Connes’s metrics for the noncommutative setting of \(C^\ast\)-algebras are applied to the geometry of the Cantor set. Since the Cantor set is \(\prod\mathbb{Z}_2\), the corresponding noncommutative space is an AF \(C^\ast\)-algebra. So the authors first study Dirac operators arising in spectral triples for AF \(C^\ast\)-algebras.
For a unital AF-algebra \(\mathcal{A}\), the Hilbert space \(H\) of the spectral triple for \(\mathcal{A}\) is the norm closure of the direct sum of finite dimensional spaces \(H_n\). Denoting the projections to \(H_n\) by \(Q_n\), the Dirac operator \(D\) of the spectral triple takes the form \(\sum_{n=1}^\infty\alpha_n Q_n\). Then for any \(p>0\), the possibility of the choice of sequence \((\alpha_n)_{n\in\mathbb{N}_0}\) such that the Fredholm module is \(p\)-summable is shown (Theorem 2.1, (ii)). The difficulty of extending this result to arbitrary \(C^\ast\)-algebras and a partial answer (Theorem 2.3) to Connes’s question whether this result characterizes AF-algebras are discussed. If \(\mathcal{A}\) is a UHF algebra of the form \(\mathcal{A}_n=\mathcal{A}_{n-1}\otimes\mathcal{M}_{d_n}\), and \(m_n=d_1\cdots d_n\), it is shown that \(D=\sum_n(\beta_n)^{-1}\sqrt{m_n}Q_n\), where \(\sum\beta_n\) converges absolutely, induces a metric on the state space for the weak \(*\)-topology and this Fredholm module is 4-summable, while \(D=\sum_n (m_n)^s Q_n\), \(s>2/p>1\), induces a metric on the state space with \(p\)-summable Fredholm module (Theorem 3.1).
The Hausdorff dimension of the Cantor set is \(\log 2/\log 3\). The corresponding Dirac operator of the spectral triple of \(C(\prod\mathbb{Z}_2)\) has been constructed by Connes (unpublished). But by successive cuttings of \(2^{n-1}\) intervals of length \((1-2\gamma)\gamma^{n-1}\), \(0<\gamma<1/2\), the space \(\mathfrak{C}_\gamma\) (or \((\prod\mathbb{Z}_2,\delta_\gamma))\) is homeomorphic to the Cantor set and has Hausdorff dimension \(-\log 2/\log\gamma\). In the paper under review, by using previous results, the corresponding Dirac operators in the spectral triples of \(C(\prod\mathbb{Z}_2)\) are constructed (Theorem 4.1). Then in §5 applying this construction, an isometric embedding of \(\prod\mathbb{Z}_2\) onto a compact subset of the Banach space \(\ell^1(\mathbb{N},\mathbb{R})\) is constructed. Denoting the image of \((\prod\mathbb{Z}_2,\delta_\gamma)\) by \(\mathcal{C}_\gamma\), the Gromov-Hausdorff distance [cf.M.A.Rieffel, Mem.Am.Math.Soc.796, 1–65, 67–91 (2004; Zbl 1043.46052)] of \(\mathcal{C}_\gamma\) and \(\mathcal{C}_\mu\) is shown to be smaller than \((2(\gamma-\mu))/(1-\gamma)\), \(0<\mu<\gamma<1\) (Proposition 6.2). In §7, the closure of \(\bigcup_{0<\gamma<1} (1-\gamma)\mathcal{C}_\gamma\) in \(\ell^1(\mathbb{N},\mathbb{R})\) is shown to be compact, containing closed Cantor sets of any Hausdorff dimension and contained in \(\{x\in\ell^1(\mathbb{N},\mathbb{R}): 0\leq x(n)\leq 4/(n+1)^2\}\) (Theorem 7.1).
§8, the last section, studies the metric on the state space \(\mathcal{S}(\mathcal{A})\), \(\mathcal{A}\) is a unital \(C^\ast\)-algebra, and constructs a representation \(\pi\) of \(\mathcal{A}\) on a Hilbert space \(H\) such that there exists a projection \(P\in B(H)\) which has the property that the norm distance on the state space is recovered exactly if this projection \(P\) plays the role of the Dirac operator (Theorem 8.3).