198 research outputs found
Extensions by simple -algebras -- Quasidiagonal extensions
Let be an amenable separable \CA and be a non-unital but
-unital simple \CA with continuous scale. We show that two essential
extensions and of by are approximately unitarily
equivalent if and only if If
is assumed to satisfy the Universal Coefficient Theorem, there is a
bijection from approximate unitary equivalence classes of the above mentioned
extensions to Using we compute exactly when
an essential extension is quasidiagonal. We show that quasidiagonal extensions
may not be approximately trivial. We also study the approximately trivial
extensions.Comment: to appear Canad J. Mat
Localizing the Elliott Conjecture at Strongly Self-absorbing C*-algebras --An Appendix
This note provides some technical support to the proof of a result of W.
Winter which shows that two unital separable simple amenable -absorbing C*-algebras with locally finite decomposition property satisfying
the UCT whose projections separate the traces are isomorphic if their
-theory is finitely generated and their Elliott invariants are the same
The Range of Approximate Unitary Equivalence Classes of Homomorphisms from AH-algebras
Let be a unital AH-algebra and be a unital simple C*-algebra with
tracial rank zero. It has been shown that two unital monomorphisms are approximately unitarily equivalent if and only if [\phi]=[\psi]
{\rm in} KL(C,A) and \tau\circ \phi=\tau\circ \psi \tforal \tau\in T(A),
where is the tracial state space of In this paper we prove the
following: Given with and with and a
continuous affine map which is compatible
with where is the convex set of all faithful
tracial states, there exists a unital monomorphism such that
[\phi]=\kappa\andeqn \tau\circ \phi(c)=\lambda(\tau)(c) for all and Denote by the set of
approximate unitary equivalence classes of unital monomorphisms. We provide a
bijective map where
is the set of compatible pairs of elements in
and continuous affine maps from to Moreover, we
realized that there are compact metric spaces , unital simple AF-algebras
and with for which there is no \hm so that
$[h]=\kappa.
Full extensions and approximate unitary equivalences
Let be a unital separable amenable \CA and be a unital \CA with
certain infinite property. We show that two full monomorphisms are approximately unitarily equivalent if and only if in
Let be a non-unital but -unital \CA for which
has the certain infinite property. We prove that two full essential extensions
are approximately unitarily equivalent if and only if they induce the same
element in The set of approximately unitarily equivalence
classes of full essential extensions forms a group. If satisfies the
Universal Coefficient Theorem, it is can be identified with $KL(A, M(B)/B).
Classification of simple C*-algebras and higher dimensional noncommutative tori
We show that unital simple C*-algebras with tracial topological rank zero
which are locally approximated by subhomogeneous C^-algebras can be classified
by their ordered -theory. We apply this classification result to show that
certain simple crossed products are isomorphic if they have the same ordered
-theory. In particular, irrational higher dimensional non-commutative tori
of the form are in fact inductive
limits of circle algebras.Comment: 24 pages published versio
Furstenberg Transformations and Approximate Conjugacy
Let and be two Furstenberg transformations on 2-torus
associated with irrational numbers integers
and Lipschitz functions and We show that and are
approximately conjugate in a measure theoretical sense if (and only if)
in Closely related to the
classification of simple amenable -algebras, we show that and
are approximately -conjugate if (and only if) in and This is also shown to be equivalent
to that the associated crossed product -algebras are isomorphic
AF-embedding of the crossed products of AH-algebras by finitely generated abelian groups
Let be a compact metric space and let be a ()
action on We give a solution to a version of Voiculescu's problem of
AF-embedding: The crossed product can be embedded
into a unital simple AF-algebra if and only if admits a strictly positive
-invariant Borel probability measure.
Let be a unital AH-algebra, let be a finitely generated abelian group
and let be a monomorphism. We show that
can be embedded into a unital simple AF-algebra if and
only if admits a faithful -invariant tracial state.Comment: 46 page
Unitaries in a Simple C*-algebra of Tracial Rank One
Let be a unital separable simple infinite dimensional \CA with tracial
rank no more than one and with the tracial state space and let be
the unitary group of Suppose that the connected component
of containing the identity.
We show that, for any \ep>0, there exists a selfadjoint element such that \|u-\exp(ih)\|<\ep. We also study the problem when
can be approximated by unitaries in with finite spectrum.
Denote by the closure of the subgroup of unitary group of
generated by its commutators. It is known that Denote by
the affine function on defined by
We show that can be approximated by unitaries
in with finite spectrum if and only if and
for all Examples are given that there are unitaries in which
can not be approximated by unitaries with finite spectrum. Significantly these
results are obtained in the absence of amenability
Kishimoto's Conjugacy Theorems in simple -algebras of tracial rank one
Let be a unital separable simple amenable -algebra with finite
tracial rank which satisfies the Universal Coefficient Theorem (UCT). Suppose
\af and \bt are two automorphisms with the Rokhlin property that {induce
the same action on the -theoretical data of .} We show that \af and
\bt are strongly cocycle conjugate and uniformly approximately conjugate,
that is, there exists a sequence of unitaries and a sequence
of strongly asymptotically inner automorphisms such that \af={\rm
Ad}\, u_n\circ \sigma_n\circ \bt\circ \sigma_n^{-1}\andeqn
\lim_{n\to\infty}\|u_n-1\|=0, and that the converse holds. {We then give a
-theoretic description as to exactly when \af and \bt are cocycle
conjugate, at least under a mild restriction. Moreover, we show that given any
-theoretical data, there exists an automorphism \af with the Rokhlin
property which has the same -theoretical data
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