Research Papers

1. Approximations of quantum tori by finite quantum tori for the quantum Gromov-Hausdorff distance
   Frederic Latremoliere (2003), Journ. of Funct. Anal., 223 (2005), 365-395, ArXiv: math.OA/0310214

   We establish that, given a compact Abelian group $G$ endowed with a continuous length function $l$ and a sequence ${(H_n)}_{n\in \mathbb{N}}$ of closed subgroups of $G$ converging to $G$ for the Hausdorff distance induced by $l$, then $C^{*}\left(\hat{G},\sigma \right)$ is the quantum Gromov-Hausdorff limit of any sequence $C^{*}{\left(\widehat{H_n},{\sigma }_n\right)}_{n\in \mathbb{N}}$ for the natural quantum metric structures and when the lifts of ${\sigma }_n$ to $\hat{G}$ converge pointwise to $\sigma$. This allows us in particular to approximate the quantum tori by finite dimensional C*-algebras for the quantum Gromov-Hausdorff distance. Moreover, We also establish that if the length function $l$ is allowed to vary, we can collapse quantum metric spaces to various quotient quantum metric spaces.

2. Bounded-Lipschitz distances on the state spaces of C*-algebras
   Frederic Latremoliere, Accepted (2005) in the Taiwanese Journal of Mathematics (June 2007). ArXiv: math.OA/0510340

   Metric noncommutative geometry, initiated by Alain Connes, has known some great recent developments under the impulsion of Rieffel and the introduction of the category of compact quantum metric spaces topologized thanks to the quantum Rieffel-Gromov-Hausdorff distance. In this paper, we undertake the first step to generalize such results and constructions to locally compact quantum metric spaces. Our present work shows how to generalize the construction of the bounded-Lipschitz metric on the state space of a C*-algebra which need not be unital, such that the topology induced by this distance on the state space is the weak* topology. In doing so we obtain some results on a state space picture of the strict topology of a C*-algebra.

3. Crossed-product C*-algebras for conformal automorphisms of the disk
   Frederic Latremoliere, Man-Duen Choi, Submitted (2005), 31 with appendix, ArXiv: math.OA/0511331

   We study the C*-algebra crossed-product of the closed unit disk by the action of one of its conformal automorphisms. When the automorphism is hyperbolic or parabolic, namely conformally equivalent to a dilation or a translation of the plane, we describe fully the crossed-product, its spectrum and all its irreducible representations. When the automorphism is elliptic, we obtain a description of the crossed-product and its representations based upon the rotation algebras.

© 2006 Frederic Latremoliere, Ph.D.