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].
Constructing new C*-algebras – Universal C*-algebras (1)
Universal C*-algebras are C*-algebras defined implicitly by relations and generators, much like group presentations. Contrary to the case with groups where the presentation can be constructed as a quotient of the free group, there is no analogous construction for C*-algebras, and we are not always guarantied the existence of a universal C*-algebras for any presentation.
This series of posts introduces the notion of universal C*-algebras. In this post I introduce the general construction of a universal C*-algebra, and a simple method for computing the the universal C*-algebra of a family of unitaries and (sufficiently nice) relations. In subsequent posts I hope to cover more general constructions with possibly non-unitary operators and more subtle relations.
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Constructing new C*-algebras – Injective and Projective limits
The injective (or direct) limit of C*-algebras is one way to construct new C*-algebras from directed system of C*-algebras (defined below), and is an essential tool in operator theory, so one may as well get acquainted with it. The projective limit (or inverse limit) is not as common it seems, but I will add it here for completeness. In this post I will try to give a definition of the construct by universal properties of colimits in the category of C*-algebras, but reducing the prerequisites from category theory to a bare minimum. The point is to highlight that similarities between direct limits of groups, rings, algebras etc., stems from the fact that they all solve the same universal problem in their respective categories, and to justify why some of these limits/colimits are preserved under certain transformations. Though the similarities may be evidenced, this is understandably (but also unfortunately) often not addressed in the classical references of operator theory, as a formal definition of a limit/colimit would be a significant digression.
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