Amenable actions and exactness for discrete groups

By Nathanial P. Brown Narutaka Ozawa. 2000

217 citations recorded by the source corpus.

Abstract It is proved that a discrete group G is exact if and only if its left translation action on the Stone–Cech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete groups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/15442b6548e043486a563bebe08ae303bf5c124e

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Citations in the corpus

Exotic crossed products and the Baum–Connes conjecture (2014, unknown): “For discrete groups, the converse is true by [33], but this is an open question in general; nonetheless, many exact groups, for example all almost connected groups [1, Proposition 3.”
The Novikov Conjecture for Linear Groups (2005, background): “(For countable groups this assertion is equivalent to the definition above [27].”
Boundaries of reduced C*-algebras of discrete groups (2014, background): “Since G is exact, the G-action on βG is amenable by a result of Ozawa [36].”
Noncommutative Riesz transforms -- a probabilistic approach (2008, unknown): “rtion follows from a well-known observation: A form bΓ( λ(g),λ(h)) = Kˆ(g,h)λ(g−1h) is completely positive if and only if Kˆ is a positive matrix. Indeed, let us assume that bΓ is positive. Following [Oza00] we deduce that Kˆ(g,h) = τ(λ(g)Γ(b λ(g),λ(h))λ(h−1)) is positive. More precisely, we have X ”
Property A and the operator norm localization property for discrete metric spaces (2012, methodology): “We thank Professor Ozawa for letting us know the paper [7].”
Coarse Non-Amenability and Coarse Embeddings (2011, background): “For a countable discrete group Γ, it is equivalent to the existence of a topological amenable action of Γ on a compact Hausdorff space and to the C∗-exactness of the reduced C∗-algebra C∗ r (Γ) [12, 8].”
Some notes on Property A (2006, background): “ [Now07b] constructs (locally ﬁnite) coarsely embeddable spaces that do not have property A. [Oll04] ﬁlls in some of the details about random groups, following the ideas given in [Gro00] and [Gro01]. [Oz00] proves that a groups is exact if and only if it has property A, via amenable actions. [Oz06] ”
Asymptotic Dimension (2007, background): “For groups, this is equivalent to the exactness of the reduced C∗algebra by a result of Ozawa [79], so it has come to be known as exactness of the group.”
Uniform embeddings of bounded geometry spaces into reflexive Banach space (2003, methodology): “This led to the verification of the Novikov conjecture for large classes of discrete groups, including, for example, linear groups [6] and, more generally, exact groups [12, 7].”
ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS (2011, result): “The result and its proof are inspired by Ozawa’s general semi-solidity theorem (Theorem 15.1.5 in [5]) and Ozawa and Popa’s Theorem B in [46], viewed through the framework developed by the first two authors in [9].”
On rigidity of Roe algebras (2011, background): “Moreover, C∗-exactness is equivalent to property A by a result of Guentner–Kaminker [14] and Ozawa [20].”
Constructions preserving Hilbert space uniform embeddability of discrete groups (2003, background): “The first indications of the relationship between uniform embeddability and exactness are found in the work of Guentner and Kaminker [13]; these preliminary steps were quickly expanded by Ozawa [16], Anantharaman-Delaroche [2], and others.”