Some Extremely Amenable Groups Related to Operator Algebras and Ergodic Theory
نویسنده
چکیده
A topological group G is called extremely amenable if every continuous action of G on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a C∗-algebra A is nuclear if and only if the unitary group U(A) with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of L(0, 1), 1 ≤ p < ∞, extending a classical result of Gromov and Milman (p = 2). We show that a measure class preserving equivalence relation R on a standard Borel space is amenable if and only if the full group [R], equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a nontrivial space in the sense of Gromov occuring in the automorphism groups of injective factors of type III.
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