There is extensive literature concerning approximately finite-dimensional (AF) C*-algebras. For example see [1, 5]. In recent years study has focused on non-self-adjoint subalgebras of AF C*-algebras ([4, 8, 9], for example) and this paper continues that theme. We define T(m, n) to be the block upper triangular subalgebra of Mm+n with self-adjoint part equal to Mm [oplus ] Mn. The class of algebras that are considered here are norm closures of ascending chains of algebras of the form T(m, n) and are referred to as uniformly T2 algebras. This class of algebras properly contains the matroid C*-algebras of Dixmier.