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 is homeomorphic to an inverse limit of metrizable ANRs , that is, . This means that is isomorphic to an inductive limit of the , that is, . 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 is a contractible, compact, metrizable space, and if is an inducitve limit of semiprojective C*-algebras, then so is . 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 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 is an inductive limit of semiprojective C*-algebras. By Example 4.6 in [3], we know that the stabilization is an inductive limit of semiprojective C*-algebras.

- [1]B. Blackadar, Shape theory for -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 -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 *-algebras, J. Noncommut. Geom. 13 (2020) 1435–1462. https://doi.org/10.4171/jncg/350.