Exactness and the Novikov conjecture

By E. Guentner, J. Kaminker. 2000

91 citations recorded by the source corpus.

Abstract In this note we will study a connection between the conjecture that C ∗ r (Γ) is exact and the Novikov conjecture for Γ. The main result states that if the inclusion of the reduced C ∗ -algebra C r ∗ (Γ) of a discrete group Γ into the uniform Roe algebra of Γ, UC ∗ (Γ) , is a nuclear map then Γ is uniformly embeddable in a Hilbert space. By a result of Yu (Invent. Math. 139 (2000) 201) this implies that Γ satisfies the Novikov conjecture. Note that the hypothesis is a slight strengthening of the usual notion of exactness since a group Γ is exact if and only if the inclusion of C r ∗ (Γ) into B (l 2 (Γ)) is nuclear.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/07fedbd2af9c93240cbbfd585876b1181631cdd2

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In Algebra and Number Theory

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Citations in the corpus

Counterexamples to the Baum—Connes conjecture (2002, result): “The same result has been obtained, in a different way, by Anantharaman-Delaroche [A], Guentner and Kaminker [GuK] and Ozawa [O].”
Uniform embeddability of relatively hyperbolic groups (2005, background): “In particular, Ozawa gave a characterization of C -exact groups, from which it directly follows that a C -exact group is uniformly embeddable [13] (see [9] for the link to uniform embeddability).”
Group C*-Algebras and K-Theory (2004, background): “The following result (see [28, 29, 15]) shows that Gromov groups fail to be exact.”
Some notes on Property A (2006, methodology): “f C∗ u(G) (we abuse notation by using Gto refer both to the group and the underlying coarse space), and exactness of C∗ r(G) are equivalent. This is a theorem of E. Guentner, J. Kaminkerand N. Ozawa, [GK02a], [GK02b] and[Oz00] (the resultsofHigsonandRoe, [HR00a], andAnantharaman-Delaroche and Renaul”
On rigidity of Roe algebras (2011, background): “Moreover, C∗-exactness is equivalent to property A by a result of Guentner–Kaminker [14] and Ozawa [20].”
Constructions preserving Hilbert space uniform embeddability of discrete groups (2003, background): “6 is motivated by the following result, inspired by [13].”
Property A, partial translation structures, and uniform embeddings in groups (2006, result): “E. Guentner and J. Kaminker, ‘Exactness and the Novikov conjecture’, Topology 41 (2002) 411–418.”
Metrics on diagram groups and uniform embeddings in a Hilbert space (2004, background): “For more applications see [17] and references therein.”
Amenable actions and exactness for discrete groups (2000, methodology): “As it was suggested in [GK], these examples of Gromov are indeed not exact since the condition (ii) in Theorem 3 assures uniform embeddings.”