1 Department of Mathematics and Computer Science (IMADA), Faculty of Science, SDU2 Faculty of Science, SDU3 Mathematics, Department of Mathematics and Computer Science (IMADA), Faculty of Science, SDU4 Korea Maritime University5 Institut of Mathematics, Polish Academy of Science6 Department of Mathematics and Computer Science (IMADA), Faculty of Science, SDU
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.
Indiana University Mathematics Journal, 2010, Vol 59, Issue 6, p. 1873-1892