THE GROUP Aut(μ) IS ROELCKE PRECOMPACT

Following a similar result of Uspenskij on the unitary group of a separable Hilbert space we show that with respect to the lower (or Roelcke) uniform structure the Polish group G = Aut(μ), of automorphisms of an atomless standard Borel probability space (X,μ), is precompact. We identify the corresponding compactification as the space of Markov operators on L2(μ) and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on G (i.e. functions on G arising from unitary representations) all coincide. Again following Uspenskij we also conclude that G is totally minimal. Let (X,μ) be an atomless standard Borel probability space. 1 We denote by Aut(μ) the Polish group of measure preserving automorphisms of (X,μ) equipped with the weak topology. If for T ∈ G we let UT : L2(μ)→ L2(μ) be the corresponding unitary operator (defined by UTf(x) = f(T −1x)), then the map T 7→ UT (the Koopman map) is a topological isomorphic embedding of the topological group G into the Polish topological group U(H) of unitary operators on the Hilbert space H = L2(μ) equipped with the strong operator topology. The image of G in U(H) under the Koopman map is characterized as the collection of unitary operators U ∈ U(H) for which U(1) = 1 and Uf ≥ 0 whenever f ≥ 0; see e.g. [5, Theorem A.11]. It is well known (and not hard to see) that the strong and weak operator topologies coincide on U(H) and that with respect to the weak operator topology, the group U(H) is dense in the unit ball Θ of the space B(H) of bounded linear operators on H. Now Θ is a compact space and as such it admits a unique uniform structure. The trace of the latter on U(H) defines a uniform structure on U(H). We denote by J the collection of Markov operators in Θ, where K ∈ Θ is Markov if K(1) = K∗(1) = 1 and Kf ≥ 0 whenever f ≥ 0. It is easy to see that J is a closed subset of Θ. Clearly the image of G in U(H) is contained in J and it is well known that this image is actually dense in J (see e.g. [6] or [7]). Thus, via the embedding of G into J we obtain also a uniform structure on G. We will denote this uniform space by (G, J). On every topological group G there are two naturally defined uniform structures L(G) and R(G). The lower or the Roelcke uniform structure on G is defined as U = L ∧ R, the greatest lower bound of the left and right uniform structures on G. If N is a base for the topology of G at the neutral element e, then with UL = {(x, y) : x−1y ∈ U} UR = {(x, y) : xy−1 ∈ U}, Date: February 24, 2009. 2000 Mathematics Subject Classification. Primary 54H11, 22A05. Secondary 37B05, 54H20.