Boundaries of reduced C*C^{*}-algebras of discrete groups

Abstract For a discrete group G, we consider the minimal C * C^{*} -subalgebra of ℓ ∞ \ell^{\infty} (G) that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. We prove that, more generally, it can be identified with the algebra C( ∂ F ⁡ G \partial_{F}G ) of continuous functions on Furstenberg’s universal G-boundary ∂ F ⁡ G {\partial_{F}G} . This operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove that G is exact precisely when the G-action on ∂ F ⁡ G {\partial_{F}G} is amenable, and use this fact to prove Ozawa’s conjecture that if G is exact, then there is an embedding of the reduced C * C^{*} -algebra C r * ⁢ ( G ) {\mathrm{C}^{*}_{r}(G)} of G into a nuclear C * C^{*} -algebra which is contained in the injective envelope of C r * ⁢ ( G ) {\mathrm{C}^{*}_{r}(G)} . The algebra C( ∂ F ⁡ G \partial_{F}G ) arises as an injective envelope in the sense of Hamana, which implies rigidity results for certain G-equivariant maps. We prove a generalization of a rigidity result of Ozawa for G-equivariant maps between spaces of functions on the hyperbolic boundary of a hyperbolic group. Our result applies to hyperbolic groups, but also to groups that are not hyperbolic or even relatively hyperbolic, including certain mapping class groups. It is a longstanding open problem to determine which groups are C * C^{*} -simple, in the sense that the algebra C r * ⁢

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