Semisimple triangular AF algebras
Necessary and sufficient conditions for a triangular AF algebra to be semisimple are given. In particular, a triangular AF algebra which can be written using the standard embedding infinitely often is semisimple; also a semisimple triangular AF algebra is given 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, two algebras with the same diagonal invariant are given, 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. Generalizations of these results to triangular subalgebras of groupoid \(C^*\)-algebras are given. It is shown that there is a unique maximal bimodule over the diagonal which does not intersect the radical. Using this, it is shown 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.
scientific article; zbMATH DE number 152045