Reflexive Subalgebras of AF Algebras

A subalgebra of a von Neumann algebra is reflexive if = Alg Lat . 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 , the reflexive algebra generated by , is contained in Alg Lat , and usually the containment is proper. Necessary and sufficient conditions are given for these algebras to coincide. In particular, if Lat is a maximal nest then Ref = Alg Lat . 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