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Michael’s noncommutative selection principle and the classification of nonsimple algebras. (Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren.)
... We will define in a forthcoming paper the KK-theory of a (not necessarily unital) precosheaf of C * -algebras, coherent with the notion of K-homology that we study here ([32]). This will yield a new approach for the K-theory of precosheaves of C * -algebras (already studied in [5] [6]) and, in a different scenario, for computing K-theoretic invariants of non-simple C * -algebras, on the research line started by Kirchberg [18] ...
Let X be a space, intended as a possibly curved spacetime, and A a precosheaf
of C*-algebras on X. Motivated by algebraic quantum field theory, we study the
Kasparov and Theta-summable K-homology of A interpreting them in terms of the
holonomy equivariant K-homology of the associated C*-dynamical system. This
yields a characteristic class for K-homology cycles of A with values in the odd
cohomology of X, that we interpret as a generalized statistical dimension.
... Using the UCT of Bonkat (see Satz 7.5.3 of [1]), we can lift this to an element x ∈ KK E (e, e ′ ). Using Kirchberg's lifting result for nuclear, stable, strongly purely infinite C * -algebras (see Hauptsatz 4.2 of [9]), we know that x can be lifted to a morphism ψ = (ψ 0 , ψ 1 , ψ 2 ) : e → e ′ of extensions. Consequently, K six (ψ) = φ. ...
We prove that the natural homomorphism from Kirchberg's ideal-related
KK-theory, KKE(e, e'), with one specified ideal, into Hom_{\Lambda}
(\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all
extensions e and e' of separable, nuclear C*-algebras in the bootstrap category
N with the K-groups of the associated cyclic six term exact sequence being
finitely generated, having zero exponential map and with the K_{1}-groups of
the quotients being free abelian groups.
This class includes all Cuntz-Krieger algebras with exactly one non-trivial
ideal. Combining our results with the results of Kirchberg, we classify
automorphisms of the stabilized purely infinite Cuntz-Krieger algebras with
exactly one non-trivial ideal modulo asymptotically unitary equivalence. We
also get a classification result modulo approximately unitary equivalence.
The results in this paper also apply to certain graph algebras.
... . It is worth noting that recent work of Kirchberg [15] has shown the significance of the topological space Prim(A) for the classification of nonsimple C * -algebras. Prim(A) has sometimes been regarded as of limited use because of the coarseness of its Jacobson topology and the fact that inequivalent irreducible representations may have the same primitive kernel. ...
It is well known that if A is a von Neumann algebra then the norm of any inner derivation ad(a) is equal to twice the distance from the element a to the centre Z(A) of the algebra. More generally, this property holds in a unital C∗-algebra if and only if the ideal P∩Q∩R is primal whenever P, Q, and R are primitive ideals of A such that P∩Z(A)=Q∩Z(A)=R∩Z(A). In this paper we give a characterization, in terms of ideal structure, of those unital C∗-algebras A for which the norm of any inner derivation ad(a) at least dominates the distance from a to the centre Z(A). In doing so, we show that if A does not have this property then it necessarily contains an element a, with ‖ad(a)‖=1, whose distance from Z(A) is greater than or equal to 3+8214. We also show how this number is related to the numbers 415 and 12+13 which have previously arisen in the study of norms of inner derivations. The techniques used in this work include spectral theory, the theory of primitive and primal ideals, and constrained geometrical optimisation.
... The interest in such algebras arises for their role in the context of crossed products by endomorphisms, and duality for non-compact groups (see [27]). Anyway, they are an interesting class also from the viewpoint of the classification of C*-algebras by KK-theoretical invariants, in the spirit of [21] [7]. We will develop here the basic properties of such CP-algebras, and give some applications for Cuntz algebra bundles; more detailed K-theoretical and classification questions will be approached in future papers. ...
We study the Cuntz–Pimsner algebra associated with the module of continuous sections of a Hilbert bundle, and prove that it is a continuous bundle of Cuntz algebras. Furthermore, we assign to bundles of Cuntz algebras carrying a global circle action a class in the representable KK-group of the zero-grade bundle. We explicitly compute such class for the Cuntz–Pimsner algebra of a vector bundle.
We initiate the study of absorbing representations of $C^\ast$-algebras with
respect to closed operator convex cones. We completely determine when such
absorbing representations exist, which leads to the question of characterising
when a representation is absorbing, as in the classical Weyl-von Neumann type
theorem of Voiculescu. In the classical case, this was proven by Elliott and
Kucerovsky who proved that a representation is nuclearly absorbing if and only
if it induces a purely large extension. By considering a related problem for
extensions of $C^\ast$-algebras, which we call the purely large problem, we ask
when a purely largeness condition similar to the one defined by Elliott and
Kucerovsky, implies absorption with respect to some given closed operator
convex cone.
We solve this question for a special type of closed operator convex cone
induced by actions of finite topological spaces on $C^\ast$-algebras. As an
application of this result, we give $K$-theoretic classification for certain
$C^\ast$-algebras containing a purely infinite, two-sided, closed ideal for
which the quotient is an AF algebra. This generalises a similar result by the
second author, S. Eilers and G. Restorff in which all extensions had to be
full.
