A revised augmented Cuntz semigroup
DOI:
https://doi.org/10.7146/math.scand.a-121016Abstract
We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $1$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.
References
Antoine, R., Perera, F., and Thiel, H., Tensor products and regularity properties of Cuntz semigroups, Mem. Amer. Math. Soc. 251 (2018), no. 1199, 191 pp. https://doi.org/10.1090/memo/1199
Blackadar, B., Robert, L., Tikuisis, A. P., Toms, A. S., and Winter, W., An algebraic approach to the radius of comparison, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3657–3674. https://doi.org/10.1090/S0002-9947-2012-05538-3
Brown, N. P. and Ciuperca, A., Isomorphism of Hilbert modules over stably finite $C^*$-algebras, J. Funct. Anal. 257 (2009), no. 1, 332–339. https://doi.org/10.1016/j.jfa.2008.12.004
Ciuperca, A., Some properties of the Cuntz semigroup and an isomorphism theorem for a certain class of non-simple C*-algebras, Ph.D. thesis, University of Toronto, 2008.
Ciuperca, A., Robert, L., and Santiago, L., The Cuntz semigroup of ideals and quotients and a generalized Kasparov stabilization theorem, J. Operator Theory 64 (2010), no. 1, 155–169.
Coward, K. T., Elliott, G. A., and Ivanescu, C., The Cuntz semigroup as an invariant for $C^*$-algebras, J. Reine Angew. Math. 623 (2008), 161–193. https://doi.org/10.1515/CRELLE.2008.075
Elliott, G. A., Gong, G., Lin, H., and Niu, Z., The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension, eprint arXiv:1712.09463 [math.OA], 2017.
Elliott, G. A., Robert, L., and Santiago, L., The cone of lower semicontinuous traces on a $C^*$-algebra, Amer. J. Math. 133 (2011), no. 4, 969–1005. https://doi.org/10.1353/ajm.2011.0027
Jacelon, B., A simple, monotracial, stably projectionless $C^\ast $-algebra, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 365–383. https://doi.org/10.1112/jlms/jds049
Robert, L., Classification of inductive limits of $1$-dimensional NCCW complexes, Adv. Math. 231 (2012), no. 5, 2802–2836. https://doi.org/10.1016/j.aim.2012.07.010
Robert, L., The Cuntz semigroup of some spaces of dimension at most two, C. R. Math. Acad. Sci. Soc. R. Can. 35 (2013), no. 1, 22–32.
Robert, L., Remarks on $\mathcal Z$-stable projectionless $\rm C^*$-algebras, Glasg. Math. J. 58 (2016), no. 2, 273–277. https://doi.org/10.1017/S0017089515000117
Rørdam, M., The stable and the real rank of $\mathcal Z$-absorbing $C^*$-algebras, Internat. J. Math. 15 (2004), no. 10, 1065–1084. https://doi.org/10.1142/S0129167X04002661
Rørdam, M. and Winter, W., The Jiang-Su algebra revisited, J. Reine Angew. Math. 642 (2010), 129–155. https://doi.org/10.1515/CRELLE.2010.039
Tikuisis, A. P. and Toms, A., On the structure of Cuntz semigroups in (possibly) nonunital $\rm C^*$-algebras, Canad. Math. Bull. 58 (2015), no. 2, 402–414. https://doi.org/10.4153/CMB-2014-040-5
Winter, W., Nuclear dimension and $\mathcal Z$-stability of pure $\rm C^*$-algebras, Invent. Math. 187 (2012), no. 2, 259–342. https://doi.org/10.1007/s00222-011-0334-7
Blackadar, B., Robert, L., Tikuisis, A. P., Toms, A. S., and Winter, W., An algebraic approach to the radius of comparison, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3657–3674. https://doi.org/10.1090/S0002-9947-2012-05538-3
Brown, N. P. and Ciuperca, A., Isomorphism of Hilbert modules over stably finite $C^*$-algebras, J. Funct. Anal. 257 (2009), no. 1, 332–339. https://doi.org/10.1016/j.jfa.2008.12.004
Ciuperca, A., Some properties of the Cuntz semigroup and an isomorphism theorem for a certain class of non-simple C*-algebras, Ph.D. thesis, University of Toronto, 2008.
Ciuperca, A., Robert, L., and Santiago, L., The Cuntz semigroup of ideals and quotients and a generalized Kasparov stabilization theorem, J. Operator Theory 64 (2010), no. 1, 155–169.
Coward, K. T., Elliott, G. A., and Ivanescu, C., The Cuntz semigroup as an invariant for $C^*$-algebras, J. Reine Angew. Math. 623 (2008), 161–193. https://doi.org/10.1515/CRELLE.2008.075
Elliott, G. A., Gong, G., Lin, H., and Niu, Z., The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension, eprint arXiv:1712.09463 [math.OA], 2017.
Elliott, G. A., Robert, L., and Santiago, L., The cone of lower semicontinuous traces on a $C^*$-algebra, Amer. J. Math. 133 (2011), no. 4, 969–1005. https://doi.org/10.1353/ajm.2011.0027
Jacelon, B., A simple, monotracial, stably projectionless $C^\ast $-algebra, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 365–383. https://doi.org/10.1112/jlms/jds049
Robert, L., Classification of inductive limits of $1$-dimensional NCCW complexes, Adv. Math. 231 (2012), no. 5, 2802–2836. https://doi.org/10.1016/j.aim.2012.07.010
Robert, L., The Cuntz semigroup of some spaces of dimension at most two, C. R. Math. Acad. Sci. Soc. R. Can. 35 (2013), no. 1, 22–32.
Robert, L., Remarks on $\mathcal Z$-stable projectionless $\rm C^*$-algebras, Glasg. Math. J. 58 (2016), no. 2, 273–277. https://doi.org/10.1017/S0017089515000117
Rørdam, M., The stable and the real rank of $\mathcal Z$-absorbing $C^*$-algebras, Internat. J. Math. 15 (2004), no. 10, 1065–1084. https://doi.org/10.1142/S0129167X04002661
Rørdam, M. and Winter, W., The Jiang-Su algebra revisited, J. Reine Angew. Math. 642 (2010), 129–155. https://doi.org/10.1515/CRELLE.2010.039
Tikuisis, A. P. and Toms, A., On the structure of Cuntz semigroups in (possibly) nonunital $\rm C^*$-algebras, Canad. Math. Bull. 58 (2015), no. 2, 402–414. https://doi.org/10.4153/CMB-2014-040-5
Winter, W., Nuclear dimension and $\mathcal Z$-stability of pure $\rm C^*$-algebras, Invent. Math. 187 (2012), no. 2, 259–342. https://doi.org/10.1007/s00222-011-0334-7
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Published
2021-02-17
How to Cite
Robert, L., & Santiago, L. (2021). A revised augmented Cuntz semigroup. MATHEMATICA SCANDINAVICA, 127(1), 131–160. https://doi.org/10.7146/math.scand.a-121016
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