93 research outputs found
On a counterexample to a conjecture by Blackadar
Blackadar conjectured that if we have a split short-exact sequence 0 -> I ->
A -> A/I -> 0 where I is semiprojective and A/I is isomorphic to the complex
numbers, then A must be semiprojective. Eilers and Katsura have found a
counterexample to this conjecture. Presumably Blackadar asked that the
extension be split to make it more likely that semiprojectivity of I would
imply semiprojectivity of A. But oddly enough, in all the counterexamples of
Eilers and Katsura the quotient map from A to A/I is split. We will show how to
modify their examples to find a non-semiprojective C*-algebra B with a
semiprojective ideal J such that B/J is the complex numbers and the quotient
map does not split.Comment: 6 page
The ordered K-theory of a full extension
Let A be a C*-algebra with real rank zero which has the stable weak
cancellation property. Let I be an ideal of A such that I is stable and
satisfies the corona factorization property. We prove that 0->I->A->A/I->0 is a
full extension if and only if the extension is stenotic and K-lexicographic. As
an immediate application, we extend the classification result for graph
C*-algebras obtained by Tomforde and the first named author to the general
non-unital case. In combination with recent results by Katsura, Tomforde, West
and the first author, our result may also be used to give a purely
K-theoretical description of when an essential extension of two simple and
stable graph C*-algebras is again a graph C*-algebra.Comment: Version IV: No changes to the text. We only report that Theorem 4.9
is not correct as stated. See arXiv:1505.05951 for more details. Since
Theorem 4.9 is an application to the main results of the paper, the main
results of this paper are not affected by the error. Version III comments:
Some typos and errors corrected. Some references adde
Levinson's theorem and higher degree traces for Aharonov-Bohm operators
We study Levinson type theorems for the family of Aharonov-Bohm models from
different perspectives. The first one is purely analytical involving the
explicit calculation of the wave-operators and allowing to determine precisely
the various contributions to the left hand side of Levinson's theorem, namely
those due to the scattering operator, the terms at 0-energy and at infinite
energy. The second one is based on non-commutative topology revealing the
topological nature of Levinson's theorem. We then include the parameters of the
family into the topological description obtaining a new type of Levinson's
theorem, a higher degree Levinson's theorem. In this context, the Chern number
of a bundle defined by a family of projections on bound states is explicitly
computed and related to the result of a 3-trace applied on the scattering part
of the model.Comment: 33 page
Relative commutants of strongly self-absorbing C*-algebras
The relative commutant of a strongly self-absorbing
algebra is indistinguishable from its ultrapower . This
applies both to the case when is the hyperfinite II factor and to the
case when it is a strongly self-absorbing C*-algebra. In the latter case we
prove analogous results for and reduced powers
corresponding to other filters on . Examples of algebras with
approximately inner flip and approximately inner half-flip are provided,
showing the optimality of our results. We also prove that strongly
self-absorbing algebras are smoothly classifiable, unlike the algebras with
approximately inner half-flip.Comment: Some minor correction
A Simple Separable Exact C*-Algebra not Anti-isomorphic to Itself
We give an example of an exact, stably finite, simple. separable C*-algebra D
which is not isomorphic to its opposite algebra. Moreover, D has the following
additional properties. It is stably finite, approximately divisible, has real
rank zero and stable rank one, has a unique tracial state, and the order on
projections over D is determined by traces. It also absorbs the Jiang-Su
algebra Z, and in fact absorbs the 3^{\infty} UHF algebra. We can also
explicitly compute the K-theory of D, namely K_0 (D) = Z[1/3] with the standard
order, and K_1 (D) = 0, as well as the Cuntz semigroup of D.Comment: 16 pages; AMSLaTeX. The material on other possible K-groups for such
an algebra has been moved to a separate paper (1309.4142 [math.OA]
Isomorphisms between Leavitt algebras and their matrix rings
Let K be any field, let Ln denote the Leavitt algebra of type (1,n – 1) having coefficients in K, and let Md(Ln) denote the ring of d × d matrices over Ln. In our main result, we show that Md(Ln) ≅ Ln if and only if d and n – 1 are coprime. We use this isomorphism to answer a question posed in [W. Paschke and N. Salinas, Matrix algebras over , Michigan Math. J. 26 (1979), 3–12.] regarding isomorphisms between various C*-algebras. Furthermore, our result demonstrates that data about the K 0 structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple K-algebras
The topological dimension of type I C*-algebras
While there is only one natural dimension concept for separable, metric
spaces, the theory of dimension in noncommutative topology ramifies into
different important concepts. To accommodate this, we introduce the abstract
notion of a noncommutative dimension theory by proposing a natural set of
axioms. These axioms are inspired by properties of commutative dimension
theory, and they are for instance satisfied by the real and stable rank, the
decomposition rank and the nuclear dimension.
We add another theory to this list by showing that the topological dimension,
as introduced by Brown and Pedersen, is a noncommutative dimension theory of
type I C*-algebras. We also give estimates of the real and stable rank of a
type I C*-algebra in terms of its topological dimension.Comment: 20 pages; minor correction
Gene Context Analysis in the Integrated Microbial Genomes (IMG) Data Management System
Computational methods for determining the function of genes in newly sequenced genomes have been traditionally based on sequence similarity to genes whose function has been identified experimentally. Function prediction methods can be extended using gene context analysis approaches such as examining the conservation of chromosomal gene clusters, gene fusion events and co-occurrence profiles across genomes. Context analysis is based on the observation that functionally related genes are often having similar gene context and relies on the identification of such events across phylogenetically diverse collection of genomes. We have used the data management system of the Integrated Microbial Genomes (IMG) as the framework to implement and explore the power of gene context analysis methods because it provides one of the largest available genome integrations. Visualization and search tools to facilitate gene context analysis have been developed and applied across all publicly available archaeal and bacterial genomes in IMG. These computations are now maintained as part of IMG's regular genome content update cycle. IMG is available at: http://img.jgi.doe.gov
Comparison theory and smooth minimal C*-dynamics
We prove that the C*-algebra of a minimal diffeomorphism satisfies
Blackadar's Fundamental Comparability Property for positive elements. This
leads to the classification, in terms of K-theory and traces, of the
isomorphism classes of countably generated Hilbert modules over such algebras,
and to a similar classification for the closures of unitary orbits of
self-adjoint elements. We also obtain a structure theorem for the Cuntz
semigroup in this setting, and prove a conjecture of Blackadar and Handelman:
the lower semicontinuous dimension functions are weakly dense in the space of
all dimension functions. These results continue to hold in the broader setting
of unital simple ASH algebras with slow dimension growth and stable rank one.
Our main tool is a sharp bound on the radius of comparison of a recursive
subhomogeneous C*-algebra. This is also used to construct uncountably many
non-Morita-equivalent simple separable amenable C*-algebras with the same
K-theory and tracial state space, providing a C*-algebraic analogue of McDuff's
uncountable family of II_1 factors. We prove in passing that the range of the
radius of comparison is exhausted by simple C*-algebras.Comment: 30 pages, no figure
Classification and realizations of type III factor representations of Cuntz-Krieger algebras associated with quasi-free states
We completely classify type III factor representations of Cuntz-Krieger
algebras associated with quasi-free states up to unitary equivalence.
Furthermore, we realize these representations on concrete Hilbert spaces
without using GNS construction. Free groups and their type
factor representations are used in these realizations.Comment: 11 page
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