Sketches of Noncommutative Topology

Abstract

This thesis thematically divided into two parts. In the first part we are mastering C*-isomorphism problem by using various techniques applied to different examples of noncommutative algebraic varieties. In the second part we apply noncommutative homotopy theory to C*-algebraic objects related to manifold theory, in such a way deriving results and formulas for such an object as differential operators. In the first article we consider C*-algebra Isom_{q_{ij}} generated by n isometries a_1, \ldots, a_n satisfying the relations a_i^* a_j = q_{ij} a_j a_i^* with \max |q_{ij}| < 1. This C*-algebra is shown to be nuclear. We prove that the Fock representation of Isom_{q_{ij}} is faithful. Further we describe an ideal in Isom_{q_{ij}} which is isomorphic to the algebra of compact operators. In the second article we consider the C*-algebra \mathcal{E}^q_{n,m}, which is a q-twist of two Cuntz-Toeplitz algebras. For the case |q| < 1, we give an explicit formula which untwists the q-deformation showing that the isomorphism class of \mathcal{E}^q_{n,m} does not depend on q. For the case |q| = 1, we give an explicit description of all ideals in \mathcal{E}^q_{n,m}. In particular, we show that \mathcal{E}^q_{n,m} contains a unique largest ideal \mathcal{M}q. We identify \mathcal{E}^q{n,m}/\mathcal{M}_q with the Rieffel deformation of \mathcal{O}_n \otimes \mathcal{O}_m and use a K-theoretical argument to show that the isomorphism class does not depend on q. The latter result holds true in a more general setting of multiparameter deformations. In the third article we consider the universal enveloping C*-algebra \mathsf{CAR}\Theta of the -algebra generated by a_1, \ldots, a_n subject to the relations a_i^ a_i + a_i a_i^* = 1, a_i^* a_j =e^{2\pi i \Theta{ij}}a_j a_i^*, a_i a_j = e^{-2\pi i \Theta_{ij}} a_j a_i for a skew-symmetric real n x n matrix \Theta. We prove that \mathsf{CAR}\Theta has a C(K_n)-structure, where K_n = [0, \frac{1}{2}]^n is the hypercube and describe the fibers. We classify irreducible representations of \mathsf{CAR}\Theta in terms of irreducible representations of a higher-dimensional noncommutative torus. We prove that for a given irrational skew-symmetric \Theta_1 there are only finitely many \Theta_2 such that \mathsf{CAR}{\Theta_1} \simeq \mathsf{CAR}{\Theta_2}. Namely, \mathsf{CAR}{\Theta_1} \simeq \mathsf{CAR}{\Theta_2} implies (\Theta_1){ij} = \pm (\Theta_2){\sigma(i,j)} for a bijection \sigma of the set {(i,j):i