Noncommutative torus from Fibonacci chains via foliation (Q2766154)

The Fibonacci chain (F-chain) is a typical example of a quasiperiodic structure. In this paper, an approximately finite-dimensional (AF) \(C^*\)-algebra on the space of F-chains and an embedding of a noncommutative torus into this algebra are constructed. This noncommutative torus can be regarded as the \(C^*\)-algebra of Kronecker foliation and a surjection from the space of F-chains to the leaf space of the foliation is constructed. It is shown there is one singular leaf which corresponds to two different classes of F-chains. The authors remark this embedding is consistent to the result of \textit{G. Landi}, \textit{F. Lizzi} and \textit{R. J. Szabo} [Commun. Math. Phys. 217, No. 1, 181-201 (2001; Zbl 0982.58004)].NEWLINENEWLINENEWLINEThe outline of the paper is as follows: In section 2, \(C^*\)-algebra on the space of F-chains is constructed and its \(K\)-theory is reviewed. An F-chain is an infinite series of two segments \(S\) and \(L\) such thatNEWLINENEWLINENEWLINE1. Any finite part of the sequence appears infinite times but none of them are consecutively repeated more than two times.NEWLINENEWLINENEWLINE2. \(SS\) is not allowed.NEWLINENEWLINENEWLINEThe deflation method to obtain this sequence starts from a finite subchain of an F-chain and repeat the deflation rule \(s\to L\) and \(L\to LS\) [\textit{B. Grünbaum} and \textit{G. Shephard}, ``Tilings and Patterns, An introduction'', New York (1989; Zbl 0746.52001)]. By using the inverse of deflation (inflation), the index sequence \(i({\mathcal F},\alpha)\) of a given segment \(\alpha\) of a F-chain \({\mathcal F}\) is defined as an infinite sequence of integers \((a_0,a_1,\dots)\), \(a_n=1\) or \(0\). The set of index sequences is denoted by \(Z\) (it is homeomorphic to the Cantor set). \((a_n)\in Z\) and \((a_n')\in Z\) are said to be equivalent if \(a_n= a_n'\), \(n> M\) for some \(M\). The quotient space \(X\) of \(Z\) by this relation is the space of F-chains and the \(C^*\)-algebra \(A\) on \(X\) is constructed following the sketch of \textit{A. Connes} [``Noncommutative Geometry'', New York (1994; Zbl 0818.46076)]. The \(K_0(A)\) is \(\mathbb{Z}\oplus\mathbb{Z}\) with the positive cone NEWLINE\[NEWLINEK^+_0(A)= \{(a, b)\in \mathbb{Z}\oplus \mathbb{Z}\mid a+\tau b\geq 0\},NEWLINE\]NEWLINE where \(\tau\) is the golden mean (\(K_1(A)\) vanishes). Compairing \(K\)-theory of noncommutative torus, one might expect to relate \(X\) and the Kronecker foliation, the foliation of the torus whose leaves are the solutions of the equation \(dy- 1/\tau dx\), which is reviewed in section 3. In section 4, F-chains are lifted to a two-dimensional hyperspace, which leads to the torus parametrization of the F-chain. By the equivalence relation of F-chains, this torus a parametrization becomes the Kronecker foliation, and a surjection from \(X\) to the leaf space is obtained. Owing to the identification of \(0\) and \(1\) in the torus, there is one singular leaf which corresponds to two different classes of F-chain. In section 5, the leaf space is extended to be isomorphic to \(X\) and embed the \(C^*\)-algebra of leaves of foliation into an AF algebra on this extended space. The equivalence relations on the extended leaf space is obtained using the equivalence relation of corresponding F-chains in the finite steps. The equivalence partitions the space to the finite intervals. The AF algebra is obtained as an inductive limit of the finite algebra on the space of finite intervals. In concluding remark (section 6), the implication for future research of the results in this paper in the properties of Penrose tiling is discussed.