Group C*-algebras as compact quantum metric spaces

Abstract

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 $C_r^{\ast }(G)$. It defines a Lipschitz seminorm on $C_r^{\ast }(G)$, which defines a metric on the state space of $C_r^{\ast }(G)$. 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 $C_r^{\ast }(G)$ 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.

53 pages, yet more minor improvements. To appear in Doc. Math

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• The Spectral Action Principle
• Metrics on State Spaces
• Metrics on states from actions of compact groups
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Cited by in corpus (9)

• The horofunction boundary of finite-dimensional normed spaces
• Quantum isometry groups of noncommutative manifolds associated to group C*-algebras
• Euler characteristics and Gysin sequences for group actions on boundaries
• Nilpotent group C*-algebras as compact quantum metric spaces
• The horofunction boundary of finite-dimensional ${\ell }_p$ spaces
• Metric compactification of infinite Sierpiński carpet graphs
• The Polish topology of the isometry group of the infinite dimensional hyperbolic space
• From C*algebra extensions to CQMS, ${SU}_q(2)$, Podles sphere and other examples
• Real Spectral Triples on Crossed Products