In their preceding publication [P. Antonini et al., J. Funct. Anal. 270, No. 1, 447–481 (2016; Zbl 1352.46063)], the authors constructed \(\Gamma\)-equivariant \(KK\)-theory \(KK^\Gamma_{\mathbb{R}}(-,-)\) with coefficients in \(\mathbb{R}\) for any discrete group \(\Gamma\). The name is motivated by the following two observations: First, all \(KK^\Gamma_{\mathbb{R}}(A,B)\) are vector spaces over \(\mathbb{R}\), because they are canonically modules over \(KK^\Gamma_{\mathbb{R}}(\mathbb{C},\mathbb{C})\) and there is a ring homomorphism \(\mathbb{R}=KK_{\mathbb{R}}(\mathbb{C},\mathbb{C})\to KK^\Gamma_{\mathbb{R}}(\mathbb{C},\mathbb{C})\) induced by the group homomorphism \(\Gamma\to 1\). And second, there are canonical natural homomorphisms \(KK^\Gamma(A,B)\to KK^\Gamma_{\mathbb{R}}(A,B)\).
One advantageous property of considering real coefficients, which is fundamental to the paper under review, is that tracial states on \(C^*\Gamma\) define elements of \(KK^\Gamma_{\mathbb{R}}(\mathbb{C},\mathbb{C})\). In particular, the element \([\tau]\in KK^\Gamma_{\mathbb{R}}(\mathbb{C},\mathbb{C})\) obtained from the canonical group trace \(\mathrm{tr}_\Gamma\colon C^*\Gamma\to\mathbb{C}\) is even an idempotent which is central in \(KK^\Gamma_{\mathbb{R}}\). Hence, the image \(KK^\Gamma_{\mathbb{R}}(A,B)_\tau\) of the action of \([\tau]\) on each \(KK^\Gamma_{\mathbb{R}}(A,B)\) is a direct summand, on which it acts as the identity, and it is called the \(\tau\)-part of \(KK^\Gamma_{\mathbb{R}}(A,B)\). Similarily, letting \([\tau]\) act via descent morphisms yields direct summands \(KK_{\mathbb{R}}(D,A\rtimes_r\Gamma)_\tau\subset KK_{\mathbb{R}}(D,A\rtimes_r\Gamma)\) and \(K_{*,\mathbb{R}}(C^*_r\Gamma)_\tau\subset K_{*,\mathbb{R}}(C^*_r\Gamma)\), also called their respective \(\tau\)-parts. It is noteworthy that the analogously defined \(\tau\)-parts for the full cross products are canonically isomorphic to those for the reduced ones. The reason for calling this procedure the localisation at the unit element is, naively speaking, that \(\tau\) takes the value of a function at the unit element.
In the paper under review, the authors build up on all of these notions and introduce the \(\tau\)-Baum-Connes map as a homomorphism \(\mu_\tau\colon K^{\mathrm{top}}_{*,\mathbb{R}}(\Gamma;A)_\tau\to K_{*,\mathbb{R}}(A\rtimes_r\Gamma)_\tau\). Corresponding to it, there is the \(\tau\)-form of the Baum-Connes conjecture with coefficients, which claims that this map is bijective. It is shown that the \(\tau\)-form of the Baum-Connes conjecture is weaker than the Baum-Connes conjecture with coefficients in the sense that if \(A\) is a \(\Gamma\)-\(C^*\)-algebra and the Baum-Connes map \(K^{\mathrm{top}}_*(\Gamma;A\otimes N)\to K_*(A\rtimes_r\Gamma\otimes N)\) for \(A\otimes N\) is injective or surjective for any choice of \(I\!I_1\)-factor \(N\), then the \(\tau\)-Baum-Connes map \(\mu_\tau\colon K^{\mathrm{top}}_{*,\mathbb{R}}(\Gamma;A)_\tau\to K_{*,\mathbb{R}}(A\rtimes_r\Gamma)_\tau\) for \(A\) is also injective or surjective, respectively.
It is furthermore shown that the \(\tau\)-Baum-Connes conjecture without coefficients still implies the strong Novikov conjecture: If \(\mu_\tau\colon K^{\mathrm{top}}_{*,\mathbb{R}}(\Gamma)_\tau\to K_{*,\mathbb{R}}(C^*_r\Gamma)_\tau\) is injective, then the Mishchenko-Kasparov assembly map \(K_*(B\Gamma)\to K_*(C^*_r\Gamma)\) is rationally injective. In fact, the \(\tau\)-Baum-Connes conjecture is even closer to the Novikov conjecture, because it is only concerned with the part of \(K^{\mathrm{top}}_*(\Gamma)\) corresponding to free and proper actions. The comparison is based on canonical isomorphisms \(K_*(B\Gamma)\otimes\mathbb{R}\simeq K_{*,\mathbb{R}}(B\Gamma)\simeq K^\Gamma_{*,\mathbb{R}}(E\Gamma)=K^\Gamma_{*,\mathbb{R}}(E\Gamma)_\tau\simeq K^{\mathrm{top}}_{*,\mathbb{R}}(\Gamma)_\tau\), where the first isomorphism holds because adding real coefficients to the \(K\)-homology of \(B\Gamma\) only discards torsion, the equality between the third and fourth group holds because the \(\tau\)-part is everything for free and proper actions and the last isomorphism is induced by \(E\Gamma\to\underline{E}\Gamma\).
Finally, the authors also show that their approach of localising at the unit element unfortunately does not solve the conceptual problem with the Baum-Connes conjecture with coefficients: Exploiting the failure of exactness once again, the counterexample presented in [N. Higson et al., Geom. Funct. Anal. 12, No. 2, 330–354 (2002; Zbl 1014.46043)] also shows that the \(\tau\)-Baum-Connes map for a Gromov monster group cannot be an isomorphism for every coefficient \(\Gamma\)-\(C^*\)-algebra \(A\).