| Title:
|
Pseudometrics on Ext-semigroups (English) |
| Author:
|
Wei, Changguo |
| Author:
|
Zhao, Xiangmei |
| Author:
|
Liu, Shudong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
435-451 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper considers certain pseudometric structures on Ext-semigroups and gives a unified characterization of several topologies on Ext-semigroups. It is demonstrated that these Ext-semigroups are complete topological semigroups. To this end, it is proved that a metric induces a pseudometric on a quotient space with respect to an equivalence relation if it has certain invariance. We give some properties of this pseudometric space and prove that the topology induced by the pseudometric coincides with the one induced by the quotient map. (English) |
| Keyword:
|
pseudometric |
| Keyword:
|
topological group |
| Keyword:
|
extension |
| Keyword:
|
Ext-group |
| MSC:
|
22A05 |
| MSC:
|
46L05 |
| idZBL:
|
07217144 |
| idMR:
|
MR4111852 |
| DOI:
|
10.21136/CMJ.2019.0352-18 |
| . |
| Date available:
|
2020-06-17T12:34:02Z |
| Last updated:
|
2022-07-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148238 |
| . |
| Reference:
|
[1] Arveson, W.: Notes on extensions of $C^*$-algebras.Duke Math. J. 44 (1977), 329-355. Zbl 0368.46052, MR 0438137, 10.1215/S0012-7094-77-04414-3 |
| Reference:
|
[2] Blackadar, B.: $K$-Theory for Operator Algebras.Mathematical Sciences Research Institute Publications 5, Cambridge University Press, Cambridge (1998). Zbl 0913.46054, MR 1656031 |
| Reference:
|
[3] Brown, L. G.: The universal coefficient theorem for Ext and quasidiagonality.Operator Algebras and Group Representations, Vol. I Monographs and Studies in Mathematics 17, Pitman, Boston (1984), 60-64. Zbl 0548.46055, MR 0731763 |
| Reference:
|
[4] Brown, L. G., Douglas, R. G., Fillmore, P. A.: Extensions of $C^*$-algebras, operators with compact self-commutators, and $K$-homology.Bull. Am. Math. Soc. 79 (1973), 973-978. Zbl 0277.46052, MR 0346540, 10.1090/S0002-9904-1973-13284-7 |
| Reference:
|
[5] Brown, L. G., Douglas, R. G., Fillmore, P. A.: Extensions of $C^*$-algebras and $K$-homology.Ann. Math. (2) 105 (1977), 265-324. Zbl 0376.46036, MR 0458196, 10.2307/1970999 |
| Reference:
|
[6] Dadarlat, M.: On the topology of the Kasparov groups and its applications.J. Func. Anal. 228 (2005), 394-418. Zbl 1088.46042, MR 2175412, 10.1016/j.jfa.2005.02.015 |
| Reference:
|
[7] Elliott, G. A., Kucerovsky, D.: An abstract Voiculescu-Brown-Douglas-Fillmore absorption theorem.Pac. J. Math. 198 (2001), 385-409. Zbl 1058.46041, MR 1835515, 10.2140/pjm.2001.198.385 |
| Reference:
|
[8] Kucerovsky, D., Ng, P. W.: The corona factorization property and approximate unitary equivalence.Houston J. Math. 32 (2006), 531-550. Zbl 1111.46050, MR 2219330 |
| Reference:
|
[9] Rosenberg, J., Schochet, C.: The Künneth theorem and the universal coefficient theorem for Kasparov's generalized $K$-functor.Duke Math. J. 55 (1987), 431-474. Zbl 0644.46051, MR 0894590, 10.1215/S0012-7094-87-05524-4 |
| Reference:
|
[10] Salinas, N.: Homotopy invariance of $ Ext(\mathcal A)$.Duke Math. J. 44 (1977), 777-794. Zbl 0391.46057, MR 0512388, 10.1215/S0012-7094-77-04435-0 |
| Reference:
|
[11] Salinas, N.: Quasitriangular extensions of $C^*$-algebras and problems on joint quasitriangularity of operators.J. Oper. Theory 10 (1983), 167-205. Zbl 0539.47011, MR 0715566 |
| Reference:
|
[12] Salinas, N.: Relative quasidiagonality and $KK$-theory.Houston J. Math. 18 (1992), 97-116. Zbl 0772.46039, MR 1159442 |
| Reference:
|
[13] Schochet, C. L.: The fine structure of the Kasparov groups I: Continuity of the $KK$-pairing.J. Func. Anal. 186 (2001), 25-61. Zbl 0990.19003, MR 1863291, 10.1006/jfan.2001.3784 |
| Reference:
|
[14] Schochet, C. L.: The fine structure of the Kasparov groups II: Topologizing the UCT.J. Func. Anal. 194 (2002), 263-287. Zbl 1029.19004, MR 1934604, 10.1006/jfan.2002.3949 |
| Reference:
|
[15] Schochet, C. L.: The fine structure of the Kasparov groups III: Relative quasidiagonality.J. Oper. Theory 53 (2005), 91-117. Zbl 1119.19006, MR 2132689 |
| Reference:
|
[16] Wei, C.: Universal coefficient theorems for the stable Ext-groups.J. Funct. Anal. 258 (2010), 650-664. Zbl 1194.46103, MR 2557950, 10.1016/j.jfa.2009.10.009 |
| Reference:
|
[17] Wei, C.: Classification of extensions of A$\mathbb T$-algebras.Int. J. Math. 22 (2011), 1187-1208. Zbl 1232.46059, MR 2826560, 10.1142/S0129167X11007227 |
| Reference:
|
[18] Wei, C.: On the classification of certain unital extensions of $C^*$-algebras.Houston J. Math. 41 (2015), 965-991. Zbl 1344.46050, MR 3423693 |
| Reference:
|
[19] Wei, C., Liu, S.: On the structure of multiplier algebras.Rocky Mt. J. Math. 47 (2017), 997-1012. Zbl 1380.46042, MR 3682159, 10.1216/RMJ-2017-47-3-997 |
| Reference:
|
[20] Wei, C., Wang, L.: Hereditary $ C^*$-subalgebras and comparison of positive elements.Sci. China, Math. 53 (2010), 1565-1570. Zbl 1200.46050, MR 2658613, 10.1007/s11425-010-4011-x |
| Reference:
|
[21] Wei, C., Wang, L.: Isomorphism of extensions of $C({\mathbb T}^2)$.Sci. China, Math. 54 (2011), 281-286. Zbl 1225.46051, MR 2771204, 10.1007/s11425-010-4132-2 |
| Reference:
|
[22] Xing, R., Wei, C., Liu, S.: Quotient semigroups and extension semigroups.Proc. Indian Acad. Sci., Math. Sci. 122 (2012), 339-350. Zbl 1264.46052, MR 2972657, 10.1007/s12044-012-0086-3 |
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