Banach Journal of Mathematical Analysis

Generalized quasidiagonality for extensions

P. W. Ng and Tracy Robin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We generalize the notion of quasidiagonality, for extensions, allowing for the case where the canonical ideal has few projections. We prove that the pointwise-norm limit of generalized quasidiagonal extensions is generalized quasidiagonal. We also provide a K-theory sufficient condition for generalized quasidiagonality of certain extensions of simple continuous-scale C-algebras, including certain continuous-scale hereditary C-subalgebras of the stabilized Jiang–Su algebra.

Article information

Banach J. Math. Anal., Volume 13, Number 3 (2019), 582-598.

Received: 28 September 2018
Accepted: 3 January 2019
First available in Project Euclid: 25 May 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Secondary: 47A10: Spectrum, resolvent 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]

$C^{*}$-algebras multiplier algebras extension theory Weyl–von Neumann–Berg theorem $KK$-theory


Ng, P. W.; Robin, Tracy. Generalized quasidiagonality for extensions. Banach J. Math. Anal. 13 (2019), no. 3, 582--598. doi:10.1215/17358787-2019-0005.

Export citation


  • [1] I. D. Berg, An extension of the Weyl–von Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365–371.
  • [2] L. G. Brown, R. G. Douglas, and P. A. Fillmore, “Unitary equivalence modulo the compact operators and extensions of $C^{*}$-algebras” in Operator Theory (Halifax, 1973), Lecture Notes in Math. 345, Springer, Berlin, 1973, 58–128.
  • [3] L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of $C^{*}$-algebras and $K$-homology, Ann. of Math. (2) 105 (1977), no. 2, 265–324.
  • [4] P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. (N.S.) 76 (1970), 887–933.
  • [5] X. Jiang and H. Su, On a simple unital projectionless $C^{*}$-algebra, Amer. J. Math. 121 (1999), no. 2, 359–413.
  • [6] G. G. Kasparov, The operator K-functor and extensions of C∗-algebras (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 44 (1980), no. 3, 571–636; English translation in Izv. Math. 16 (1981), no. 3, 513–572.
  • [7] H. Lin, Generalized Weyl–von Neumann theorems, Internat. J. Math. 2 (1991), no. 6, 725–739.
  • [8] H. Lin, Extensions by $C^{*}$-algebras of real rank zero, II, Proc. Lond. Math. Soc. (3) 71 (1995), no. 3, 641–674.
  • [9] H. Lin, Generalized Weyl–von Neumann theorems, II, Math. Scand. 77 (1995), no. 1, 129–147.
  • [10] H. Lin, Extensions by $C^{*}$-algebras of real rank zero, III, Proc. Lond. Math. Soc. (3) 76 (1998), no. 3, 634–666.
  • [11] H. Lin, Extensions by simple $C^{*}$-algebras: Quasidiagonal extensions, Canad. J. Math. 57 (2005), no. 2, 351–399.
  • [12] H. Lin and P. W. Ng, The corona algebra of stabilized Jiang–Su algebra, J. Funct. Anal. 270 (2016), no. 3, 1220–1267.
  • [13] P. W. Ng, Nonstable absorption, Houston J. Math. 44 (2018), no. 3, 975–1017.
  • [14] G. K. Pedersen, $C^{*}$-Algebras and Their Automorphism Groups, London Math. Soc. Monogr. 14, Academic Press, London, 1979.
  • [15] S. Razak, On the classification of simple stably projectionless $C^{*}$-algebras, Canad. J. Math. 54 (2002), no. 1, 138–224.
  • [16] D. Voiculescu, A noncommutative Weyl–von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113.
  • [17] S. Zhang, A Riesz decomposition property and ideal structure of multiplier algebras, J. Operator Theory 24 (1990), no. 2, 209–225.
  • [18] S. Zhang, $K_{1}$-groups, quasidiagonality, and interpolation by multiplier projections, Trans. Amer. Math. Soc. 325 (1991), no. 2, 793–818.
  • [19] S. Zhang, Certain $C^{*}$-algebras with real rank zero and their corona and multiplier algebras, I, Pacific J. Math. 155 (1992), no. 1, 169–197.