##
**Classification of inductive limits of 1-dimensional NCCW complexes.**
*(English)*
Zbl 1268.46041

This paper deals with non-simple \(C^*\)-algebras that are given as inductive limits of certain building blocks. It is shown that such algebras are classified by an invariant that is a variation of the well-known Cuntz semigroup.

The building blocks are one-dimensional NCCW-complexes, as introduced by S. Eilers et al. [J. Reine Angew. Math. 499, 101–143 (1998; Zbl 0897.46056)], with vanishing \(K_1\)-group. Let \(\mathcal{C}\) be the class of \(C^*\)-algebras that are stably isomorphic to an inductive limit of the considered building blocks. It is not known, but quite conceivable, that a \(C^*\)-algebra lies in \(\mathcal{C}\) if it is an inductive limit of one-dimensional NCCW-complexes and its \(K_1\)-group vanishes (one has to show that a trivial \(K_1\)-group of the inductive limit algebra makes it possible to find a suitable inductive limit structure such that each building block has trivial \(K_1\)-group).

The Cuntz semigroup of a \(C^*\)-algebra is an ordered, abelian semigroup given by certain equivalence classes of positive elements (much like the Murray-von Neumann semigroup \(V(A)\) is given by equivalence classes of projections). It was shown that the Cuntz semigroup satisfies many additional properties which were used to define a category \(\mathrm{Cu}\) in [K. T. Coward et al., J. Reine Angew. Math. 623, 161–193 (2008; Zbl 1161.46029)]. The assignment of the Cuntz semigroup \(\mathrm{Cu}(A)\) to a \(C^*\)-algebra \(A\) is a functor that preserves sequential inductive limits. This paper introduces a new functor, denoted by \(\mathrm{Cu}\tilde{\;}\), from stable rank one \(C^*\)-algebras to \(\mathrm{Cu}\).

For simple \(C^*\)-algebras that are inductive limits of one-dimensional NCCW-complexes, it is shown in Section 6 that the new invariant \(\mathrm{Cu}\tilde{\;}\) contains the same information as the ordered \(K_0\)-group, the cone of traces and their pairing. If one additionally assumes that the \(K_1\)-group vanishes (which is, for instance, satisfied for the simple algebras in the class \(\mathcal{C}\)), then \(\mathrm{Cu}\tilde{\;}\) is equivalent to the Elliott invariant. One has to distinguish the two cases that the simple \(C^*\)-algebra \(A\) contains a projection or is projectionless. In the latter case, it is shown that the usual Cuntz semigroup \(\mathrm{Cu}(A)\) contains no information about \(K_0(A)\), which makes apparent the advantage of the new invariant \(\mathrm{Cu}\tilde{\;}(A)\). In the unital case, however, the new invariant is determined by the usual Cuntz semigroup.

Given a \(C^*\)-algebra \(A\) in the class \(\mathcal{C}\) and an arbitrary \(C^*\)-algebra \(B\) of stable rank one, the main result of the paper (Theorem 1.0.1) classifies *-homomorphisms from \(A\) to \(B\) up to approximate unitary equivalence in terms of \(\mathrm{Cu}\tilde{\;}\). Using an approximate intertwining argument, one obtains a classification of the algebras in \(\mathcal{C}\) up to stable isomorphism by the new invariant \(\mathrm{Cu}\tilde{\;}\). By additionally recording the position of a strictly positive element, one achieves classification up to isomorphism.

The article gives two interesting applications. Firstly, it is shown that the Jiang-Su algebra \(\mathcal{Z}\) unitally embeds into the reduced group \(C^*\)-algebra \(C^*(F_\infty)\) of the free group of infinitely many generators. This is done by computing the Cuntz semigroups of both algebras, which turn out to be isomorphic. Using that the Jiang-Su algebra is an element of the class \(\mathcal{C}\) and that \(C^*(F_\infty)\) has stable rank one, the main result allows to lift the map at the level of Cuntz semigroups to a *-homomorphism. Secondly, the paper considers quasi-free actions of the real numbers on the Cuntz algebra \(\mathcal{O}_2\). Such an action is determined by a parameter \(\lambda\), and it is shown that, for a dense set of positive parameters, the resulting crossed products are all pairwise isomorphic.

