770 research outputs found
Quantization of edge currents for continuous magnetic operators
For a magnetic Hamiltonian on a half-plane given as the sum of the Landau
operator with Dirichlet boundary conditions and a random potential, a
quantization theorem for the edge currents is proven. This shows that the
concept of edge channels also makes sense in presence of disorder. Moreover,
Gaussian bounds on the heat kernel and its covariant derivatives are obtained
Stable finiteness of ample groupoid algebras, traces and applications
In this paper we study stable finiteness of ample groupoid algebras with
applications to inverse semigroup algebras and Leavitt path algebras,
recovering old results and proving some new ones. In addition, we develop a
theory of (faithful) traces on ample groupoid algebras, mimicking the
-algebra theory but taking advantage of the fact that our functions are
simple and so do not have integrability issues, even in the non-Hausdorff
setting. The theory of traces is closely connected with the theory of invariant
means on Boolean inverse semigroups. We include an appendix on stable
finiteness of more general semigroup algebras, improving on an earlier result
of Munn, which is independent of the rest of the paper
The Identity Correspondence Problem and its Applications
In this paper we study several closely related fundamental problems for words
and matrices. First, we introduce the Identity Correspondence Problem (ICP):
whether a finite set of pairs of words (over a group alphabet) can generate an
identity pair by a sequence of concatenations. We prove that ICP is undecidable
by a reduction of Post's Correspondence Problem via several new encoding
techniques.
In the second part of the paper we use ICP to answer a long standing open
problem concerning matrix semigroups: "Is it decidable for a finitely generated
semigroup S of square integral matrices whether or not the identity matrix
belongs to S?". We show that the problem is undecidable starting from dimension
four even when the number of matrices in the generator is 48. From this fact,
we can immediately derive that the fundamental problem of whether a finite set
of matrices generates a group is also undecidable. We also answer several
question for matrices over different number fields. Apart from the application
to matrix problems, we believe that the Identity Correspondence Problem will
also be useful in identifying new areas of undecidable problems in abstract
algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the
published journal version of this article, see footnote 3 on page 1
- …