Continue reading “A short rant about power cables in Norway”
A note on torsion in K-theory
In this post we will look at the correspondence between torsion in cohomology of finite CW-complexes and torsion in their K-theory. The title could just as well have been “a use case for Atiyah-Hirzebruch spectral sequence” as this will be our primary tool.
Continue reading “A note on torsion in K-theory”Homomorphisms of transformation group C*-algebras – A worked example
In this blog post we work through some examples of computations with crossed product C*-algebras, particularly we look at characterizing *-homomorphisms from a transformation group C*-algebra to a commutative C*-algebra. Some familiarity with C*-crossed products will be assume. Trying to determine all homomorphisms between two given C*-algebra, their algebraic structure, their homotopy classes and so forth, can be quite challenging in general. Most of the standard references on crossed products show how to construct a *-homomorphism of crossed products from an equivariant *-homomorphism of the underlying C*-algebra, but rarely show examples of morphisms that don’t arise in this way.
Continue reading “Homomorphisms of transformation group C*-algebras – A worked example”More hyperbolic spaces with isomorphic boundary C*-algebras
In this blog post, we are going to show that neither the fundamental group, nor the corresponding action on the geodesic boundary on a hyperbolic manifold, are uniquely determined by the boundary C*-algebra. We will finish the post by an outlook of possible paths for the non-commutative version of the Mostow rigidity theorem discussed in a previous post.
Continue reading “More hyperbolic spaces with isomorphic boundary C*-algebras”Hyperbolic surfaces with isomorphic boundary C*-algebras
Given a manifold 
with boundary 
, and isometry 
, for each subgroup 
of one can form the crossed product C*-algebra 
. This post is a one in a series of posts that investigates the non-commutative analogues of the Mostow rigidity theorem, which shows that for certain locally symmetric spaces the fundamental group is a complete isomorphism (i.e. isometry) invariant for the locally symmetric spaces, setteling the Borel conjecture for such spaces.
In this post we look at the failure of the theorem in the special case of 
.
A note on induced representations of groupoid C*-algebras
In this post, which is based on a seminar series on groupoids held in Leiden in late 2022, I collect some of the basic properties of induced representations of groupoid C*-algebras. Virtually everything in this post can be found in chapter 5 of the book of Williams (cited below) with more examples, fewer mistakes and better english. The main focus will be the full groupoid C*-algebra, and we barely mention the definition of unitary representations of groupoids. In the sequel, unless stated otherwise, 
will denote a second countable locally compact Hausdorff groupoid with a Haar system 
.