1 results
The Group Aut (μ) is Roelcke Precompact
-
- Journal:
- Canadian Mathematical Bulletin / Volume 55 / Issue 2 / 01 June 2012
- Published online by Cambridge University Press:
- 20 November 2018, pp. 297-302
- Print publication:
- 01 June 2012
-
- Article
-
- You have access
- Export citation
-
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
$G\,=\,\text{Aut(}\mu \text{)}$ of automorphisms of an atomless standard Borel probability space 
$(X,\,\mu )$ is precompact. We identify the corresponding compactification as the space of Markov operators on 
${{L}_{2}}(\mu )$ 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.