The building blocks are one-dimensional NCCW-complexes, as introduced by S. Eilers et al. [J. Reine Angew. Math. 499, 101–143 (1998; Zbl 0897.46056)], with vanishing \(K_1\)-group. Let \(\mathcal{C}\) be the class of \(C^*\)-algebras that are stably isomorphic to an inductive limit of the considered building blocks. It is not known, but quite conceivable, that a \(C^*\)-algebra lies in \(\mathcal{C}\) if it is an inductive limit of one-dimensional NCCW-complexes and its \(K_1\)-group vanishes (one has to show that a trivial \(K_1\)-group of the inductive limit algebra makes it possible to find a suitable inductive limit structure such that each building block has trivial \(K_1\)-group).

The Cuntz semigroup of a \(C^*\)-algebra is an ordered, abelian semigroup given by certain equivalence classes of positive elements (much like the Murray-von Neumann semigroup \(V(A)\) is given by equivalence classes of projections). It was shown that the Cuntz semigroup satisfies many additional properties which were used to define a category \(\mathrm{Cu}\) in [K. T. Coward et al., J. Reine Angew. Math. 623, 161–193 (2008; Zbl 1161.46029)]. The assignment of the Cuntz semigroup \(\mathrm{Cu}(A)\) to a \(C^*\)-algebra \(A\) is a functor that preserves sequential inductive limits. This paper introduces a new functor, denoted by \(\mathrm{Cu}\tilde{\;}\), from stable rank one \(C^*\)-algebras to \(\mathrm{Cu}\).

For simple \(C^*\)-algebras that are inductive limits of one-dimensional NCCW-complexes, it is shown in Section 6 that the new invariant \(\mathrm{Cu}\tilde{\;}\) contains the same information as the ordered \(K_0\)-group, the cone of traces and their pairing. If one additionally assumes that the \(K_1\)-group vanishes (which is, for instance, satisfied for the simple algebras in the class \(\mathcal{C}\)), then \(\mathrm{Cu}\tilde{\;}\) is equivalent to the Elliott invariant. One has to distinguish the two cases that the simple \(C^*\)-algebra \(A\) contains a projection or is projectionless. In the latter case, it is shown that the usual Cuntz semigroup \(\mathrm{Cu}(A)\) contains no information about \(K_0(A)\), which makes apparent the advantage of the new invariant \(\mathrm{Cu}\tilde{\;}(A)\). In the unital case, however, the new invariant is determined by the usual Cuntz semigroup.

Given a \(C^*\)-algebra \(A\) in the class \(\mathcal{C}\) and an arbitrary \(C^*\)-algebra \(B\) of stable rank one, the main result of the paper (Theorem 1.0.1) classifies *-homomorphisms from \(A\) to \(B\) up to approximate unitary equivalence in terms of \(\mathrm{Cu}\tilde{\;}\). Using an approximate intertwining argument, one obtains a classification of the algebras in \(\mathcal{C}\) up to stable isomorphism by the new invariant \(\mathrm{Cu}\tilde{\;}\). By additionally recording the position of a strictly positive element, one achieves classification up to isomorphism.

The article gives two interesting applications. Firstly, it is shown that the Jiang-Su algebra \(\mathcal{Z}\) unitally embeds into the reduced group \(C^*\)-algebra \(C^*(F_\infty)\) of the free group of infinitely many generators. This is done by computing the Cuntz semigroups of both algebras, which turn out to be isomorphic. Using that the Jiang-Su algebra is an element of the class \(\mathcal{C}\) and that \(C^*(F_\infty)\) has stable rank one, the main result allows to lift the map at the level of Cuntz semigroups to a *-homomorphism. Secondly, the paper considers quasi-free actions of the real numbers on the Cuntz algebra \(\mathcal{O}_2\). Such an action is determined by a parameter \(\lambda\), and it is shown that, for a dense set of positive parameters, the resulting crossed products are all pairwise isomorphic.

