An analytic Novikov conjecture for semigroups. (English) Zbl 1403.19006
The author formulates an analytic assembly map for semigroups and conjectures that the map is injective for torsion-free semigroups, which is an analytic Novikov conjecture for semigroups. The author also shows that the decent argument from the coarse Baum-Connes conjecture works in the same way as it does for groups. In order to carry it out, the author generalizes parts of equivariant homology theory for group actions to the semigroup case. The paper also includes some examples where the decent argument works.
Reviewer: Tatsuki Seto (Nagoya)
MSC:
| 19K56 | Index theory |
References:
| [1] | P. Baum, A. Connes and N. Higson, Classifying spaces for proper actions and \(K\)-theory of group \(C^⁎\)-algebras, Contemp. Math. 167 (1994), 241–291. · Zbl 0830.46061 |
| [2] | A.H. Clifford and G.B. Preston, The algebraic theory of semigroups, Math. Surv. 7, American Mathematical Society, 1961. · Zbl 0111.03403 |
| [3] | J. Cuntz, \(C^⁎\) algebras associated with the \(ax+b\) semigroup over \(\mathbb N\), in \(K\)-theory and noncommutative geometry, Europ. Math. Soc. (2008), 201–216. · Zbl 1162.46036 |
| [4] | J. Cuntz, S. Echterhoff and X. Li, \(K\)-theory of semigroup \(C^⁎\)-algebras, arXiv:1205.5412v1, 2012. |
| [5] | N. Higson, V. Lafforgue and G. Skandalis, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), 330–354. · Zbl 1014.46043 · doi:10.1007/s00039-002-8249-5 |
| [6] | N. Higson and J. Roe, Analytic \(K\)-homology, Oxford Math. Mono., Oxford University Press, Oxford, 2000. · Zbl 0968.46058 |
| [7] | X. Li, Semigroup \(C^⁎\)-algebras and amenability of semigroups, arXiv: 1055.5539v1, 2011. |
| [8] | J. Milnor, Construction of universal bundles, II, Ann. Math. 63 (1956), 430–436. · Zbl 0071.17401 |
| [9] | P.D. Mitchener, Coarse homology theories, Alg. Geom. Topol. 1 (2001), 271–297. · Zbl 0978.58011 · doi:10.2140/agt.2001.1.271 |
| [10] | ——–, Addendum to Coarse homology theories, Alg. Geom. Topol. 3 (2003), 1089–1101. |
| [11] | ——–, The general notion of descent in coarse geometry, Alg. Geom. Topol. 10 (2010), 2149–2450. · Zbl 1452.55004 |
| [12] | J. Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Reg. Conf. Ser. Math. 90 (1996). · Zbl 0853.58003 |
| [13] | ——–, Lectures on coarse geometry, Univ. Lect. Ser. 31, American Mathematical Society, 2003. · Zbl 1042.53027 |
| [14] | M. Rørdam, F. Larsen and N. Laustsen, An introduction to \(K\)-theory for \(C^⁎\)-algebras, Lond. Math. Soc. 49 (2000). · Zbl 0967.19001 |
| [15] | E.H. Spanier, Algebraic topology, Springer, New York, 1994. · Zbl 0810.55001 |
| [16] | N.E. Wegge-Olsen, \(K\)-theory and \(C^⁎\)-algebras, Oxford University Press, Oxford, 1994. · Zbl 0835.06011 |
| [17] | G.L. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), 201–240. · Zbl 0956.19004 · doi:10.1007/s002229900032 |
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