  We study the triangular subalgebras of UHF algebras which provide new
examples of algebras with the Dirichlet property and the Ando property. This in
turn allows us to describe the semicrossed product by an isometric
automorphism. We also study the isometric automorphism group of these algebras
and prove that it decomposes into the semidirect product of an abelian group by
a torsion free group. Various other structure results are proven as well.
