[For the entire collection see Zbl 0699.00028.]
Let \({\mathfrak A}\) be a unital AF algebra and \({\mathcal D}\) a masa of the Stratila-Voiculescu type. A \({\mathcal D}\)-bimodule T is a vector subspace of A containing \({\mathcal D}\) such that \({\mathcal D}{\mathcal T}\subset {\mathcal T}\) and \({\mathcal T}{\mathcal D}\subset {\mathcal T}\). \({\mathcal T}\) is a triangular AF algebra with diagonal \({\mathcal D}\) if \({\mathcal T}\) is a \({\mathcal D}\)-bimodule which is also a Banach subalgebra of \({\mathfrak A}\) such that \({\mathcal T}\cap {\mathcal T}^*={\mathcal D}.\)
The authors give a survey of some results and examples concerning TAF algebras. Some proofs are included, but full results appear in [J. Peters, Y. Poon, and B. Wagner, Triangular AF algebras, J. Operator Theory, to appear]. Sample results:
Proposition 8: Let \({\mathcal S}\), \({\mathcal T}\) be TAF algebras with diagonals \({\mathcal D}\) and \({\mathcal E}\), respectively. If \(\phi: {\mathcal P}({\mathcal D})\to {\mathcal P}({\mathcal E})\) is an order isomorphism, then \(\phi\) (Lat \({\mathcal S})=Lat {\mathcal T}.\)
Example 9 shows that the converse of Proposition 8 is not necessarily true.
Theorem 12: Let \({\mathfrak A}\), \({\mathfrak B}\) be AF algebras and \({\mathcal S}\subset {\mathfrak A}\), \({\mathcal T}\subset {\mathfrak B}\) be strongly maximal TAF algebras with diagonals \({\mathcal D}\), \({\mathcal E}\) respectively. If \(\Phi: {\mathcal S}\to {\mathcal T}\) is an isometric algebra isomorphism, then there is an extension \({\hat \Phi}\) of \(\Phi\).