Gupta, Ved Prakash; Luthra, Preeti Operator system nuclearity via \(C^\ast\)-envelopes. (English) Zbl 1368.46044 J. Aust. Math. Soc. 101, No. 3, 356-375 (2016). Summary: We prove that an operator system is (min, ess)-nuclear if its \(C^\ast\)-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of a countable discrete group by D. Farenick et al. [Commun. Math. Phys. 329, No. 1, 207–238 (2014; Zbl 1304.46050)] is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (respectively, a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to three (respectively, every component of the graph is complete). Cited in 4 Documents MSC: 46L06 Tensor products of \(C^*\)-algebras 46L07 Operator spaces and completely bounded maps 47L25 Operator spaces (= matricially normed spaces) 46M05 Tensor products in functional analysis 46L35 Classifications of \(C^*\)-algebras Keywords:operator systems; amenable groups; graphs; nuclearity; \(C^\ast\)-envelopes; tensor products Citations:Zbl 1304.46050 PDFBibTeX XMLCite \textit{V. P. Gupta} and \textit{P. Luthra}, J. Aust. Math. Soc. 101, No. 3, 356--375 (2016; Zbl 1368.46044) Full Text: DOI arXiv References: [1] DOI: 10.1017/CBO9781107360235 · doi:10.1017/CBO9781107360235 [2] DOI: 10.1007/s00020-013-2087-8 · Zbl 1294.46049 · doi:10.1007/s00020-013-2087-8 [3] DOI: 10.1016/j.laa.2015.03.022 · Zbl 1327.46053 · doi:10.1016/j.laa.2015.03.022 [4] DOI: 10.1016/0022-1236(73)90021-9 · Zbl 0252.46065 · doi:10.1016/0022-1236(73)90021-9 [5] DOI: 10.1006/jfan.1997.3226 · Zbl 0940.46038 · doi:10.1006/jfan.1997.3226 [6] DOI: 10.7146/math.scand.a-15225 · Zbl 1273.46038 · doi:10.7146/math.scand.a-15225 [7] DOI: 10.1016/j.aim.2012.05.025 · Zbl 1325.46060 · doi:10.1016/j.aim.2012.05.025 [8] DOI: 10.1007/s00220-014-2037-6 · Zbl 1304.46050 · doi:10.1007/s00220-014-2037-6 [9] DOI: 10.1016/j.jfa.2011.03.014 · Zbl 1235.46051 · doi:10.1016/j.jfa.2011.03.014 [10] DOI: 10.7900/jot.2011oct07.1938 · Zbl 1299.46060 · doi:10.7900/jot.2011oct07.1938 [11] DOI: 10.7900/jot.2011nov16.1977 · Zbl 1349.46060 · doi:10.7900/jot.2011nov16.1977 [12] Effros, Operator Spaces (2000) [13] DOI: 10.1016/j.jfa.2011.04.009 · Zbl 1223.46053 · doi:10.1016/j.jfa.2011.04.009 [14] DOI: 10.1016/j.jmaa.2011.05.070 · Zbl 1242.46067 · doi:10.1016/j.jmaa.2011.05.070 [15] DOI: 10.1016/0022-1236(77)90052-0 · Zbl 0341.46049 · doi:10.1016/0022-1236(77)90052-0 [16] DOI: 10.2977/prims/1195187876 · Zbl 0436.46046 · doi:10.2977/prims/1195187876 [17] Brown, C*-Algebras and Finite-Dimensional Approximations Vol. 88 (2008) · Zbl 1160.46001 [18] DOI: 10.1007/978-3-319-16718-3 · Zbl 1431.81007 · doi:10.1007/978-3-319-16718-3 [19] DOI: 10.1007/BF02392388 · Zbl 0194.15701 · doi:10.1007/BF02392388 [20] DOI: 10.7153/oam-09-19 · Zbl 1321.46061 · doi:10.7153/oam-09-19 [21] Paulsen, Completely Bounded Maps and Operator Algebras Vol. 78 (2002) · Zbl 1029.47003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.