Reflexive Subalgebras of AF Algebras

A subalgebra T of a von Neumann algebra is reflexive if T = Alg Lat 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, we consider another natural notion of reflexivity which does not rely on projections. In this setting Ref T, the reflexive algebra generated by T, is contained in Alg Lat T, and usually the containment is proper. Necessary and sufficient conditions are given for these algebras to coincide. In particular, if Lat T is a maximal nest then Ref T = Alg Lat T. We show 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.