Topologizing UCTs for unital extensions


In this paper, we study the topological structure on Ext-groups of unital extensions. It is proved that the stable strong Ext-group and the stable weak Ext-group for unital extensions are pseudopolish groups if A is a unital \(C^*\)-algebra in the bootstrap class \({{\mathcal {N}}}\) and B is a stable separable \(C^*\)-algebra. Furthermore, we topologize certain UCTs and prove that these UCTs are exact sequences as topological groups.

This is a preview of subscription content, log in to check access.


  1. 1.

    The UCT2 was proved in [27] under the assumption that B has an approximate identity of projections. We are grateful to a referee for his providing the following reference in which a proof of the UCT2 without this condition was given for separable \(C^*\)-algebras: [8].

  2. 2.

    The completeness of \(E^{ua}(A,B)\) was proved in [16, Remark 3.1] and a similar argument holds for the completeness of \(E^{a}(A,B)\).

  3. 3.

    In [17, P. 49] C. Schochet adopted a different approach to show that \(E \text {xt}(A,B)\) is a separable complete pseudometric groups under the additional assumption that B is also separable.


  1. 1.

    Blackadar, B.: \(K\)-Theory for Operator Algebras, 2nd edn. Mathematical Sciences Research Institute Publications, vol. 5. Cambridge University Press, Cambridge (1998)

  2. 2.

    Brown, L.G.: Operator algebras and algebraic K-theory. Bull. Am. Math. Soc. 81(6), 1119–1121 (1975)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brown, L.G.: The Universal Coefficient Theorem for \({\rm Ext}\) and Quasidiagonality, Operator Algebras and Group Representations, vol. I (Neptun, 1980), pp. 60–64. Monographs and Studies in Mathematics, vol. 17. Pitman, Boston, MA (1984)

  4. 4.

    Brown, L.G., Dadarlat, M.: Extensions of \(C^*\)-algebras and quasidiagonality. J. Lond. Math. Soc. 53(2), 582–600 (1996)

    MathSciNet  Article  Google Scholar 

  5. 5.

    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, 973–978 (1973)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Brown, L.G., Douglas, R.G., Fillmore, P.A.: Extensions of \(C^*\)-algebras and \(K\)-homology. Ann. Math. 105, 265–324 (1977)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dadarlat, M.: On the topology of the Kasparov groups and its applications. J. Funct. Anal. 228(2), 394–418 (2005)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Gabe, J., Ruiz, E.: The unital Ext-groups and classification of \(C^*\)-algebras. Glasg. Math. J. 62, 201–231 (2020)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kuratowski, C.: Topologie, vol. 1, 3rd edn. PWN, Warsaw (1952)

  10. 10.

    Lin, H.: Unitary equivalences for essential extensions of \(C^*\)-algebras. Proc. Am. Math. Soc. 137(10), 3407–3420 (2009)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Moore, C.C.: Group extensions and cohomology for locally compact groups. III. Trans. Am. Math. Soc. 221, 1–33 (1976)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Rordam, M., Larsen, F., Laustsen, N.: An Introduction to \(K\)-Theory for \(C^*\)-Algebras. London Mathematical Society Student Texts, vol. 49. Cambridge University Press, Cambridge (2000)

  13. 13.

    Rosenberg, J., Schochet, C.: The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized \(K\)-functor. Duke Math. J. 55(2), 431–474 (1987)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Salinas, N.: Homotopy invariance of Ext(A). Duke Math. J. 44(4), 777–794 (1977)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Salinas, N.: Quasitriangular extensions of \(C^*\)-algebras and problems on joint quasitriangularity of operators. J. Oper. Theory 10, 167–205 (1983)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Salinas, N.: Relative quasidiagonality and KK-theory. Houst. J. Math. 18, 97–116 (1992)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Schochet, C.: The fine structure of the Kasparov groups I: continuity of the KK-pairing. J. Funct. Anal. 186(1), 25–61 (2001)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Schochet, C.: The fine structure of the Kasparov groups II: topologizing the UCT. J. Funct. Anal. 194, 263–287 (2002)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Schochet, C.: The fine structure of the Kasparov groups III: relative quasidiagonality. J. Operat. Theory 53(1), 91–117 (2005)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Skandalis, G.: On the strong Ext bifunctor. Preprint (1983)

  21. 21.

    Skandalis, G.: Kasparovs bivariant K-theory and applications. Expos. Math. 9, 193–250 (1991)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Thomsen, K.: On absorbing extensions. Proc. Am. Math. Soc. 129(5), 1409–1417 (2001)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Wei, C.: Universal coefficient theorems for the stable Ext-groups. J. Funct. Anal. 258, 650–664 (2010)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Wei, C.: Classification of extensions of AT-algebras. Int. J. Math. 22(8), 1187–1208 (2011)

    Article  Google Scholar 

  25. 25.

    Wei, C.: Classification of extensions of torus algebra II. Sci. China Math. 55(1), 179–186 (2012)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Wei, C.: Classification of unital extensions of AT-algebras. J. Math. Anal. Appl. 397(2), 757–765 (2013)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Wei, C.: On the classification of certain unital extensions of \(C^*\)-algebras. Houst. J. Math. 41(3), 965–991 (2015)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Wei, C., Liu, S.: On the structure of multiplier algebras. Rocky Mt. J. Math. 47(3), 997–1012 (2017)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Wei, C., Wang, L.: Hereditary subalgebras and comparisons of positive elements. Sci. China Math. 53(6), 1565–1570 (2010)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Wei, C., Wang, L.: Isomorphism of extensions of \(C(T^2)\). Sci. China Math. 54(2), 281–286 (2011)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Wei, C., Zhao, X., Liu, S.: Pseudometrics on Ext-semigroups. Czech. Math. J. (2019).

  32. 32.

    Xing, R., Wei, C., Liu, S.: Quotient semigroups and extension semigroups. Proc. Indian Acad. Sci. Math. Sci. 122(3), 339–350 (2012)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Xue, Y.: The Ext-group of unitary equivalence classes of unital extensions. Acta Math. Sin. Engl. Ser. 27(12), 2329–2342 (2011)

    MathSciNet  Article  Google Scholar 

Download references


The authors are very grateful to the referees for their suggestions. This work was supported by the Fundamental Research Funds for the Central Universities (Grant no. 3016000841964007) and the Shandong Provincial Natural Science Foundation (Grant no. ZR2018MA006).

Author information



Corresponding author

Correspondence to Changguo Wei.

Additional information

Communicated by Michael Frank.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wei, C., Liu, S. & Liu, J. Topologizing UCTs for unital extensions. Banach J. Math. Anal. (2020).

Download citation


  • Polish group
  • Extension
  • Ext-group
  • UCT

Mathematics Subject Classification

  • 46L05
  • 22A05