Distance Between Unitary Orbits of Self-AdjointElements in C*-Algebras of Tracial Rank One

The note studies certain distance between unitary orbits. A result about Riesz interpolation property is proved in the first place. Weyl (1912) shows that dist(U(x), U(y)) = δ(x, y) for self-adjoint elements in matrixes. The author generalizes the result to C^*-algebras of tracial rank one. It is proved that dist(U(x), U(y)) = D_c(x, y) in unital AT -algebras and in unital simple C^*-algebras of tracial rank one, where x, y are self-adjoint elements and D_c (x, y) is a notion generalized from δ(x, y).