Skip to main content Accessibility help

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



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.



Hide All
[1] Arici, F. and Rennie, A.. Cuntz–Pimsner extension and mapping cone exact sequences. Preprint, 2016,arXiv:1605.08593.
[2] 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), 875878.
[3] Bellissard, J., Marcolli, M. and Reihani, K.. Dynamical systems on spectral metric spaces. Preprint, 2010,arXiv:1008.4617.
[4] 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), 161191.
[5] 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), 59109.
[6] 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), 339380.
[7] Cuntz, J. and Krieger, W.. A class of C -algebras and topological Markov chains. Invent. Math. 56(3) (1980), 251268.
[8] Dadarlat, M.. The C -algebra of a vector bundle. J. Reine Angew. Math. 670 (2012), 121144.
[9] Deaconu, V.. Generalized solenoids and C -algebras. Pacific J. Math. 190(2) (1999), 247260.
[10] Deeley, R. J., Goffeng, M., Mesland, B. and Whittaker, M.. Wieler solenoids, Cuntz–Pimsner algebras and $K$ -theory. Preprint, 2016, arXiv:1606.05449.
[11] Gabriel, O. and Grensing, M.. Spectral triples and generalized crossed products. Preprint, 2013,arXiv:1310.5993.
[12] Goffeng, M. and Mesland, B.. Spectral triples and finite summability on Cuntz–Krieger algebras. Doc. Math. 20 (2015), 89170.
[13] 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), 149.
[14] Kaminker, J. and Putnam, I.. K-theoretic duality for shifts of finite type. Comm. Math. Phys. 187(3) (1997), 509522.
[15] Kasparov, G. G.. The operator K-functor and extensions of C -algebras. Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 571636. English translation, Math. USSR-Izv. 16 (1981), 513–572.
[16] Katsura, T.. On C -algebras associated with C -correspondences. J. Funct. Anal. 217 (2004), 366401.
[17] Kucerovsky, D.. The KK -product of unbounded modules. K-Theory 11 (1997), 1734.
[18] Mesland, B.. Unbounded bivariant K-theory and correspondences in noncommutative geometry. J. Reine Angew. Math. 691 (2014), 101172.
[19] 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. 189212.
[20] Renault, J.. A Groupoid Approach to C -Algebras (Lecture Notes in Mathematics, 793) . Springer, Berlin, 1980.
[21] Renault, J.. Cuntz-like algebras. Operator Theoretical Methods (Timisoara, 1998). Theta Found, Bucharest, 2000, pp. 371386.
[22] 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.
[23] Szymański, W.. Bimodules for Cuntz–Krieger algebras of infinite matrices. Bull. Aust. Math. Soc. 62 (2000), 8794.
[24] Vasselli, E.. The C -algebra of a vector bundle and fields of Cuntz algebras. J. Funct. Anal. 222 (2005), 491502.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed