Group C*-algebras as compact quantum metric spaces

Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on \bell^2(G). Following Connes, $M_{\ell}$ 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 $G = {\mathbb Z}^d$ when $\ell$ is a word-length, or the restriction to ${\mathbb Z}^d$ of a norm on ${\mathbb R}^d$. 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.Comment: 53 pages, yet more minor improvements. To appear in Doc. Mat

Leibniz seminorms for "Matrix algebras converge to the sphere"

In an earlier paper of mine relating vector bundles and Gromov-Hausdorff distance for ordinary compact metric spaces, it was crucial that the Lipschitz seminorms from the metrics satisfy a strong Leibniz property. In the present paper, for the now non-commutative situation of matrix algebras converging to the sphere (or to other spaces) for quantum Gromov-Hausdorff distance, we show how to construct suitable seminorms that also satisfy the strong Leibniz property. This is in preparation for making precise certain statements in the literature of high-energy physics concerning "vector bundles" over matrix algebras that "correspond" to monopole bundles over the sphere. We show that a fairly general source of seminorms that satisfy the strong Leibniz property consists of derivations into normed bimodules. For matrix algebras our main technical tools are coherent states and Berezin symbols.Comment: 46 pages. Scattered very minor improvement

A Long-Term Study of Sons of Alcoholics.

Men with a family history of alcoholism appear to have a lower intensity reaction to alcohol's effects than those without this family history. This study investigated whether a lower reaction could encourage greater alcohol consumption among family history-positive (FHP) subjects, predisposing them to develop alcohol-related problems. A family history of alcoholism was associated with increased risk of alcohol dependence and abuse among study subjects. Likewise, over half of the FHP's whose reactions to alcohol were low had developed alcoholism at a 10-year followup