  In this paper we show that certain simple locally recursive subhomogeneous
(RSH) C*-algebras are tracially approximately interval algebras after tensoring
with the universal UHF algebra. This involves a linear algebraic encoding of
the structure of the local RSH algebra allowing us to find a path through the
algebra which looks like a discrete version of [0,1] and exhausts most of the
algebra. We produce an actual copy of the interval and use properties of
C*-algebras tensored with UHF algebras to move the honest interval underneath
the discrete version. It follows from our main result that such C*-algebras are
classifiable by Elliott invariants. Our theorem requires finitely many tracial
states that all induce the same state on the K_0-group; in particular we do not
require that projections separate tracial states. We apply our results to
classify some examples of C*-algebras constructed by Elliott to exhaust the
invariant. We also give an alternate way to classify examples of Lin and Matui
of C*-algebras of minimal dynamical systems. In this way our result can be
viewed as a first step towards removing the requirement that projections
separate tracial states in the classification theorem for C*-algebras of
minimal dynamical systems given by Toms and the second named author.
