Let \(\Gamma\) be a finitely generated group. In this paper, it is shown \(\Gamma\) satisfies the Novikov conjecture if the inclusion of the reduced \(C^*\)-algebra \(C^*_r(\Gamma)\) into the uniform Roe algebra \(U C^*(\Gamma)\) is a nuclear map (Theorem 1.2). Since \(\Gamma\) satisfies the Novikov conjecture if \(\Gamma\) is uniformly embeddable in a Hilbert space [G. Yu, Invent. Math. 139, No. 1, 201-240 (2000; Zbl 0956.19004)], to show Theorem 1.2, it is sufficient to show that \(\Gamma\) is uniformly embeddable in a Hilbert space under the assumption of the Theorem. For this purpose, uniformly embeddable conditions for a discrete metric space are given in Section 2. Applying these conditions, Theorem 1.2 is proved in Section 3. The authors say that a discrete group \(\Gamma\) is exact, i.e., its reduced \(C^*\)-algebra \(C^*_r (\Gamma)\) is an exact \(C^*\)-algebra, if and only if the inclusion of \(C^*_r (\Gamma)\) into \({\mathcal B}(l^2 (\Gamma))\) given by the left regular representation is a nuclear embedding [E. Kirchberg J. Funct. Anal. 129, No. 1, 35-63 (1995; Zbl 0912.46059)] and the uniform Roe algebra of \(\Gamma\) is a subalgebra of \({\mathcal B}(l^2(\Gamma))\) [cf. N. Higson and J. Roe, J. Reine angew. Math. 519, 143-153 (2000; Zbl 0964.55015)]. Theorem 1.2 shows intimate relation between exactness and the Novikov conjecture.
In the last Section, related uniform embeddability conditions of \(\Gamma\) are given (Theorem 4.1).