Hostname: page-component-5d84bcc8dc-swm2b Total loading time: 0 Render date: 2026-09-14T08:11:39.069Z Has data issue: false hasContentIssue false
  • English
  • Français

Shift–tail equivalence and an unbounded representative of the Cuntz–Pimsner extension

Published online by Cambridge University Press:  08 November 2016

MAGNUS GOFFENG ,
BRAM MESLAND  and
ADAM RENNIE
MAGNUS GOFFENG
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden
BRAM MESLAND
Affiliation:
Institut für Analysis, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
ADAM RENNIE
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Northfields Ave, Wollongong, NSW 2522, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

We show how the fine structure in shift–tail equivalence, appearing in the non-commutative geometry of Cuntz–Krieger algebras developed by the first two listed authors, has an analogue in a wide range of other Cuntz–Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz–Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third listed author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz and Cuntz–Krieger algebras and for Cuntz–Pimsner algebras associated to vector bundles twisted by an equicontinuous $\ast$ -automorphism.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 4 , June 2018 , pp. 1389 - 1421
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arici, F. and Rennie, A.. Cuntz–Pimsner extension and mapping cone exact sequences. Preprint, 2016,arXiv:1605.08593.Google Scholar
Baaj, S. and Julg, P.. Théorie bivariante de Kasparov et opérateurs non bornées dans les C ∗ -modules hilbertiens. C. R. Acad. Sci. Paris 296 (1983), 875–878.Google Scholar
Bellissard, J., Marcolli, M. and Reihani, K.. Dynamical systems on spectral metric spaces. Preprint, 2010,arXiv:1008.4617.Google Scholar
Carey, A. L., Neshveyev, S., Nest, R. and Rennie, A.. Twisted cyclic theory, equivariant KK -theory and KMS states. J. Reine Angew. Math. 650 (2011), 161–191.Google Scholar
Carey, A., Phillips, J. and Rennie, A.. Noncommutative Atiyah–Patodi–Singer boundary conditions and index pairings in KK -theory. J. Reine Angew. Math. 643 (2010), 59–109.Google Scholar
Carey, A., Phillips, J. and Rennie, A.. Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. J. K-Theory 6(2) (2010), 339–380.Google Scholar
Cuntz, J. and Krieger, W.. A class of C ∗ -algebras and topological Markov chains. Invent. Math. 56(3) (1980), 251–268.Google Scholar
Dadarlat, M.. The C ∗ -algebra of a vector bundle. J. Reine Angew. Math. 670 (2012), 121–144.Google Scholar
Deaconu, V.. Generalized solenoids and C ∗ -algebras. Pacific J. Math. 190(2) (1999), 247–260.Google Scholar
Deeley, R. J., Goffeng, M., Mesland, B. and Whittaker, M.. Wieler solenoids, Cuntz–Pimsner algebras and $K$ -theory. Preprint, 2016, arXiv:1606.05449.Google Scholar
Gabriel, O. and Grensing, M.. Spectral triples and generalized crossed products. Preprint, 2013,arXiv:1310.5993.Google Scholar
Goffeng, M. and Mesland, B.. Spectral triples and finite summability on Cuntz–Krieger algebras. Doc. Math. 20 (2015), 89–170.Google Scholar
Kajiwara, T., Pinzari, C. and Watatani, Y.. Jones index theory for Hilbert C ∗ -bimodules and its equivalence with conjugation theory. J. Funct. Anal. 215 (2004), 1–49.Google Scholar
Kaminker, J. and Putnam, I.. K-theoretic duality for shifts of finite type. Comm. Math. Phys. 187(3) (1997), 509–522.Google Scholar
Kasparov, G. G.. The operator K-functor and extensions of C ∗ -algebras. Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 571–636. English translation, Math. USSR-Izv. 16 (1981), 513–572.Google Scholar
Katsura, T.. On C ∗ -algebras associated with C ∗ -correspondences. J. Funct. Anal. 217 (2004), 366–401.Google Scholar
Kucerovsky, D.. The KK -product of unbounded modules. K-Theory 11 (1997), 17–34.Google Scholar
Mesland, B.. Unbounded bivariant K-theory and correspondences in noncommutative geometry. J. Reine Angew. Math. 691 (2014), 101–172.Google Scholar
Pimsner, M.. A class of C ∗ -algebras generalising both Cuntz–Krieger algebras and crossed products by ℤ. Free Probability Theory (Fields Institute Communications, 12) . Ed. Voiculescu, D.. American Mathematical Society, Providence, RI, 1997, pp. 189–212.Google Scholar
Renault, J.. A Groupoid Approach to C ∗ -Algebras (Lecture Notes in Mathematics, 793) . Springer, Berlin, 1980.Google Scholar
Renault, J.. Cuntz-like algebras. Operator Theoretical Methods (Timisoara, 1998). Theta Found, Bucharest, 2000, pp. 371–386.Google Scholar
Robertson, D., Rennie, A. and Sims, A.. The extension class and KMS states for Cuntz–Pimsner algebras of some bi-Hilbertian bimodules. J. Topol. Anal. to appear, arXiv:1501:05363.Google Scholar
Szymański, W.. Bimodules for Cuntz–Krieger algebras of infinite matrices. Bull. Aust. Math. Soc. 62 (2000), 87–94.Google Scholar
Vasselli, E.. The C ∗ -algebra of a vector bundle and fields of Cuntz algebras. J. Funct. Anal. 222 (2005), 491–502.Google Scholar