An elementary proof of the Russo-Dye theorem

Abstract

PROOF. It suffices to show that if x E AI, the open unit ball of A, then x E co(U), the closed convex hull of U. It is easy to reduce this to showing that for every u E U, y = (x + u)/2 E co(U): For then U C 2 co(U) x, which is closed and convex, so co(U) C 2 co(U)-x, or (x +co(U))/2 Cco(U); if x0 E U, and x n += (x + xn)/2, xn co(U) and xn -* x. But note thaty = ((xu-1 + 1)/2)u, so (since l1xu-111 = lixii < 1) IIYl < 1, and y is invertible. Thus y = v Iy I, with v unitary and (y*y)l/2 =y I= (W + w*)/2, where W = YI + i(l IY 12)1/2 is also unitary. This concludes the proof and indeed proves the stronger result: AI U C U + U. NOTE ADDED IN PROOF. C. K. Fong has pointed out that essentially the same argument proves also that Al C co(U): Extend slightly the segment from u through x to x' E Al, and apply the given argument to x' instead of x; then x' C co(U), and for large n, x lies between x' and u.