The -tau-Baum--Connes conjecture with coefficients

Let $\Gamma$ be a discrete group, let $A$ be a $\Gamma$-algebra, and let $\tau$ be the idempotent used to define the corresponding $\tau$-parts of real topological and analytic $K$-theory. The $\tau$-Baum--Connes map is

$$\mu_\tau: K^{\operatorname{top}}_{*,{\mathbin{{\mathbb {R}}}}}(\Gamma;A)_\tau\longrightarrow K_{*,{\mathbin{{\mathbb {R}}}}}(A\rtimes_r\Gamma)_\tau.$$

The $\tau$-Baum--Connes conjecture with coefficients. The map $\mu_\tau$ is bijective. This is the $\tau$-localised form of the Baum--Connes conjecture with coefficients, asserting that the assembly map becomes an isomorphism after restricting to the part determined by $\tau$.