Amenable Actions and Exactness for Discrete Groups
نویسنده
چکیده
It is proved that a discrete group G is exact if and only if its left translation action on the Stone-Čech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete groups. In [KW], Kirchberg and Wassermann discussed exactness for groups. A discrete group G is said to be exact if its reduced group C-algebra C λ(G) is exact. Throughout this paper, G always means a discrete group and we identify G with the corresponding convolution operators on l2(G). Amenability of a group action was discussed by Anantharaman-Delaroche and Renault in [ADR]. The left translation action of a group G on its Stone-Čech compactification βG was considered by Higson and Roe in [HR]. This action is amenable if and only if the uniform Roe algebra UC(G) := C∗(l∞(G), G) = span{sl∞(G) : s ∈ G} ⊂ B(l2(G)) is nuclear. Since a C-subalgebra of an exact C-algebra is exact, C λ(G) is exact if UC (G) is nuclear. In this article, we will prove the converse. A function u : G×G → C is called a positive definite kernel if the matrix [u(si, sj)] ∈ Mn is positive for any n and s1, . . . , sn ∈ G. If u is a positive definite kernel on G×G such that u(s, s) ≤ 1 for all s ∈ G, then the Schur multiplier θu on B(l2(G)) defined by θu(x) = [u(s, t)xs,t]s,t∈G for x = [xs,t] ∈ B(l2(G)) is a well-defined completely positive contraction. Lemma 1 (Section 5 in [Pa]). Let B be a C-algebra and n ∈ N. Then, the map CP(B,Mn) ∋ φ 7→ fφ ∈ (Mn(B)) ∗ + defined by fφ(X) = ∑
منابع مشابه
Amenable actions and applications
We will give a brief account of (topological) amenable actions and exactness for countable discrete groups. The class of exact groups contains most of the familiar groups and yet is manageable enough to provide interesting applications in geometric topology, von Neumann algebras and ergodic theory. Mathematics Subject Classification (2000). Primary 46L35; Secondary 20F65, 37A20, 43A07.
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