In this paper, we give three functors $mathfrak{P}$, $[cdot]_K$ and $mathfrak{F}$ on the category of C$^ast$-algebras. The functor $mathfrak{P}$ assigns to each C$^ast$-algebra $mathcal{A}$ a pre-C$^ast$-algebra $mathfrak{P}(mathcal{A})$ with completion $[mathcal{A}]_K$. The functor $[cdot]_K$ assigns to each C$^ast$-algebra $mathcal{A}$ the Cauchy extension $[mathcal{A}]_K$ of $mathcal{A}$ by ...

We study the category O for a general Coxeter system using a formulation of Fiebig. The translation functors, the Zuckerman functors and the twisting functors are defined. We prove the fundamental properties of these functors, the duality of Zuckerman functor and generalization of Verma’s result about homomorphisms between Verma modules.

2 A bimodule realization of the Temperley-Lieb two-category 8 2.1 Ring A and two-dimensional cobordisms . . . . . . . . . . . . 8 2.2 Flat tangles and the Temperley-Lieb category . . . . . . . . . 10 2.3 The Temperley-Lieb 2-category . . . . . . . . . . . . . . . . . 12 2.4 The ring H . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Projective H-modules . . . . . . . . . . . . . . ....