Reflexive subalgebras of AF algebras (Q1328304)

A subalgebra \(\mathcal T\) of a von Neumann algebra is reflexive if \({\mathcal T}= \text{Alg Lat }{\mathcal T}\). This notion of reflexivity is in general not well-suited for subalgebras of \(C^*\)-algebras, since a \(C^*\)-algebra may not contain enough projections. For subalgebras of simple AF algebras, the authors consider another natural notion of reflexivity which does not rely on projections. In this setting \(\text{Ref }{\mathcal T}\), the reflexive algebra generated by \(\mathcal T\), is contained in \(\text{Alg Lat }{\mathcal T}\), and usually the containment is proper. Necessary and sufficient conditions are given for these algebras to coincide. In particular, if \(\text{Lat }{\mathcal T}\) is a maximal nest then \(\text{Rel }{\mathcal T}= \text{Alg Lat }{\mathcal T}\). It is shown that analytic subalgebras must be either reflexive or transitive and that reflexive analytic algebras are trivially analytic. In addition, a strongly maximal algebra is semisimple if and only if it is transitive.