For a discrete group G, we consider the minimal C∗-subalgebra of ℓ∞(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) of continuous functions on Furstenberg's universal G-boundary ∂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 ∂FG 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∗-algebra C∗ r(G) of G into a nuclear C∗-algebra which is contained in the injective envelope of C∗ r(G). The algebra C(∂FG) 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∗-simple, in the sense that the algebra C∗ r (G) is simple. We prove that this problem can be reformulated as a problem about the structure of the G-action on the Furstenberg boundary. Specifically, we prove that a discrete group G is C∗-simple if and only if the G-action on the Furstenberg boundary is topologically free. We apply this result to prove that Tarski monster groups are C∗-simple. This provides another solution to a problem of de la Harpe (recently answered by Olshanskii and Osin) about the existence of C∗-simple groups with no free subgroups.

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Journal Journal fur die Reine und Angewandte Mathematik
Kalantar, M. (Mehrdad), & Kennedy, M. (Matthew). (2014). Boundaries of reduced c∗-algebras of discrete groups. Journal fur die Reine und Angewandte Mathematik, 2014, 1–21. doi:10.1515/crelle-2014-0111