The purpose of this chapter is to write about Quot and Hilbert functors and to prove that these are algebraic spaces provided certain technical conditions are satisfied. In this chapter we will discuss this in the setting of algebraic space. A reference is Grothendieck’s lectures, see [Gro95a], [Gro95b], [Gro95e], [Gro95f], [Gro95c], and [Gro95d]. Another reference is the paper [OS03]; this pap...

We study the Hilbert polynomials of non-standard graded algebras R, that are finitely generated on generators not all of degree one. Given an expression P (R, t) = a(t)/(1 − t) for the Poincaré series of R as a rational function, we study for 0 ≤ i ≤ l the graded subspaces ⊕kRkl+i (which we denote R[l; i]) of R, in particular their Poincaré series and Hilbert functions. For example, we prove th...

A C* algebra is a closed self-adjoint subalgebra of B(H), the bounded operators on a Hilbert space H. Kirchberg asked whether every C* algebra embeds into an ultrapower of a particular C* algebra, the Cuntz algebra O_2. In this talk, I will explain the model-theoretic content of this problem. In particular, I will show how model-theoretic forcing can be used to give a weak local reformulation o...

In 1999 A. Connes and D. Kreimer have discovered a Hopf algebra structure on the Feynman graphs of scalar field theory. They have found that the renormalization can be interpreted as a solving of some Riemann — Hilbert problem. In this work the generalization of their scheme to the case of nonabelian gauge theories is proposed. The action of the gauge group on the Hopf algebra of diagrams is de...

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that as...

3 Koszul duality 22 3.1 Quadratic algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Dual coalgebra and algebra . . . . . . . . . . . . . . . . . . . . . 23 3.3 Some more examples . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.1 The free algebra . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.2 An example from Fröberg . . . . . . . . . . . . . . . . . . 28 3.3.3 ...

In this note we prove, for groups G fulfilling the Atiyah conjecture, a lemma relating arbitrary G-modules to free group-von-Neumann-algebra modules. As an immediate application we get properties of finitely generated pro-jective G-modules and G-modules. Other related arguments are envisaged. G; it can be defined as algebra of all (left) G-equivariant bounded linear operators on the Hilbert spa...

A class of vector coherent states is derived with multiple of matrices as vectors in a Hilbert space, where the Hilbert space is taken to be the tensor product of several other Hilbert spaces. As examples vector coherent states with multiple of quaternions and octonions are given. The resulting generalized oscillator algebra is briefly discussed. Further, vector coherent states for a tensored H...

We consider C-algebras generated by a single Hilbert bimodule (Pimsner-Toeplitz algebras) and by a product systems of Hilbert bimodules. We give a new proof of a theorem of Pimsner, which states that any representation of the generating bimodule gives rise to a representation of the Pimsner-Toeplitz algebra. Our proof does not make use of the conditional expectation onto the subalgebra invarian...

We display methods that allow for computations of spectra, pseudospectra and resolvents of linear operators on Hilbert spaces and also elements in unital Banach algebras. The paper considers two different approaches, namely, pseudospectral techniques and polynomial numerical hull theory. The former is used for Hilbert space operators whereas the latter can handle the general case of elements in...