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
of automorphisms of an atomless standard Borel probability space
is precompact. We identify the corresponding compactification as the space of Markov operators on
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
, i.e., functions on
arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that
is totally minimal.