The Jacobson radical of triangular AF algebras

We give several necessary and sufficient conditions for a triangular AF algebra to be semisimple. In particular, a triangular AF algebra which can be written using the standard embedding infinitely often is semisimple; we also give a semisimple triangular AF algebra which does not have a presentation of this form. If two triangular AF algebras have the same Peters-Poon-Wagner diagonal invariant, then either both are semisimple or both are not. However, we give two algebras with the same diagonal invariant where one has Jacobson radical equal to the strong radical and the other does not. Semisimplicity can be characterized in terms of Power's fundamental relation. We give generalizations of these results to triangular subalgebras of groupoid C...