Length function

Abstract: Let l be a length function on a group G, and let M l denote the operator of pointwise multiplication by l on l 2 (G). Following Connes, M l can be used as a Dirac operator for C* r (G). It defines a Lipschitz seminorm on C* r (G), which defines a metric on the state space of C* r (G). We investigate whether the topology from this metric coincides with the weak-* topology (our definition of a compact quantum metric space). We give an affirmative answer for G = Z d when l is a word-length, or the restriction to Z d of a norm on R d . This works for C* r (G) twisted by a 2-cocycle, and thus for non-commutative tori. Our approach involves Connes' cosphere algebra, and an interesting compactification of metric spaces which is closely related to geodesic rays.

Abstract: We study a linear representation ρ:Bn? GLm(Z[q±1,t±1]) with m=n(n-1)/2. We will show that for n=4, this representation is faithful. We prove a relation with the new Charney length function. We formulate a conjecture implying that ρ is faithful for all n.

Abstract: In this article we prove that quasi-multiplicative (with respect to the usual length function) mappings on the permutation group $\SSn$ (or, more generally, on arbitrary amenable Coxeter groups), determined by self-adjoint contractions fulfilling the braid or Yang-Baxter relations, are completely positive. We point out the connection of this result with the construction of a Fock representation of the deformed commutation relations $d_id_j^*-\sum_{r,s} t_{js}^{ir} d_r^*d_s=\delta_{ij}\id$, where the matrix $t_{js}^{ir}$ is given by a self-adjoint contraction fulfilling the braid relation. Such deformed commutation relations give examples for operator spaces as considered by Effros, Ruan and Pisier. The corresponding von Neumann algebras, generated by $G_i=d_i+d_i^*$, are typically not injective.