Hyperbolic Group C -Algebras and Free-Product C -Algebras as Compact Quantum Metric Spaces

By Nathanial P. Brown Narutaka Ozawa, Marc Rieffel. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2003

117 citations recorded by the source corpus.

Abstract Let $\ell $ be a length function on a group $G$ , and let ${{M}_{\ell }}$ denote the operator of pointwise multiplication by $\ell $ on ${{\ell }^{2}}\left( G \right)$ . Following Connes, ${{M}_{\ell }}$ can be used as a “Dirac” operator for $C_{r}^{*}\left( G \right)$ . It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$ , which defines a metric on the state space of $C_{r}^{*}\left( G \right)$ . We show that if $G$ is a hyperbolic group and if $\ell $ is a word-length function on $G$ , then the topology from this metric coincides with the weak- $*$ topology (our definition of a “compact quantum metric space”). We show that a convenient framework is that of filtered ${{C}^{*}}$ -algebras which satisfy a suitable “Haagerup-type” condition. We also use this framework to prove an analogous fact for certain reduced free products of ${{C}^{*}}$ -algebras.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7e954cfd014205134407c564eeeb02175296ce0b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Nilpotent Group C*-algebras as Compact Quantum Metric Spaces (2002, methodology): “Let us compare this proposition with the main result of section 2 of [12].”
The Quantum Gromov-Hausdorff Propinquity (2013, methodology): “Thus, if one wishes to work within the context of C*-algebras and (strong) Leibniz seminorms, then some very non-trivial changes must be made.”
Noncommutative martingale deviation and Poincaré type inequalities with applications (2012, background): “y. Corollary 0.4. Under the assumptions above W1(f,EFixf) ≤ C(α) p τ(flnf) for all normal states φf(x) = τ(fx). The Wasserstein distance has been extensively studied in the noncommutative setting; see[7,44,52]. This probabilistic connection which provides universal upper bound given by the entropy f”
Quantum Ultrametrics on AF Algebras and The Gromov-Hausdorff Propinquity (2015, unknown): “ver e 6∈F. Rieffelinitiatedthesystematicstudyofquantumcompactmetricspacewiththe followingcharacterizationofthesespaces,whichcanbeseenasanoncommutative form of the Arzéla-Ascoli theorem. Theorem 2.5 ( [28,30,31]). Let Abe a unital C*-algebra and L a seminorm deﬁned on a dense subspace of sa(A), such ”
Noncommutative Riesz transforms -- a probabilistic approach (2008, methodology): “ order t−3/2 and hence property (R 6). According to Varopoulos’ deﬁnition [Var85] this means dimension d= n/2 = 3, as predicted by P. Biane [Bia]. Remark 6.7. According to the work of Rieﬀel and Ozawa[OR05] hyperbolic groups satisﬁes rapid decay with s= 1 and d= 3. Proof of Corollary 0.2. According ”
Dynamical Systems on Spectral Metric Spaces (2010, background): “mutant A0 D = fa2A; [D;a] = 0gis trivial, namely it is reduced to multiples of the unit. In particular this condition requires that the representation of Aon the Hilbert space Hbe faithful. Result 2 ([53, 61, 62, 63, 52]). Let X= (A;H;D) be a spectral triple with Aunital. In addition, let Xbe such t”