Boundaries of reduced C*-algebras of discrete groups

By Mehrdad Kalantar, M. Kennedy. 2014

164 citations recorded by the source corpus.

For a discrete group G, we consider the minimal C*-subalgebra of $\ell^\infty(G)$ 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 $C(\partial_F G)$ of continuous functions on Furstenberg's universal G-boundary $\partial_F G$. 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 $\partial_F G$ 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*-algebra $\mathrm{C}_r^*(G)$ of G into a nuclear C*-algebra which is contained in the injective envelope of $\mathrm{C}_r^*(G)$. It is a longstanding open problem to determine which groups are C*-simple, in the sense that the algebra $\mathrm{C}_r^*(G)$ 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/91bef5e2bd583638bef49b09662429a2929fc5d6

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In Algebra and Number Theory

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Citations in the corpus

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems (2018, background): “xtension Zof πadmits a Γ-conditional expectation C0(Y) → C0(Z). It is interesting to compare Theorem 2.3with the ideal structure results of the reduced crossed product C∗-algebras [33], [3] (see also [30], [7]). Unlike these results, our argument requires the freeness rather than just the topologica”
A New Look at C∗-Simplicity and the Unique Trace Property of a Group (2015, background): “G) is a simple C∗-algebra, and it is said to have the unique trace property if the canonical trace on C∗ λ(G), here denoted by τ0, is the only tracial state on C∗ λ(G). Kalantar and Kennedy proved in [8] that a group Gis C∗-simple if and only if it has a topologically free boundary action on some co”