The smallest primitive ideal spaces for which there exist counterexamples to the classification of non-simple, purely infinite, nuclear, separable C*-algebras using filtrated K-theory, are four-point spaces. In this article, we therefore restrict to real rank zero C*-algebras with four-point primitive ideal spaces. Up to homeomorphism, there are ten different connected T0-spaces with exactly four points. We show that filtrated K-theory classifies real rank zero, tight, stable, purely infinite, nuclear, separable C*-algebras that satisfy that all simple subquotients are in the bootstrap class for eight out of ten of these spaces.
We prove a strong classification result for a certain class of
$C^{*}$-algebras with primitive ideal space $\widetilde{\mathbb{N}}$, where
$\widetilde{\mathbb{N}}$ is the one-point compactification of $\mathbb{N}$.
This class contains the class of graph $C^{*}$-algebras with primitive ideal
space $\widetilde{\mathbb{N}}$. Along the way, we prove a universal coefficient
theorem with ideal-related $K$-theory for $C^{*}$-algebras over
$\widetilde{\mathbb{N}}$ whose $\infty$ fiber has torsion-free $K$-theory.
We show that the K-theory cosheaf is a complete invariant for separable
continuous fields with vanishing boundary maps over a finite-dimensional
compact metrizable topological space whose fibers are stable Kirchberg algebras
with rational K-theory groups satisfying the universal coefficient theorem. We
provide a range result for fields in this class with finitely generated
K-theory. There are versions of both results for unital continuous fields.
We describe the status quo of the classification problem of graph C*-algebras
with four primitive ideals or less.
We prove that all unital separable continuous fields of C*-algebras over [0,1] with fibers isomorphic to the Cuntz algebra
On (2 £ n £ ¥){\mathcal{O}}_n \, (2 \leq n \leq \infty) are trivial. More generally, we show that if A is a separable, unital or stable, continuous field over [0,1] of Kirchberg C*-algebras satisfying the UCT and having finitely
generated K-theory groups, then A is isomorphic to a trivial field if and only if the associated K-theory presheaf is trivial. For fixed d Î {0,1}d\in \{0,1\} we also show that, under the additional assumption that the fibers have torsion free K
d
-group and trivial K
d+1-group, the K
d
-sheaf is a complete invariant for separable stable continuous fields of Kirchberg algebras.
Let X be a finite dimensional compact metrizable space. We study a technique which employs semiprojectivity as a tool to produce approximations of C(X)-algebras by C(X)-subalgebras with controlled complexity. The following applications are given. All unital separable continuous fields of C*-algebras over X with fibers isomorphic to a fixed Cuntz algebra On, n∈{2,3,…,∞}, are locally trivial. They are trivial if n=2 or n=∞. For n⩾3 finite, such a field is trivial if and only if (n−1)[A1]=0 in K0(A), where A is the C*-algebra of continuous sections of the field. We give a complete list of the Kirchberg algebras D satisfying the UCT and having finitely generated K-theory groups for which every unital separable continuous field over X with fibers isomorphic to D is automatically locally trivial or trivial. In a more general context, we show that a separable unital continuous field over X with fibers isomorphic to a KK-semiprojective Kirchberg C*-algebra is trivial if and only if it satisfies a K-theoretical Fell type condition.
In this paper we study the C*-algebras associated to continuous fields over locally compact metrizable zero-dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants.
Local and global definitions of pure infiniteness for a C∗-algebra A are compared, and equivalence between them is obtained if the primitive ideal space of A is Hausdorff and of finite dimension, if A has real rank zero, or if A is approximately divisible. Sufficient criteria are given for local pure infiniteness of tensor products. They yield that exact simple tensorially non-prime C∗-algebras are purely infinite if they have no semi-finite lower semi-continuous trace. One obtains that A is isomorphic to if A is (1-)purely infinite, separable, stable, nuclear and Prim(A) is a Hausdorff space (not necessarily of finite dimension).
We define the filtrated K-theory of a C*-algebra over a finite topological
space X and explain how to construct a spectral sequence that computes the
bivariant Kasparov theory over X in terms of filtrated K-theory. For finite
spaces with totally ordered lattice of open subsets, this spectral sequence
becomes an exact sequence as in the Universal Coefficient Theorem, with the
same consequences for classification. We also exhibit an example where
filtrated K-theory is not yet a complete invariant. We describe a space with
four points and two C*-algebras over this space in the bootstrap class that
have isomorphic filtrated K-theory but are not KK(X)-equivalent. For this
particular space, we enrich filtrated K-theory by another K-theory functor, so
that there is again a Universal Coefficient Theorem. Thus the enriched
filtrated K-theory is a complete invariant for purely infinite, stable
C*-algebras with this particular spectrum and belonging to the appropriate
bootstrap class.
An overview about C*-algebra bundles with a Z-grading is presented, with
particular emphasis on classification questions. In particular, we discuss the
role of the representable KK(X ; -, -)-bifunctor introduced by Kasparov. As an
application, we consider Cuntz-Pimsner algebras associated with vector bundles,
and give a classification in terms of K-theoretical invariants in the case in
which the base space is an n-sphere.
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