Reviewer: Hannes Thiel (Münster)

### MSC:

46L35 | Classifications of \(C^*\)-algebras |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

46L85 | Noncommutative topology |

### Keywords:

classification of \(C^*\)-algebras; Cuntz semigroup; noncommutative CW-complexes; inductive limit; \(K_1\)-group; stable rank; Jiang-Su algebra### References:

[1] | Antoine, R.; Bosa, J.; Perera, F., Completions of monoids with applications to the Cuntz semigroup, Internat. J. math., 22, 6, 837-861, (2011) · Zbl 1239.46042 |

[2] | Antoine, R.; Perera, F.; Santiago, L., Pullbacks, \(C(X)\)-algebras, and their Cuntz semigroup, J. funct. anal., 260, 10, 2844-2880, (2011) · Zbl 1255.46030 |

[3] | Ara, P.; Perera, F.; Toms, A.S., \(K\)-theory for operator algebras. classification of \(C^\ast\)-algebras, (), 1-71 · Zbl 1219.46053 |

[4] | Brown, N.P.; Ciuperca, A., Isomorphism of Hilbert modules over stably finite \(C^\ast\)-algebras, J. funct. anal., 257, 1, 332-339, (2009) · Zbl 1173.46038 |

[5] | Brown, N.P.; Toms, A.S., Three applications of the Cuntz semigroup, Int. math. res. not. IMRN, 19, (2007), Art. ID rnm068, 14 · Zbl 1134.46040 |

[6] | Ciuperca, A.; Elliott, G.A., A remark on invariants for \(\operatorname{C}^\ast\)-algebras of stable rank one, Int. math. res. not. IMRN, 5, (2008) |

[7] | Ciuperca, A.; Elliott, G.A.; Santiago, L., On inductive limits of type-I \(C^\ast\)-algebras with one-dimensional spectrum, Int. math. res. not. IMRN, 11, 2577-2615, (2011) · Zbl 1232.46051 |

[8] | Ciuperca, A.; Robert, L.; Santiago, L., The Cuntz semigroup of ideals and quotients and a generalized kasparov stabilization theorem, J. oper. theory, 64, 1, 155-169, (2010) · Zbl 1212.46084 |

[9] | Coward, K.T.; Elliott, G.A.; Ivanescu, C., The Cuntz semigroup as an invariant for \(C^\ast\)-algebras, J. reine angew. math., 623, 161-193, (2008) · Zbl 1161.46029 |

[10] | Dean, A., A continuous field of projectionless \(C^\ast\)-algebras, Canad. J. math., 53, 1, 51-72, (2001) · Zbl 0981.46050 |

[11] | Dykema, K.; Haagerup, U.; Rørdam, M., The stable rank of some free product \(C^\ast\)-algebras, Duke math. J., 90, 1, 95-121, (1997) · Zbl 0905.46036 |

[12] | Dykema, K.J.; Rørdam, Mikael, Projections in free product \(C^\ast\)-algebras. II, Math. Z., 234, 1, 103-113, (2000) · Zbl 1014.46037 |

[13] | Dykema, K.J.; Rørdam, M., Projections in free product \(C^\ast\)-algebras, Geom. funct. anal., 8, 1, 1-16, (1998) · Zbl 0907.46045 |

[14] | Eilers, S.; Loring, T.A.; Pedersen, G.K., Stability of anticommutation relations: an application of noncommutative CW complexes, J. reine angew. math., 499, 101-143, (1998) · Zbl 0897.46056 |

[15] | Elliott, G.A., An invariant for simple \(C^\ast\)-algebras, (), 61-90, English, with English and French summaries · Zbl 1206.46046 |

[16] | Elliott, G.A., A classification of certain simple \(C^\ast\)-algebras, () |

[17] | Elliott, G.A., Hilbert modules over a \(\operatorname{C}^\ast\)-algebra of stable rank one, C. R. math. acad. sci. soc. R. can., 29, 2, 48-51, (2007), English, with English and French summaries · Zbl 1151.46043 |

