One of the most fundamental questions one can ask in an operator algebra is: when are two operators unitarily equivalent? The present talk addresses the related problem of computing the distance d_U between unitary orbits in terms of spectral information. In 2008,  Elliott and Ciuperca defined two distances for any C*-algebra A that has stable rank one: one is d_U, which is defined between approximate unitary equivalence classes of *-homomorphisms from C_0(0,1] to A, and the other is d_W, defined on the semigroup of morphisms between the Cuntz semigroups, from Cu(C_0(0,1]) to Cu(A). (The distance d_W is based on the work of Weyl.) The above distances can also be regarded as  distances (pseudo-metrics) between positive elements. In this talk, we will show that the two distances are equal for C*-algebras with stable rank one and real rank zero.

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