On Sums of Projections in C*-Algebras

For an arbitrary unital C*-algebra A, we consider the problem of describing all λ∈ℝ such that the element λ1A can be represented as the sum of n self-adjoint projections. We obtain a description of the set of such λ for the class of uniformly hyperfinite algebras and for C*-algebras of type I. An example of a C*-algebra for which this set is not closed is constructed. We also study some properties of the universal C*-algebra generated by n projections summing to λ.