26 research outputs found
A short proof of an index theorem, II
We introduce a slight modification of the usual equivariant -theory. We
use this to give a -theoretical proof of an equivariant index theorem for
Dirac-Schrodinger operators on a non-compact manifold of nowhere positive
curvature. We incidentally show that the boundary of Dirac is Dirac;
generalizing earlier work of Baum and coworkers, and a result of Higson and
Roe.Comment: Corrected typo in abstract. No changes to conten
Remarks on computing the Grothendieck rings of C*-algebras
In this paper, we present a captivating construction by Grothendieck,
originally formulated for algebraic varieties, and adapt it to the realm of
C*-algebras. Our main objective is to investigate the conditions under which
this particular class of C*-algebras possesses a nontrivial Grothendieck ring.
To achieve this, we explore the existence of nontrivial characters, which
significantly enriches our understanding of these algebras. In particular, we
conduct a detailed study of rings of C*-algebras over ,
, and
On an intriguing distributional identity
For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω1 → Ω2 a continuous, strictly increasing function, such that Ω1∩Ω2⊇(a, b), but otherwise arbitrary, we establish that the random variables F(X) − F(g(X)) and F(g−1(X)) − F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U − ψ(U) = dψ−1(U) − U for U ∼ U(0, 1), where ψ is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation
The noncommutative geometry of Yang-Mills fields
We generalize to topologically non-trivial gauge configurations the
description of the Einstein-Yang-Mills system in terms of a noncommutative
manifold, as was done previously by Chamseddine and Connes. Starting with an
algebra bundle and a connection thereon, we obtain a spectral triple, a
construction that can be related to the internal Kasparov product in unbounded
KK-theory. In the case that the algebra bundle is an endomorphism bundle, we
construct a PSU(N)-principal bundle for which it is an associated bundle. The
so-called internal fluctuations of the spectral triple are parametrized by
connections on this principal bundle and the spectral action gives the
Yang-Mills action for these gauge fields, minimally coupled to gravity.
Finally, we formulate a definition for a topological spectral action.Comment: 14 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
Classification of graph C*-algebras with no more than four primitive ideals
We describe the status quo of the classification problem of graph C*-algebras
with four primitive ideals or less
The bulk-edge correspondence for the quantum Hall effect in Kasparov theory
We prove the bulk-edge correspondence in -theory for the quantum Hall
effect by constructing an unbounded Kasparov module from a short exact sequence
that links the bulk and boundary algebras. This approach allows us to represent
bulk topological invariants explicitly as a Kasparov product of boundary
invariants with the extension class linking the algebras. This paper focuses on
the example of the discrete integer quantum Hall effect, though our general
method potentially has much wider applications.Comment: 16 pages. Minor corrections and introduction expanded. To appear in
Letters in Mathematical Physic
-regularity and the corona factorization property
Stability is an important and fundamental property of -algebras. Given a short exact sequence of -algebras where the ends are stable, the middle algebra may or may not be stable. We say that the first algebra, , is -regular if every extension of by a stable algebra has a stable extension algebra, . Rördam has given a sufficient condition for -regularity. We define a new condition, weaker than Rördam's, which we call the corona factorization property, and we show that the corona factorization property implies -regularity. The corona factorization property originated in a study of the Kasparov group of extensions, however, we obtain our results without explicit reference to -theory.
Our main result is that for a separable stable -algebra the first two of the following properties (which we define later) are equivalent, and both imply the third. With additional hypotheses on the -algebra, all three properties are equivalent.
has the corona factorization property.
Stability is a stable property for full hereditary subalgebras of .
is -regular.
We also show that extensions of separable stable -algebras with the corona factorization property give extension algebras with the corona factorization property, extending the results of [9]