1,192 research outputs found
Endomorphisms of the Cuntz Algebras
This mainly expository article is devoted to recent advances in the study of
dynamical aspects of the Cuntz algebras O_n, with n finite, via their
automorphisms and, more generally, endomorphisms. A combinatorial description
of permutative automorphisms of O_n in terms of labeled, rooted trees is
presented. This in turn gives rise to an algebraic characterization of the
restricted Weyl group of O_n. It is shown how this group is related to certain
classical dynamical systems on the Cantor set. An identification of the image
in Out(O_n) of the restricted Weyl group with the group of automorphisms of the
full two-sided n-shift is given, for prime n, providing an answer to a question
raised by Cuntz in 1980. Furthermore, we discuss proper endomorphisms of O_n
which preserve either the canonical UHF-subalgebra or the diagonal MASA, and
present methods for constructing exotic examples of such endomorphisms.Comment: 2 figures, uses pictex, to appear in the Proceedings of the Workshop
on Noncommutative Harmonic Analysis, Bedlewo 201
Lifting endomorphisms to automorphisms
Normal endomorphisms of von Neumann algebras need not be extendable to
automorphisms of a larger von Neumann algebra, but they always have asymptotic
lifts. We describe the structure of endomorphisms and their asymptotic lifts in
some detail, and apply those results to complete the identification of
asymptotic lifts of unital completely positive linear maps on von Neumann
algebras in terms of their minimal dilations to endomorphisms.Comment: 7 pages, minor clarification of the introduction, new referenc
Noncommutative topological entropy of endomorphisms of Cuntz algebras II
A study of noncommutative topological entropy of gauge invariant
endomorphisms of Cuntz algebras began in our earlier work with Joachim
Zacharias is continued and extended to endomorphisms which are not necessarily
of permutation type. In particular it is shown that if H is an N-dimensional
Hilbert space, V is an irreducible multiplicative unitary on the tensor product
of H with itself and F is the tensor flip, then the Voiculescu entropy of the
Longo's canonical endomorphism associated with the unitary VF is equal to log
N.Comment: 8 page
A Garden of Eden theorem for Anosov diffeomorphisms on tori
Let be an Anosov diffeomorphism of the -dimensional torus
and a continuous self-mapping of
commuting with . We prove that is surjective if and only if the
restriction of to each homoclinicity class of is injective.Comment: 9 page
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