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 
.
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 
(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”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
Wedderburn’s theorem and finite dimensional C*-algebras
The classical Wedderburn theorem, dating back to 1908, is a milestone in abstract algebra. This article is devoted to the proof the this beautiful theorem over arbitrary fields, after which we specialize to 
and 
-algebras.
The hard Lefschetz Theorem and Lefschetz Decomposition.
The hard Lefschetz theorem is a theorem in complex differential geometry, more specifically Kahler geometry, which determines an isomorphisms between certain cohomology spaces. Together with the Poincare duality, Serre duality it shows just how rigid the cohomology groups of a compact Kahler manifold are. In this post the theorem and its proof will be stated, with a view towards the more rudimentary setting of symplectic geometry.
Continue reading “The hard Lefschetz Theorem and Lefschetz Decomposition.”