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 ofMarkov operators on L ₂(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.