Distance between unitary orbits of normal elements in simple C⁎C⁎-algebras of real rank zero

Let x,yx,y be two normal elements in a unital simple C⁎C⁎-algebra A  . We introduce a function Dc(x,y)Dc(x,y) and show that in a unital simple AF-algebra there is a constant 1>C>01>C>0 such thatC⋅Dc(x,y)≤dist(U(x),U(y))≤Dc(x,y),C⋅Dc(x,y)≤dist(U(x),U(y))≤Dc(x,y), where U(x)U(x) and U(y)U(y) are the closures of the unitary orbits of x and of y  , respectively. We also generalize this to unital simple C⁎C⁎-algebras with real rank zero, stable rank one and weakly unperforated K0K0-group. More complicated estimates are given in the presence of non-trivial K1K1-information.