Russo–Dye theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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]:44 The theorem was published by B. Russo and H. A. Dye in 1966.[2]

Other formulations and generalizations[edit]

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.[1]:73

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]:98

Applications[edit]

This example is due to Russo & Dye,[2] 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

\sup_{U \in U(A)} ||f(U)||.

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

Further reading[edit]

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

Notes[edit]

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