965 research outputs found
Logic and operator algebras
The most recent wave of applications of logic to operator algebras is a young
and rapidly developing field. This is a snapshot of the current state of the
art.Comment: A minor chang
Finitely generated nilpotent group C*-algebras have finite nuclear dimension
We show that group C*-algebras of finitely generated, nilpotent groups have
finite nuclear dimension. It then follows, from a string of deep results, that
the C*-algebra generated by an irreducible representation of such a group
has decomposition rank at most 3. If, in addition, satisfies the universal
coefficient theorem, another string of deep results shows it is classifiable by
its Elliott invariant and is approximately subhomogeneous. We give a large
class of irreducible representations of nilpotent groups (of arbitrarily large
nilpotency class) that satisfy the universal coefficient theorem and therefore
are classifiable and approximately subhomogeneous.Comment: Fixed typos. Question 5.1 of the previous version was already
answered in the literature; we have provided the appropriate referenc
The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length
Theories of classification distinguish classes with some good structure
theorem from those for which none is possible. Some classes (dense linear
orders, for instance) are non-classifiable in general, but are classifiable
when we consider only countable members. This paper explores such a notion for
classes of computable structures by working out a sequence of examples.
We follow recent work by Goncharov and Knight in using the degree of the
isomorphism problem for a class to distinguish classifiable classes from
non-classifiable. In this paper, we calculate the degree of the isomorphism
problem for Abelian -groups of bounded Ulm length. The result is a sequence
of classes whose isomorphism problems are cofinal in the hyperarithmetical
hierarchy. In the process, new back-and-forth relations on such groups are
calculated.Comment: 15 page
The nuclear dimension of C*-algebras
We introduce the nuclear dimension of a C*-algebra; this is a noncommutative
version of topological covering dimension based on a modification of the
earlier concept of decomposition rank. Our notion behaves well with respect to
inductive limits, tensor products, hereditary subalgebras (hence ideals),
quotients, and even extensions. It can be computed for many examples; in
particular, it is finite for all UCT Kirchberg algebras. In fact, all classes
of nuclear C*-algebras which have so far been successfully classified consist
of examples with finite nuclear dimension, and it turns out that finite nuclear
dimension implies many properties relevant for the classification program.
Surprisingly, the concept is also linked to coarse geometry, since for a
discrete metric space of bounded geometry the nuclear dimension of the
associated uniform Roe algebra is dominated by the asymptotic dimension of the
underlying space.Comment: 33 page
K-Theoretic Characterization of C*-Algebras with Approximately Inner Flip
The author is supported by an NSERC PDF.Peer reviewedPostprin
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