Inductive limits of semiprojective C*-algebras

Question: Is every separable C*-algebra an inductive limit of semiprojective C*-algebras?

This question was first raised by Blackadar in [1]. If we think of C*-algebras as noncommutative topological spaces, then semiprojective C*-algebras are noncommutative absolute neighborhood retracts (ANRs). It is a classical result from shape theory that every metrizable space $X$ is homeomorphic to an inverse limit of metrizable ANRs $X_n$ , that is, $X\cong\varprojlim_n X_n$ . This means that $C(X)$ is isomorphic to an inductive limit of the $C(X_n)$ , that is, $C(X)\cong\varinjlim_n C(X_n)$ . The question is if the noncommutative analog of this result also holds.

It has been verified for many classes of C*-algebras that they are inductive limits of semiprojective C*-algebras. For example, Enders showed in [2] that every UCT-Kirchberg algebra is an inductive limit of semiprojective C*-algebras. In [3], it was shown that the class of C*-algebras that are inductive limits of semiprojective C*-algebras is closed under shape domination, and in particular under homotopy equivalence. One deduces, for example, that if $X$ is a contractible, compact, metrizable space, and if $A$ is an inducitve limit of semiprojective C*-algebras, then so is $C(X,A)$ . It also follows that every contractible C*-algebra is an inductive limit of semiprojective C*-algebras – in fact, even of projective C*-algebras, as was shown in [4].

The commutative C*-algebra $C(S^2)$ is probably the easiest C*-algebra where it is currently unknown if it is an inductive limit of semiprojective C*-algebras. Equivalently, it is unknown if $C_0(\mathbb{R}^2)$ is an inductive limit of semiprojective C*-algebras. By Example 4.6 in [3], we know that the stabilization $C_0(\mathbb{R}^2)\otimes\mathcal{K}$ is an inductive limit of semiprojective C*-algebras.

[1]
B. Blackadar, Shape theory for $C^*$ -algebras, MATH. SCAND. 56 (1985) 249–275. https://doi.org/10.7146/math.scand.a-12100.
[2]
D. Enders, Semiprojectivity for Kirchberg algebras, ArXiv Preprint ArXiv:1507.06091. (2015).
[3]
H. Thiel, Inductive limits of semiprojective $C^*$ -algebras, Advances in Mathematics. 347 (2019) 597–618. https://doi.org/10.1016/j.aim.2019.02.030.
[4]
H. Thiel, Inductive limits of projective $C$ *-algebras, J. Noncommut. Geom. 13 (2020) 1435–1462. https://doi.org/10.4171/jncg/350.

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