On invariant MASAs for endomorphisms of the Cuntz algebras

The problem of existence of standard (i.e. product-type) invariant MASAs for endomorphisms of the Cuntz algebras O_n is studied. In particular, endomorphisms which preserve the canonical diagonal MASA D_n are investigated. Conditions on a unitary w equivalent to the fact that the corresponding endomorphism λ_w preserves D_n are found, and it is shown that they may be satisfied by unitaries which do not normalize D_n. Unitaries giving rise to endomorphisms which leave all standard MASAs invariant and have identical actions on them are characterized. Finally, some properties of examples of finite-index endomorphisms of O_n given by Izumi and related to sector theory are discussed and it is shown that they lead to an endomorphism of O_2 associated to a matrix unitary which does not preserve any standard MASA.