Russo–Dye theorem
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.
{{cite book | last = Doran | first = Robert S. |author2=Victor A. Belfi | title = Characterizations of C*-Algebras: The Gelfand–Naimark Theorems | publisher = Marcel Dekker | location = New York | year = 1986 | isbn = 0-8247-7569-4 }}
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The theorem was published by B. Russo and H. A. Dye in 1966.
{{cite journal |last=Russo |first=B. |author2=H. A. Dye |year=1966|title=A Note on Unitary Operators in C*-Algebras |journal=Duke Mathematical Journal |volume=33 |pages=413–416 |doi=10.1215/S0012-7094-66-03346-1 |issue=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.{{Rp|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.
{{cite book | last = Pedersen | first = Gert K. | title = Analysis Now | publisher = Springer-Verlag | location = Berlin | year = 1989 | isbn = 0-387-96788-5 }}
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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: {{cite journal |last=Gardner |first=L. T. |year=1984 |title=An elementary proof of the Russo–Dye theorem |jstor=2044692 |journal=Proceedings of the American Mathematical Society |volume=90 |issue=1|pages=171 |doi=10.2307/2044692 }}