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Abstract bivariant Cuntz semigroups II

Ramon Antoine, Francesc Perera ORCID logo and Hannes Thiel
From the journal Forum Mathematicum


We previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications. We further analyze the structure of not necessarily commutative Cu-semirings, and we obtain, under mild conditions, a new characterization of solid Cu-semirings R by the condition that RR,R.

Communicated by Siegfried Echterhoff

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM2014-53644-P

Award Identifier / Grant number: MTM2017-83487-P

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: SFB 878 Groups

Award Identifier / Grant number: Geometry & Actions

Funding statement: The two first named authors were partially supported by MINECO (grants MTM2014-53644-P and MTM2017-83487-P), and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The third named author was partially supported by the Deutsche Forschungsgemeinschaft (SFB 878 Groups, Geometry & Actions).


This work was initiated during a research in pairs (RiP) stay at the Oberwolfach Research Institute for Mathematics (MFO) in March 2015. The authors would like to thank the MFO for financial support and for providing inspiring working conditions. Part of this research was conducted while the third named author was visiting the Universitat Autònoma de Barcelona (UAB) in September 2015 and June 2016, and while the first and second named authors visited Münster Universität in June 2015 and 2016. Part of the work was also completed while the second and third named authors were attending the Mittag-Leffler institute during the 2016 program on Classification of Operator Algebras: Complexity, Rigidity, and Dynamics. They would like to thank all the involved institutions for their kind hospitality. The authors also thank the anonymous referee for her or his careful reading of the paper.


[1] R. Antoine, J. Bosa and F. Perera, The Cuntz semigroup of continuous fields, Indiana Univ. Math. J. 62 (2013), no. 4, 1105–1131. 10.1512/iumj.2013.62.5071Search in Google Scholar

[2] R. Antoine, M. Dadarlat, F. Perera and L. Santiago, Recovering the Elliott invariant from the Cuntz semigroup, Trans. Amer. Math. Soc. 366 (2014), no. 6, 2907–2922. 10.1090/S0002-9947-2014-05833-9Search in Google Scholar

[3] R. Antoine, F. Perera and H. Thiel, Abstract bivariant Cuntz semigroups, Int. Math. Res. Not. IMRN (2018), 10.1093/imrn/rny143. 10.1093/imrn/rny143Search in Google Scholar

[4] R. Antoine, F. Perera and H. Thiel, Tensor products and regularity properties of Cuntz semigroups, Mem. Amer. Math. Soc. 251 (2018), no. 1199, 1–191. 10.1090/memo/1199Search in Google Scholar

[5] R. Antoine, F. Perera and H. Thiel, Cuntz semigroups of ultraproducts, preprint (2019), Search in Google Scholar

[6] B. Blackadar, K-theory for Operator Algebras, 2nd ed., Math. Sci. Res. Inst. Publ. 5, Cambridge University, Cambridge, 1998. Search in Google Scholar

[7] N. P. Brown, F. Perera and A. S. Toms, The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras, J. Reine Angew. Math. 621 (2008), 191–211. 10.1515/CRELLE.2008.062Search in Google Scholar

[8] N. P. Brown and A. S. Toms, Three applications of the Cuntz semigroup, Int. Math. Res. Not. IMRN 2007 (2007), no. 19, Article ID rnm068. 10.1093/imrn/rnm068Search in Google Scholar

[9] K. T. Coward, G. A. Elliott and C. Ivanescu, The Cuntz semigroup as an invariant for C*-algebras, J. Reine Angew. Math. 623 (2008), 161–193. 10.1515/CRELLE.2008.075Search in Google Scholar

[10] J. Cuntz, Dimension functions on simple C*-algebras, Math. Ann. 233 (1978), no. 2, 145–153. 10.1007/BF01421922Search in Google Scholar

[11] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, Continuous Lattices and Domains, Encyclopedia Math. Appl. 93, Cambridge University, Cambridge, 2003. 10.1017/CBO9780511542725Search in Google Scholar

[12] B. Jacelon, A simple, monotracial, stably projectionless C-algebra, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 365–383. 10.1112/jlms/jds049Search in Google Scholar

[13] X. Jiang and H. Su, On a simple unital projectionless C*-algebra, Amer. J. Math. 121 (1999), no. 2, 359–413. 10.1353/ajm.1999.0012Search in Google Scholar

[14] G. M. Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. 2005 (2005), no. 10, 1–136. Search in Google Scholar

[15] F. Perera and A. S. Toms, Recasting the Elliott conjecture, Math. Ann. 338 (2007), no. 3, 669–702. 10.1007/s00208-007-0093-3Search in Google Scholar

[16] L. Robert, The cone of functionals on the Cuntz semigroup, Math. Scand. 113 (2013), no. 2, 161–186. 10.7146/math.scand.a-15568Search in Google Scholar

[17] A. S. Toms, On the classification problem for nuclear C-algebras, Ann. of Math. (2) 167 (2008), no. 3, 1029–1044. 10.4007/annals.2008.167.1029Search in Google Scholar

Received: 2018-11-23
Revised: 2019-06-26
Published Online: 2019-09-11
Published in Print: 2020-01-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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