Having had a hard time finding this material in English I decided to make a post on it myself. Here is a proof of the existence of a universal compactification of 
-space of various types.
Exactness of the reduced crossed product functor
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.”The Grassmannian (1)
Introduction
Given a vector space 
of dimension 
over an algebraically closed field 
, the Grassmannian 
is the collection of all 
-dimensional subspaces of with the quotient topology. It is immediate that 
, hence we may think of it as a generalization of the usual projective spaces. In fact is itself a projective variety (see the Plücker imbedding).
Hilbert C*-modules
An increasingly prominent tool in operator theory is the Hilbert C*-module, which are (loosely speaking) Hilbert spaces where the inner product takes values in a C*-algebra. The next level of generalization is that of Hilbert modules over locally C*-algebras (we briefly mentioned locally C*-algebras in this post), and much of the following theory extends to this setting as well.
Here I give the definition of a Hilbert C*-module and collect some of it’s properties, mostly as a reference for personal use. I will likely update this post with new material later on, hopefully without making it too bloated. The theory is now well developed in the literature so the proofs will kept to a bare minimum. For references I will mostly use [1] and [2].