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.^[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.