Semisimple 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 the 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*-algebras. It is shown that there is a unique maximal bimodule over the diagonal which does not intersect the radical. Using this, we show that the Wedderburn principal theorem does not hold for all triangular AF algebras. A necessary condition for the theorem to hold is that the above mentioned unique bimodule is a subalgebra.