Topologizing UCTs for unital extensions

Abstract

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.

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Notes

  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.

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Acknowledgements

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).

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Correspondence to Changguo Wei.

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Communicated by Michael Frank.

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Wei, C., Liu, S. & Liu, J. Topologizing UCTs for unital extensions. Banach J. Math. Anal. (2020). https://doi.org/10.1007/s43037-020-00075-w

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Keywords

  • Polish group
  • Extension
  • Ext-group
  • UCT

Mathematics Subject Classification

  • 46L05
  • 22A05