That Can't Be Right – Page 2 – Microblog for personal use

August 8, 2023April 25, 2024

Karpelevich’s compactification and extensions of Poisson integrals

In this short blog post, we study to what extent the Poisson integral of continuous functions on the Furstenberg boundary ${G/P_0}$ of a symmetric space of non-compact type can be extended continuously to the whole Karpelevich boundary or what parts thereof it can be extended. The interest in such integrals comes from the fact that the Poisson integrals of functions in ${L^{\infty}(G/P_0)}$ exhaust all bounded harmonic functions on ${X}$ , a deep result of Furstenberg. The study of such extensions hence has its origins in classical harmonic analysis and solutions to Dirichlet problems on the Poincare disk.

April 19, 2023October 15, 2023

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.

January 10, 2023April 19, 2023

Hyperbolic surfaces with isomorphic boundary C*-algebras

Given a manifold $\overlien{X}$ with boundary $\partial X$ , and isometry $G$ , for each subgroup $\Gamm$ of $G$ one can form the crossed product C*-algebra $C(\partial X)\rtimes_r\Gamma$ . 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 $SL_2(\R)$ .

November 10, 2022November 11, 2022

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, ${G}$ will denote a second countable locally compact Hausdorff groupoid with a Haar system ${\lambda}$ .

August 28, 2022February 2, 2024

Theorem of the month – The Ext semi-group for C*-algebras

In this blog post, we look at some examples of extensions of C*-algebras and introduce the ${Ext}$ (semi-)group. We only scrape the surface of what is known at Brown-Douglas-Fillmore theory, which as far as I know was the first place where the dual of the topological K-theory group (the ${K^1}$ -group) got its additive structure.

April 25, 2022October 6, 2023

Theorem of the month – Dauns-Hofmann theorem

In this installment of THEOREM OF THE MONTH! we will look at another big theorem in operator theory, namely the Dauns-Hofmann theorem. To set the scene, we will first introduce the notion of the primitive ideal spectrum (or space), its topology and list without proofs some of its most important properties. The theorem extends beautifully the spectral theory for commutative C*-algebras by treating the resulting algebra as sections of certain bundles of C*-algebras over the primitive ideal spectrum rather than the more familiar spectrum of a commutative C*-algebra.