A bicategorical interpretation for relative Cuntz-Pimsner algebras
DOI:
https://doi.org/10.7146/math.scand.a-112630Abstract
We interpret the construction of relative Cuntz-Pimsner algebras of correspondences in terms of the correspondence bicategory, as a reflector into a certain sub-bicategory. This generalises a previous characterisation of absolute Cuntz-Pimsner algebras of proper correspondences as colimits in the correspondence bicategory.
References
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Albandik, S. and Meyer, R., Colimits in the correspondence bicategory, Münster J. Math. 9 (2016), no. 1, 51–76.
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Buss, A. and Meyer, R., Inverse semigroup actions on groupoids, Rocky Mountain J. Math. 47 (2017), no. 1, 53–159. https://doi.org/10.1216/RMJ-2017-47-1-53
Buss, A., Meyer, R., and Zhu, C., Non-Hausdorff symmetries of $C^* $-algebras, Math. Ann. 352 (2012), no. 1, 73–97. https://doi.org/10.1007/s00208-010-0630-3
Buss, A., Meyer, R., and Zhu, C., A higher category approach to twisted actions on $C^* $-algebras, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2, 387–426. https://doi.org/10.1017/S0013091512000259
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Fowler, N. J., Discrete product systems of Hilbert bimodules, Pacific J. Math. 204 (2002), no. 2, 335–375. https://doi.org/10.2140/pjm.2002.204.335
Gray, J. W., Formal category theory: adjointness for $2$-categories, Lecture Notes in Mathematics, vol. 391, Springer-Verlag, Berlin-New York, 1974.
Gurski, N., Biequivalences in tricategories, Theory Appl. Categ. 26 (2012), no. 14, 349–384.
Katsura, T., On $C^* $-algebras associated with $C^* $-correspondences, J. Funct. Anal. 217 (2004), no. 2, 366–401. https://doi.org/10.1016/j.jfa.2004.03.010
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Kwaśniewski, B. K. and Meyer, R., Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles, Studia Math. 241 (2018), no. 3, 257–303. https://doi.org/10.4064/sm8762-5-2017
Lance, E. C., Hilbert $C^* $-modules: A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511526206
Muhly, P. S. and Solel, B., Tensor algebras over $C^* $-correspondences: representations, dilations, and $C^* $-envelopes, J. Funct. Anal. 158 (1998), no. 2, 389–457. https://doi.org/10.1006/jfan.1998.3294
Muhly, P. S. and Solel, B., On the Morita equivalence of tensor algebras, Proc. London Math. Soc. (3) 81 (2000), no. 1, 113–168. https://doi.org/10.1112/S0024611500012405
Pimsner, M. V., A class of $C^* $-algebras generalizing both Cuntz-Krieger algebras and crossed products by $\mathbb{Z} $, in “Free probability theory (Waterloo, ON, 1995)'', Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189--212.
Schweizer, J., Crossed products by $C^* $-correspondences and Cuntz-Pimsner algebras, in “$C^* $-algebras (Münster, 1999)'', Springer, Berlin, 2000, pp. 203--226.
Sehnem, C. F., On $C^* $-algebras associated to product systems, Ph.D. thesis, Universität Göttingen, 2018, http://hdl.handle.net/11858/00-1735-0000-002E-E3EC-A.
Street, R., Fibrations in bicategories, Cahiers Topologie Géom. Différentielle 21 (1980), no. 2, 111–160.
Albandik, S. and Meyer, R., Colimits in the correspondence bicategory, Münster J. Math. 9 (2016), no. 1, 51–76.
Bénabou, J., Introduction to bicategories, in “Reports of the Midwest Category Seminar”, Springer, Berlin, 1967, pp. 1--77.
Brown, N. P. and Ozawa, N., $C^* $-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. https://doi.org/10.1090/gsm/088
Buss, A. and Meyer, R., Inverse semigroup actions on groupoids, Rocky Mountain J. Math. 47 (2017), no. 1, 53–159. https://doi.org/10.1216/RMJ-2017-47-1-53
Buss, A., Meyer, R., and Zhu, C., Non-Hausdorff symmetries of $C^* $-algebras, Math. Ann. 352 (2012), no. 1, 73–97. https://doi.org/10.1007/s00208-010-0630-3
Buss, A., Meyer, R., and Zhu, C., A higher category approach to twisted actions on $C^* $-algebras, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2, 387–426. https://doi.org/10.1017/S0013091512000259
Echterhoff, S., Kaliszewski, S., Quigg, J., and Raeburn, I., A categorical approach to imprimitivity theorems for $C^* $-dynamical systems, Mem. Amer. Math. Soc. 180 (2006), no. 850, 169 pp. https://doi.org/10.1090/memo/0850
Fiore, T. M., Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory, Mem. Amer. Math. Soc. 182 (2006), no. 860, 171 pp. https://doi.org/10.1090/memo/0860
Fowler, N. J., Discrete product systems of Hilbert bimodules, Pacific J. Math. 204 (2002), no. 2, 335–375. https://doi.org/10.2140/pjm.2002.204.335
Gray, J. W., Formal category theory: adjointness for $2$-categories, Lecture Notes in Mathematics, vol. 391, Springer-Verlag, Berlin-New York, 1974.
Gurski, N., Biequivalences in tricategories, Theory Appl. Categ. 26 (2012), no. 14, 349–384.
Katsura, T., On $C^* $-algebras associated with $C^* $-correspondences, J. Funct. Anal. 217 (2004), no. 2, 366–401. https://doi.org/10.1016/j.jfa.2004.03.010
Katsura, T., Ideal structure of $C^* $-algebras associated with $C^* $-correspondences, Pacific J. Math. 230 (2007), no. 1, 107–145. https://doi.org/10.2140/pjm.2007.230.107
Kwaśniewski, B. K. and Meyer, R., Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles, Studia Math. 241 (2018), no. 3, 257–303. https://doi.org/10.4064/sm8762-5-2017
Lance, E. C., Hilbert $C^* $-modules: A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511526206
Muhly, P. S. and Solel, B., Tensor algebras over $C^* $-correspondences: representations, dilations, and $C^* $-envelopes, J. Funct. Anal. 158 (1998), no. 2, 389–457. https://doi.org/10.1006/jfan.1998.3294
Muhly, P. S. and Solel, B., On the Morita equivalence of tensor algebras, Proc. London Math. Soc. (3) 81 (2000), no. 1, 113–168. https://doi.org/10.1112/S0024611500012405
Pimsner, M. V., A class of $C^* $-algebras generalizing both Cuntz-Krieger algebras and crossed products by $\mathbb{Z} $, in “Free probability theory (Waterloo, ON, 1995)'', Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189--212.
Schweizer, J., Crossed products by $C^* $-correspondences and Cuntz-Pimsner algebras, in “$C^* $-algebras (Münster, 1999)'', Springer, Berlin, 2000, pp. 203--226.
Sehnem, C. F., On $C^* $-algebras associated to product systems, Ph.D. thesis, Universität Göttingen, 2018, http://hdl.handle.net/11858/00-1735-0000-002E-E3EC-A.
Street, R., Fibrations in bicategories, Cahiers Topologie Géom. Différentielle 21 (1980), no. 2, 111–160.
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Published
2019-08-29
How to Cite
Meyer, R., & Sehnem, C. F. (2019). A bicategorical interpretation for relative Cuntz-Pimsner algebras. MATHEMATICA SCANDINAVICA, 125(1), 84–112. https://doi.org/10.7146/math.scand.a-112630
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