Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures

We define a simple criterion for a homogeneous, complete metric structure $X$ that implies that the automorphism group $Aut(X)$ satisfies all the main consequences of the existence of ample generics: it has the small index property, the automatic continuity property, and uncountable cofinality for non-open subgroups. Then we verify it for the Urysohn space $U$, the Lebesgue probability measure algebra $MALG$, and the Hilbert space ${\mathrm{\ell }}_2$, thus proving that $Iso(U)$, $Aut(MALG)$, $U({\mathrm{\ell }}_2)$, and $O({\mathrm{\ell }}_2)$ share these properties. We also formulate a condition for $X$ which implies that every homomorphism of $Aut(X)$ into a separable group $K$ with a left-invariant, complete metric, is trivial, and we verify it for $U$, and ${\mathrm{\ell }}_2$.