Czechoslovak Mathematical Journal, Vol. 70, No. 2, pp. 435-451, 2020


Pseudometrics on Ext-semigroups

Changguo Wei, Xiangmei Zhao, Shudong Liu

Received July 29, 2018.   Published online December 10, 2019.

Abstract:  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.
Keywords:  pseudometric; topological group; extension; Ext-group
Classification MSC:  46L05, 22A05
DOI:  10.21136/CMJ.2019.0352-18


References:
[1] W. Arveson: Notes on extensions of $C^*$-algebras. Duke Math. J. 44 (1977), 329-355. DOI 10.1215/S0012-7094-77-04414-3 | MR 0438137 | Zbl 0368.46052
[2] B. Blackadar: $K$-Theory for Operator Algebras. Mathematical Sciences Research Institute Publications 5, Cambridge University Press, Cambridge (1998). MR 1656031 | Zbl 0913.46054
[3] L. G. Brown: 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. MR 0731763 | Zbl 0548.46055
[4] L. G. Brown, R. G. Douglas, P. A. Fillmore: Extensions of $C^*$-algebras, operators with compact self-commutators, and $K$-homology. Bull. Am. Math. Soc. 79 (1973), 973-978. DOI 10.1090/S0002-9904-1973-13284-7 | MR 0346540 | Zbl 0277.46052
[5] L. G. Brown, R. G. Douglas, P. A. Fillmore: Extensions of $C^*$-algebras and $K$-homology. Ann. Math. (2) 105 (1977), 265-324. DOI 10.2307/1970999 | MR 0458196 | Zbl 0376.46036
[6] M. Dadarlat: On the topology of the Kasparov groups and its applications. J. Func. Anal. 228 (2005), 394-418. DOI 10.1016/j.jfa.2005.02.015 | MR 2175412 | Zbl 1088.46042
[7] G. A. Elliott, D. Kucerovsky: An abstract Voiculescu-Brown-Douglas-Fillmore absorption theorem. Pac. J. Math. 198 (2001), 385-409. DOI 10.2140/pjm.2001.198.385 | MR 1835515 | Zbl 1058.46041
[8] D. Kucerovsky, P. W. Ng: The corona factorization property and approximate unitary equivalence. Houston J. Math. 32 (2006), 531-550. MR 2219330 | Zbl 1111.46050
[9] J. Rosenberg, C. Schochet: The Künneth theorem and the universal coefficient theorem for Kasparov's generalized $K$-functor. Duke Math. J. 55 (1987), 431-474. DOI 10.1215/S0012-7094-87-05524-4 | MR 0894590 | Zbl 0644.46051
[10] N. Salinas: Homotopy invariance of $ Ext(\mathcal A)$. Duke Math. J. 44 (1977), 777-794. DOI 10.1215/S0012-7094-77-04435-0 | MR 0512388 | Zbl 0391.46057
[11] N. Salinas: Quasitriangular extensions of $C^*$-algebras and problems on joint quasitriangularity of operators. J. Oper. Theory 10 (1983), 167-205. MR 0715566 | Zbl 0539.47011
[12] N. Salinas: Relative quasidiagonality and $KK$-theory. Houston J. Math. 18 (1992), 97-116. MR 1159442 | Zbl 0772.46039
[13] C. L. Schochet: The fine structure of the Kasparov groups I: Continuity of the $KK$-pairing. J. Func. Anal. 186 (2001), 25-61. DOI 10.1006/jfan.2001.3784 | MR 1863291 | Zbl 0990.19003
[14] C. L. Schochet: The fine structure of the Kasparov groups II: Topologizing the UCT. J. Func. Anal. 194 (2002), 263-287. DOI 10.1006/jfan.2002.3949 | MR 1934604 | Zbl 1029.19004
[15] C. L. Schochet: The fine structure of the Kasparov groups III: Relative quasidiagonality. J. Oper. Theory 53 (2005), 91-117. MR 2132689 | Zbl 1119.19006
[16] C. Wei: Universal coefficient theorems for the stable Ext-groups. J. Funct. Anal. 258 (2010), 650-664. DOI 10.1016/j.jfa.2009.10.009 | MR 2557950 | Zbl 1194.46103
[17] C. Wei: Classification of extensions of A$\mathbb T$-algebras. Int. J. Math. 22 (2011), 1187-1208. DOI 10.1142/S0129167X11007227 | MR 2826560 | Zbl 1232.46059
[18] C. Wei: On the classification of certain unital extensions of $C^*$-algebras. Houston J. Math. 41 (2015), 965-991. MR 3423693 | Zbl 1344.46050
[19] C. Wei, S. Liu: On the structure of multiplier algebras. Rocky Mt. J. Math. 47 (2017), 997-1012. DOI 10.1216/RMJ-2017-47-3-997 | MR 3682159 | Zbl 1380.46042
[20] C. Wei, L. Wang: Hereditary $ C^*$-subalgebras and comparison of positive elements. Sci. China, Math. 53 (2010), 1565-1570. DOI 10.1007/s11425-010-4011-x | MR 2658613 | Zbl 1200.46050
[21] C. Wei, L. Wang: Isomorphism of extensions of $C({\mathbb T}^2)$. Sci. China, Math. 54 (2011), 281-286. DOI 10.1007/s11425-010-4132-2 | MR 2771204 | Zbl 1225.46051
[22] R. Xing, C. Wei, S. Liu: Quotient semigroups and extension semigroups. Proc. Indian Acad. Sci., Math. Sci. 122 (2012), 339-350. DOI 10.1007/s12044-012-0086-3 | MR 2972657 | Zbl 1264.46052

Affiliations:   Changguo Wei (corresponding author), Xiangmei Zhao, School of Mathematical Sciences, Ocean University of China, 238 Songling Road, Qingdao, 266100, P. R. China, e-mail: weicgqd@163.com, 1661337829@qq.com; Shudong Liu, School of Mathematical Sciences, Qufu Normal University, 57 Jingxuan West Road, Qufu, Shandong, 273165, P. R. China, e-mail: lshd008@163.com


 
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