Spectral triples for subshifts

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• 作者：Antoine Julien^a ; ^{antoine.julien@math.ntnu.no" class="auth_mail" title="E-mail the corresponding author} ; Ian Putnam^b ; ^{ifputnam@uvic.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 46L87 ; secondary ; 37B10 ; 54H20 ; 46L55
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 February 2016
• 年：2016
• 卷：270
• 期：3
• 页码：1031-1063
• 全文大小：530 K

文摘

We propose a construction for spectral triple on algebras associated with subshifts. One-dimensional subshifts provide concrete examples of Z$\mathbb{Z}$-actions on Cantor sets. The C^⁎$C^{⁎}$-algebra of this dynamical system is generated by functions in C(X)$C(X)$ and a unitary element u   implementing the action. Building on ideas of Christensen and Ivan, we give a construction of a family of spectral triples on the commutative algebra C(X)$C(X)$. There is a canonical choice of eigenvalues for the Dirac operator D   which ensures that [D,u]$[D,u]$ is bounded, so that it extends to a spectral triple on the crossed product.

We study the summability of this spectral triple, and provide examples for which the Connes distance associated with it on the commutative algebra is unbounded, and some for which it is bounded. We conjecture that our results on the Connes distance extend to the spectral triple defined on the noncommutative algebra.