In this blog post, we look at some examples of extensions of C*-algebras and introduce the 
(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 
-group) got its additive structure.
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.
Continue reading “Theorem of the month – Dauns-Hofmann theorem”Theorem of the month – Bott Periodicity for operator K-theory
In this post we review the proof of the Bott periodicity theorem for C*-algebras, generalizing the original Bott periodicity theorem from homotopy theory (and topological K-theory).
Continue reading “Theorem of the month – Bott Periodicity for operator K-theory”The rigid world of symmetric spaces of noncompact type
The current post is based on a talk I gave at the university of Leiden which unfortunately I completely butchered due, in part, to the sudden realization that I had not chance to get through this material in the given time. Nothing here is new or proved, though some sketches are added when I find it relevant.
The (unattained) goal of the presentation was to show how much of the geometry of a symmetric space of noncompact type can be determined from it boundary sphere alone (in rank 1 cases) and a simplicial complex called the boundary at infinity (in higher rank cases) by following the evolution of the Mostow rigidity theorem from its origins (closed quotients of real hyperbolic spaces) to higher ranks.
Continue reading “The rigid world of symmetric spaces of noncompact type”Actions of certain groups on the Gromov boundary of their Cayley graphs
The object of study of this very short post are groups of the form 
where 
. We will stick to the standard choice of generators 
and look at their associated (undirected) Cayley graph 
and boundary space 
. Recall that the Cayley graph of 
is the graph given the
Equivalent ways to define topology
In every course in basic topology one learns the standard way to define a topology on a set 
, how to induce it from a basis, ambient space or a collection of functions into or out of the set. There are however other ways to associate a topology to a given set which may come up in practice. Here is a list of some of the once I have come across, let me know if you think there should be more elements to the list, as I am not an expert on this stuff.