[18] | Elliott, G.A., The classification problem for amenable \(C^\ast\)-algebras, (), 922-932 · Zbl 0946.46050 |

[19] | Elliott, G.A., Towards a theory of classification, Adv. math., 223, 1, 30-48, (2010) · Zbl 1186.18001 |

[20] | Elliott, G.A.; Robert, L.; Santiago, L., The cone of lower semicontinuous traces on a \(C^\ast\)-algebra, Amer. J. math., 133, 4, 969-1005, (2011) · Zbl 1236.46052 |

[21] | Evans, D.E., On \(O_n\), Publ. res. inst. math. sci., 16, 3, 915-927, (1980) · Zbl 0461.46042 |

[22] | B. Jacelon, A simple, self-absorbing, stably projectionless \(\operatorname{C}^\ast\)-algebra, 2010, http://arxiv.org/abs/1006.5397. |

[23] | Jiang, X.; Su, H., A classification of simple limits of splitting interval algebras, J. funct. anal., 151, 1, 50-76, (1997) · Zbl 0921.46057 |

[24] | Kirchberg, E.; Winter, W., Covering dimension and quasidiagonality, Internat. J. math., 15, 1, 63-85, (2004) · Zbl 1065.46053 |

[25] | Kishimoto, A., Simple crossed products of \(C^\ast\)-algebras by locally compact abelian groups, Yokohama math. J., 28, 1-2, 69-85, (1980) · Zbl 0467.46042 |

[26] | Kishimoto, A.; Kumjian, A., Simple stably projectionless \(C^\ast\)-algebras arising as crossed products, Canad. J. math., 48, 5, 980-996, (1996) · Zbl 0865.46054 |

[27] | Ortega, E.; Rørdam, M.; Thiel, H., The Cuntz semigroup and comparison of open projections, J. funct. anal., 260, 12, 3474-3493, (2011) · Zbl 1222.46043 |

[28] | Pedersen, G.K., Unitary extensions and polar decompositions in a \(C^\ast\)-algebra, J. oper. theory, 17, 2, 357-364, (1987) · Zbl 0646.46053 |

[29] | Razak, S., On the classification of simple stably projectionless \(C^\ast\)-algebras, Canad. J. math., 54, 1, 138-224, (2002) · Zbl 1038.46051 |

[30] | Robert, L.; Santiago, L., Classification of \(C^\ast\)-homomorphisms from \(C_0(0, 1]\) to a \(C^\ast\)-algebra, J. funct. anal., 258, 3, 869-892, (2010) · Zbl 1192.46057 |

[31] | Rørdam, M., On the structure of simple \(C^\ast\)-algebras tensored with a UHF-algebra. II, J. funct. anal., 107, 2, 255-269, (1992) · Zbl 0810.46067 |

[32] | Rørdam, M., The stable and the real rank of \(\mathcal{Z}\)-absorbing \(C^\ast\)-algebras, Internat. J. math., 15, 10, 1065-1084, (2004) · Zbl 1077.46054 |

[33] | Rørdam, M.; Winter, W., The jiang – su algebra revisited, J. reine angew. math., 642, 129-155, (2010) · Zbl 1209.46031 |

[34] | L. Santiago, Classification of non-simple \(\operatorname{C}^\ast\)-algebras: inductive limits of splitting interval algebras, Ph.D. Thesis, University of Toronto, 2008. |

[35] | Toms, A.S., Comparison theory and smooth minimal \(C^\ast\)-dynamics, Comm. math. phys., 289, 2, 401-433, (2009) · Zbl 1173.46043 |

[36] | Toms, A.S., K-theoretic rigidity and slow dimension growth, Invent. math., 183, 2, 225-244, (2011) · Zbl 1237.19009 |

[37] | Toms, A.S.; Winter, W., Strongly self-absorbing \(C^\ast\)-algebras, Trans. amer. math. soc., 359, 8, 3999-4029, (2007) · Zbl 1120.46046 |

[38] | Winter, W., Decomposition rank and \(\mathcal{Z}\)-stability, Invent. math., 179, 2, 229-301, (2010) · Zbl 1194.46104 |

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.