2,315 research outputs found
Real rank and property (SP) for direct limits of recursive subhomogeneous algebras
Let A be a unital simple direct limit of recursive subhomogeneous C*-algebras
with no dimension growth. We give criteria which specify exactly when A has
real rank zero, and exactly when A has the Property (SP): every nonzero
hereditary subalgebra of A contains a nonzero projection. Specifically, A has
real rank zero if and only if the natural map from K_0 (A) to the continuous
affine functions on the tracial state space has dense range, A has the Property
(SP) if and only if the range of this map contains strictly positive functions
with arbitrarily small norm. By comparison with results for unital simple
direct limit of homogeneous C*-algebras with no dimension growth, one might
hope that weaker conditions might suffice. We give examples to show that
several plausible weaker conditions do not suffice for the results above.
If A has real rank zero and at most countably many extreme tracial states, we
apply results of H. Lin to show that A has tracial rank zero and is
classifiable.Comment: 23 pages, AMSLaTe
Crossed products by finite cyclic group actions with the tracial Rokhlin property
We define the tracial Rokhlin property for actions of finite cyclic groups on
stably finite simple unital C*-algebras. We prove that when the algebra is in
addition simple and has tracial rank zero, then the crossed product again has
tracial rank zero. Under a kind of weak approximate innerness assumption and
one other technical condition, we prove that if the action has the the tracial
Rokhlin property and the crossed product has tracial rank zero, then the
original algebra has tracial rank zero. We give examples showing how the
tracial Rokhlin property differs from the Rokhlin property of Izumi.
We use these results, together with work of Elliott-Evans and Kishimoto, to
give an inductive proof that every simple higher dimensional noncommutative
torus is an AT algebra. We further prove that the crossed product of every
simple higher dimensional noncommutative torus by the flip is an AF algebra,
and that the crossed products of irrational rotation algebras by the standard
actions of the cyclic groups of orders 3, 4, and 6 are simple AH algebras with
real rank zero.Comment: 90 pages, AMSLaTe
The tracial Rokhlin property is generic
We prove several results of the following general form: automorphisms of (or
actions of on) certain kinds of simple separable unital
C*-algebras which have a suitable version of the Rokhlin property are
generic among all automorphisms (or actions), or in a suitable class of
automorphisms. That is, the ones with the version of the Rokhlin property
contain a dense -subset of the set of all such automorphisms (or
actions).
Specifically, we prove the following. If is stable under tensoring with
the Jiang-Su algebra and has tracial rank zero, then automorphisms with
the tracial Rokhlin property are generic. If has tracial rank zero, or,
more generally, is tracially approximately divisible together with a
technical condition, then automorphisms with the tracial Rokhlin property are
generic among the approximately inner automorphisms. If is stable under
tensoring with the Cuntz algebra or with a UHF algebra
of infinite type, then actions of on with the Rokhlin
property are generic among all actions of We further give a
related but more restricted result for actions of finite groups.Comment: AMSLaTeX, 33 page
Large subalgebras
We define and study large and stably large subalgebras of simple unital
C*-algebras. The basic example is the orbit breaking subalgebra of a crossed
product by Z, as follows. Let X be an infinite compact metric space, let h be a
minimal homeomorphism of X, and let Y be a closed subset of X. Let u be the
standard unitary in C* (Z, X, h). The Y-orbit breaking subalgebra is the
subalgebra of C* (Z, X, h) generated by C (X) and all elements f u for f in C
(X) such that f vanishes on Y. If intersects each orbit of h at most once, then
the Y-orbit breaking subalgebra is large in C* (Z, X, h). Large subalgebras
obtained via generalizations of this construction have appeared in a number of
places, and we unify their theory in this paper.
We prove the following results for an infinite dimensional simple unital
C*-algebra A and a stably large subalgebra B of A:
B is simple and infinite dimensional.
If B is stably finite then so is A, and if B is purely infinite then so is A.
The restriction maps from the tracial states of A to the tracial states of B
and from the normalized 2-quasitraces on A to the normalized 2-quasitraces on B
are bijective.
When A is stably finite, the inclusion of B in A induces an isomorphism on
the semigroups that remain after deleting from the Cuntz semigroups of A and B
all the classes of nonzero projections.
B and A have the same radius of comparison.Comment: 54 pages; AMSLaTe
Examples of different minimal diffeomorphisms giving the same C*-algebras
We give examples of minimal diffeomorphisms of compact connected manifolds
which are not topologically orbit equivalent, but whose transformation group
C*-algebras are isomorphic. The examples show that the following properties of
a minimal diffeomorphism are not invariants of the transformation group
C*-algebra: having topologically quasidiscrete spectrum; the action on singular
cohomology (when the manifolds are diffeomorphic); the homotopy type of the
manifold (when the manifolds have the same dimension); and the dimension of the
manifold.
These examples also give examples of nonconjugate isomorphic Cartan
subalgebras, and of nonisomorphic Cartan subalgebras, of simple separable
nuclear unital C*-algebras with tracial rank zero and satisfying the Universal
Coefficient Theorem.Comment: AMSLaTeX; 21 pages, no figure
Recursive subhomogeneous algebras
We introduce and characterize a particularly tractable class of unital type 1
C*-algebras with bounded dimension of irreducible representations. Algebras in
this class are called recursive subhomogeneous algebras, and they have an
inductive description (through iterated pullbacks) which allows one to carry
over from algebras of the form C (X, M_n) many of the constructions relevant in
the study of the stable rank and K-theory of simple direct limits of
homogeneous C*-algebras. Our characterization implies in particular that if A
is a separable C*-algebra whose irreducible representations all have dimension
at most N (for some finite N), and if for each n the space of n-dimensional
irreducible representations has finite covering dimension, then A is a
recursive subhomogeneous algebra. We demonstrate the good properties of this
class by proving subprojection and cancellation theorems in it.Comment: 29 pages, AMSLaTe
Every simple higher dimensional noncommutative torus is an AT algebra
We prove that every simple higher dimensional noncommutative torus is an AT
algebra.Comment: AMSLaTeX; 22 pages. This paper replaces Sections 5 through 7 of the
unpublished long preprint arXiv:math.OA/0306410. A number of minor
improvements have been made, particularly near the en
Crossed products of the Cantor set by free minimal actions of Z^d
Let d be a positive integer, let X be the Cantor set, and let Z^d act freely
and minimally on X. We prove that the crossed product C* (Z^d, X) has stable
rank one, real rank zero, and cancellation of projections, and that the order
on K_0 (C* (Z^d, X)) is determined by traces. We obtain the same conclusion for
the C*-algebras of various kinds of aperiodic tilings.Comment: 41 pages, AMSLaTe
When are crossed products by minimal diffeomorphisms isomorphic?
This is a survey which discusses the isomorphism problem for both C* and
smooth crossed products by minimal diffeomorphisms. For C* crossed products,
examples demonstrate the failure of the obvious analog of the
Giordano-Putnam-Skau Theorem on minimal homeomorphisms of the Cantor set. For
smooth crossed products, there are many open problems.Comment: 23 pages, AMSLaTeX. To appear in the proceedings of the Conference on
Operator Algebras and Mathematical Physics, Constanta, Romania (2001
A simple separable C*-algebra not isomorphic to its opposite algebra
We give an example of a simple separable C*-algebra which is not isomorphic
to its opposite algebra. Our example is nonnuclear and stably finite, has real
rank zero and stable rank one, and has a unique tracial state. It has trivial
K_1, and its K_0-group is order isomorphic to a countable subgroup of the real
numbers.Comment: 8 pages, AMSLaTe
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