 . 413–416 . 10.1215/S0012-7094-66-03346-1 . 2 .
3. Book: Pedersen, Gert K. . Analysis Now . Springer-Verlag . Berlin . 1989 . 0-387-96788-5 .