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A CHARACTERISATION OF TRACIALLY NUCLEAR C*-ALGEBRAS

  • DON HADWIN (a1), WEIHUA LI (a2), WENJING LIU (a3) and JUNHAO SHEN (a4)

Abstract

We give two characterisations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for all tracially nuclear C*-algebras. When the algebra is separable, we prove the sufficiency.

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The first three authors are, respectively, supported by a Collaboration Grant from the Simons Foundation, a Faculty Development Grant from Columbia College Chicago and a Dissertation Year Fellowship from the University of New Hampshire.

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[1] Ciuperca, A., Giordano, T., Ng, P. W. and Niu, Z., ‘Amenability and uniqueness’, Adv. Math. 240 (2013), 325345.
[2] Connes, A., ‘Classification of injective factors’, Ann. of Math. (2) 104 (1976), 73115.
[3] Ding, H. and Hadwin, D., ‘Approximate equivalence in von Neumann algebras’, Sci. China Ser. A 48(2) (2005), 239247.
[4] Dostál, M. and Hadwin, D., ‘An alternative to free entropy for free group factors’, Acta Math. Sin. (Engl. Ser.) 19(3) (2003), 419472. International Workshop on Operator Algebra and Operator Theory (Linfen, 2001).
[5] Hadwin, D. and Li, W., ‘The similarity degree of some C*-algebras’, Bull. Aust. Math. Soc. 89(1) (2014), 6069.
[6] Hadwin, D. and Liu, W., ‘Approximate unitary equivalence relative to ideals in semi-finite von Neumann algebras’, Preprint, 2018.
[7] Hadwin, D. and Shulman, T., ‘Tracial stability for C*-algebras’, Integral Equations Operator Theory 90(1) (2018), available at https://doi.org/10.1007/s00020-018-2430-1.
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