625 research outputs found
Labeled Trees and Localized Automorphisms of the Cuntz Algebras
We initiate a detailed and systematic study of automorphisms of the Cuntz
algebras \O_n which preserve both the diagonal and the core -subalgebra.
A general criterion of invertibility of endomorphisms yielding such
automorphisms is given. Combinatorial investigations of endomorphisms related
to permutation matrices are presented. Key objects entering this analysis are
labeled rooted trees equipped with additional data. Our analysis provides
insight into the structure of {\rm Aut}(\O_n) and leads to numerous new
examples. In particular, we completely classify all such automorphisms of
for the permutation unitaries in . We show that
the subgroup of {\rm Out}(\O_2) generated by these automorphisms contains a
copy of the infinite dihedral group .Comment: 35 pages, slight changes, to appear on Trans. Amer. Math. So
Noncommutative Balls and Mirror Quantum Spheres
Noncommutative analogues of n-dimensional balls are defined by repeated
application of the quantum double suspension to the classical low-dimensional
spaces. In the `even-dimensional' case they correspond to the Twisted Canonical
Commutation Relations of Pusz and Woronowicz. Then quantum spheres are
constructed as double manifolds of noncommutative balls. Both C*-algebras and
polynomial algebras of the objects in question are defined and analyzed, and
their relations with previously known examples are presented. Our construction
generalizes that of Hajac, Matthes and Szymanski for `dimension 2', and leads
to a new class of quantum spheres (already on the C*-algebra level) in all
`even-dimensions'.Comment: 20 page
Endomorphisms of O_n which preserve the canonical UHF-subalgebra
Unital endomorphisms of the Cuntz algebra O_n which preserve the canonical
UHF-subalgebra F_n of O_n are investigated. We give examples of such
endomorphisms for which the associated unitary element in O_n does not belong
to F_n. One such example, in the case where n=2, arises from a construction of
a unital endomorphism on O_2 which preserves the canonical UHF-subalgebra and
where the relative commutant of the image in O_2 contains a copy of O_2.Comment: 1 figure, requires prepictex.tex, pictex.tex, postpictex.te
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.Comment: 22 page
On conjugacy of maximal abelian subalgebras and the outer automorphism group of the Cuntz algebra
We investigate the structure of the outer automorphism group of the Cuntz algebra
and the closely related problem of conjugacy of maximal abelian subalgebras in On.
In particular, we exhibit an uncountable family of maximal abelian subalgebras,
conjugate to the standard maximal abelian subalgebra Dn via Bogolubov
automorphisms, that are not inner conjugate to Dn
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