Rosenthal's space revisited

Let $E$ be a rearrangement invariant (r.i.) function space on $[0,1]$, and let $Z_E$ consist of all measurable functions $f$ on $(0,\mathrm{\infty })$ such that $f^{\ast }{\chi }_{[0,1]}\in E$ and $f^{\ast }{\chi }_{[1,\mathrm{\infty })}\in L^2$. We reveal close connections between properties of the generalized Rosenthal’s space, corresponding to the space $Z_E$, and the behaviour of independent symmetrically distributed random variables in $E$. The results obtained are applied to consider the problem of the existence of isomorphisms between r.i.
spaces on $[0,1]$ and $(0,\mathrm{\infty })$. Exploiting particular properties of disjoint sequences, we identify a rather wide new class of r.i.\ spaces on $[0,1]$ ``close’’ to $L^{\mathrm{\infty }}$, which fail to be isomorphic to r.i.\ spaces on $(0,\mathrm{\infty })$. In particular, this property is shared by the Lorentz spaces ${\mathrm{\Lambda }}_2({\log }^{-\alpha }(e\mathrm{/}u))$, with $0<\alpha \le 1$.