Convex Combination Of C, ~ * Algebra Of Unitary Elements

Let (?) be a C^*-algebra and let T be an element in (?) with‖T‖< 1.If T is self-adjoint,then T can be expressed as the mean of two unitaries. For every T with‖T‖<1,it can be expressed as the sum of three unitaries U₁,U₂,U₃, and sp(U₁^*U₂),sp(U₂^*U₃),sp(U₃^*U₁)(?){e^iv|0≤v≤π}. Particularly,if‖T‖< 1, U₁ can be chosen freely and U₂, U₃ are uniquely determined by the conditons relating the spectrum just stated. By strengthening this result and proving a lemma, U.Haagerup improves R.V.Kadison and G.K.Pederson's result as follows: if‖T‖≤1-2/n ,then T is the mean of n unitaries. Letα(T) denote the distance from T to (?)_inv,the group of invertibles of (?).Let u(T) be the least integer n such that T is a convex combinations of n unitaries.For eachβ>1, definewhere n∈N is given by n -1 <β≤n.DefineM.Rodam shows that if T is non-invertible andα(T) <1, then V(T) = [β, +∞) or (β, +∞), whereβ= 2(1 -α(T))^-1.In this paper,we will give some new results as follows.Murray and von Neumann expressed any self-adjoint element in the closed unit ball of a C^*-algebra (?) as the mean of two unitaries. We will give an example to show that this expression is not unique.Applying M.Rodam's result, we prove that every self-adjoint element in the closed ball of C^*-algebra (?) can be approximated by a sequence of invertibles in (?).We give a new proof of R.V.Kadison and G.K.Pederson's theorem(thm 4.1) by means of M.Rodam's result.We also partly solve the following problem: when T is not invertible,α(T) < 1,V(T) = [β, +∞) or (β, +∞)? We prove that if ||A||=α(A) = 1, T = (1-2/β)A, whereβ∈(2, 3)orβ≥3is an integer, then V(T) = [β, +∞).