Exactness and the Novikov conjecture. (English) Zbl 0992.58002
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).
In the last Section, related uniform embeddability conditions of \(\Gamma\) are given (Theorem 4.1).
Reviewer: Akira Asada (Takarazuka)