Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hyperbolic group $C^*$-algebras and free-product $C^*$-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 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.

16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph