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Operator system nuclearity via \(C^\ast\)-envelopes. (English) Zbl 1368.46044

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).

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

Citations:

Zbl 1304.46050
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[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
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