On isomorphic embeddings in the class of disjointly homogeneous rearrangement invariant spaces (Q6610175)

A rearrangement invariant space (r.i. space in short) \(X\) on \([0,1]\) is said to be \textit{uniformly disjointly homogeneous} if every two disjoint normalized sequences in \(X\) have \(C\)-equivalent subsequences for some absolute constant \(C \in [1, +\infty)\). The main result asserts that, if the Haar system in a separable r.i. space \(X\) on \([0,1]\) is equivalent to a disjoint sequence in a uniformly disjointly homogeneous r.i. space \(Y\) on \([0,1]\) then \(X = L_2\) and the norms on \(X\) and \(L_2\) are equivalent. The above result was previously known under the more restrictive assumption on \(Y\) not to contain uniformly isomorphic copies of \(\ell_\infty^n\) for all \(n \in \mathbb N\). As corollaries, the author obtained the following two properties of a uniformly disjointly homogeneous r.i. space \(Y\) on \([0,1]\). (1) If an r.i. space \(X\) on \([0,1]\) is isomorphic to a complemented subspace of \(Y\) then either \(X = L_2\) or \(X = Y\) (with the equivalence of norms). (2) \(Y\) has a unique r.i. representation on \([0,1]\). There is also a consequence to r.i. representations of disjointly homogeneous r.i. spaces on \((0,+\infty)\).