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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 algebra O_n is studied. In particular endomorphisms which preserve the canonical diagonal MASA D_n are investigated. Conditions on a unitary in O_n equivalent to the fact that the corresponding endomorphism 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.

preprint2010arXivOpen access
Jeong Hee HongAdam SkalskiWojciech Szymanski
math.OAmath.DSmath.FA
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Jeong Hee HongAdam SkalskiWojciech Szymanski

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