Let $ell$ be a length function on a group $G$, and let $M_{ell}$denote theoperator of pointwise multiplication by $ell$ on $ell^2(G)$.Following Connes,$M_{ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines aLipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of$C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $ell$ isa word-length function on $G$, then the topology from this metriccoincides with theweak-$*$ topology (our definition of a ``compact quantum metricspace''). We show that a convenient framework is that of filtered$C^*$-algebras which satisfy a suitable ``Haagerup-type'' condition. Wealso use thisframework to prove an analogous fact for certain reducedfree products of $C^*$-algebras.
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机译：假设$ ell $是组$ G $的长度函数，并让$ M_ {ell} $表示在$ ell ^ 2（G）$上的点数乘法乘以$ ell $的运算符。在Connes之后，$ M_ {ell} $可以用作$ C_r ^ *（G）$的``狄拉克''运算符。它在$ C_r ^ *（G）$上定义了一个Lipschitz半范数，后者在$ C_r ^ *（G）$的状态空间上定义了一个度量。我们表明，如果$ G $是一个双曲组，并且如果$ ell $是$ G $上的字长函数，则该度量的拓扑与弱$ * $拓扑（我们对``紧凑量子度量空间''的定义）一致'）。我们显示了一个便利的框架是满足合适的``Haagerup型''条件的已过滤$ C ^ * $-代数的框架。我们还使用该框架来证明某些$ C ^ * $-代数的减少约数乘积的相似事实。
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