## Generalized quasidiagonality for extensions.(English)Zbl 1425.46040

The authors suggest a generalization for quasidiagonality of $$C^*$$-algebras that avoids the use of projections.
Let $$B$$ be a $$\sigma$$-unital non-unital $$C^*$$-algebra. A sequence $$\{b_n\}_{n\in\mathbb N}$$ of norm 1 positive elements of $$B$$ is a system of quasidiagonal units if $$b_nb_m=0$$ for $$n\neq m$$ and $$\sum_{n\in\mathbb N}x_n$$ is strictly convergent in the multiplier algebra $$M(B)$$ for any bounded sequence $$x_n\in b_nBb_n$$. Further, $$x\in M(B)$$ is generalized block-diagonal if $$x=\sum_{n\in\mathbb N} x_n$$ for some $$x_n\in b_nBb_n$$, and a $$C^*$$-subalgebra $$A\subset M(B)$$ is generalized quasidiagonal if any $$a\in A$$ can be written as $$a=x+b$$, where $$x$$ is generalized block-diagonal and $$b\in B$$.
It is shown that some results on quasidiagonal $$C^*$$-algebras hold also for the generalized ones. E.g., if $$B$$ is simple, then a pointwise-norm limit of generalized quasidiagonal extensions is generalized quasidiagonal.
Some $$K$$-theory conditions are given that provide generalized quasidiagonality of extensions of a commutative $$C^*$$-algebra by a continuous-scale $$C^*$$-algebra.

### MSC:

 46L05 General theory of $$C^*$$-algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory)
Full Text:

### References:

 [1] I. D. Berg, An extension of the Weyl – von Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365-371. · Zbl 0212.15903 [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. · Zbl 0204.15001 [5] X. Jiang and H. Su, On a simple unital projectionless $$C^{*}$$-algebra, Amer. J. Math. 121 (1999), no. 2, 359-413. · Zbl 0923.46069 [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. · Zbl 0448.46051 [7] H. Lin, Generalized Weyl – von Neumann theorems, Internat. J. Math. 2 (1991), no. 6, 725-739. · Zbl 0768.46035 [8] H. Lin, Extensions by $$C^{*}$$-algebras of real rank zero, II, Proc. Lond. Math. Soc. (3) 71 (1995), no. 3, 641-674. · Zbl 0837.46042 [9] H. Lin, Generalized Weyl – von Neumann theorems, II, Math. Scand. 77 (1995), no. 1, 129-147. · Zbl 0852.46051 [10] H. Lin, Extensions by $$C^{*}$$-algebras of real rank zero, III, Proc. Lond. Math. Soc. (3) 76 (1998), no. 3, 634-666. · Zbl 0911.46043 [11] H. Lin, Extensions by simple $$C^{*}$$-algebras: Quasidiagonal extensions, Canad. J. Math. 57 (2005), no. 2, 351-399. · Zbl 1087.46039 [12] H. Lin and P. W. Ng, The corona algebra of stabilized Jiang-Su algebra, J. Funct. Anal. 270 (2016), no. 3, 1220-1267. · Zbl 1410.46047 [13] P. W. Ng, Nonstable absorption, Houston J. Math. 44 (2018), no. 3, 975-1017. · Zbl 1436.46057 [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. · Zbl 1038.46051 [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. · Zbl 0747.46043 [18] S. Zhang, $$K_{1}$$-groups, quasidiagonality, and interpolation by multiplier projections, Trans. Amer. Math. Soc. 325 (1991), no. 2, 793-818. · Zbl 0673.46050 [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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.