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
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
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
\( {\displaystyle \sup _{U\in U(A)}||f(U)||.} \)
In other words, the norm of an operator can be calculated using only the unitary elements of the algebra.
Further reading
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
Doran, Robert S.; Victor A. Belfi (1986). Characterizations of C*-Algebras: The Gelfand–Naimark Theorems. New York: Marcel Dekker. ISBN 0-8247-7569-4.
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.
Pedersen, Gert K. (1989). Analysis Now. Berlin: Springer-Verlag. ISBN 0-387-96788-5.
vte
Functional analysis (topics – glossary)
Spaces
Hilbert space Banach space Fréchet space topological vector space
Theorems
Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem
Operators
bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator
Algebras
Banach algebra C*-algebra spectrum of a C*-algebra operator algebra group algebra of a locally compact group von Neumann algebra
Open problems
invariant subspace problem Mahler's conjecture
Applications
Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis
Advanced topics
locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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