Exactness and the Novikov conjecture

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 Cr∗(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 Cr∗(Γ) into B(l2(Γ)) is nuclear.