Stable homotopy, $1$-dimensional NCCW complexes, and Property (H)
In this paper, we show that the homomorphisms between two unital one-dimensional NCCW complexes with the same KK-class are stably homotopic, that is, after adding on a common homomorphism (with finite dimensional image), they are homotopic. As a consequence, any one-dimensional NCCW complex has the Property (H).
An, Q., Liu, Z., and Zhang, Y., On the classification of certain real rank zero $\mathrm C^*$-algebras, eprint arXiv:1902.00124v2 [math.OA], 2019.
Blackadar, B., The homotopy lifting theorem for semiprojective $C^*$-algebras, Math. Scand. 118 (2016), no. 2, 291–302. https://doi.org/10.7146/math.scand.a-23691
Dădărlat, M., Reduction to dimension three of local spectra of real rank zero $C^*$-algebras, J. Reine Angew. Math. 460 (1995), 189–212. https://doi.org/10.1515/crll.1995.460.189
Dădărlat, M. and Gong, G., A classification result for approximately homogeneous $C^*$-algebras of real rank zero, Geom. Funct. Anal. 7 (1997), no. 4, 646–711. https://doi.org/10.1007/s000390050023
Eilers, S., Loring, T. A., and Pedersen, G. K., Stability of anticommutation relations: an application of noncommutative CW complexes, J. Reine Angew. Math. 499 (1998), 101–143.
Eilers, S., Loring, T. A., and Pedersen, G. K., Fragility of subhomogeneous $C^*$-algebras with one-dimensional spectrum, Bull. London Math. Soc. 31 (1999), no. 3, 337–344. https://doi.org/10.1112/S002460939800558X
Elliott, G. A., An invariant for simple $C^*$-algebras, in “Canadian Mathematical Society. 1945–1995, Vol. 3”, Canadian Math. Soc., Ottawa, ON, 1996, pp. 61–90.
Elliott, G. A. and Thomsen, K., The state space of the $K_0$-group of a simple separable $C^*$-algebra, Geom. Funct. Anal. 4 (1994), no. 5, 522–538. https://doi.org/10.1007/BF01896406
Gong, G., Lin, H., and Niu, Z., Classification of finite simple amenable $\mathcal Z$-stable $C^*$-algebras, eprint arXiv:1501.00135v6 [math.OA], 2015.
Liu, Z., Injectivity of the connecting homomorphisms in inductive limits of Elliott-Thomsen algebras, Canad. Math. Bull. 62 (2019), no. 1, 131–148. https://doi.org/10.4153/cmb-2018-020-2
Thomsen, K., From trace states to states on the $K_0$ group of a simple $C^*$-algebra, Bull. London Math. Soc. 28 (1996), no. 1, 66–72. https://doi.org/10.1112/blms/28.1.66