This interesting paper studies the problems of isomorphic embeddings between non-commutative \(L_p\)-spaces. One striking result asserts that for all \(0<q<p<2\), all \(\varepsilon> 0\) and any natural number \(n\), there exists an isomorphic embedding \(u\) of the finite-dimensional Schatten \(p\)-class \(S^n_p\) (= non-commutative \(L_p\)-space associated with von Neumann algebra of all complex \(n\times n\) matrices) into \(S^m_q\), such that \[ (1-\varepsilon)2^{-(2/q+1/p)}\|x\|_p\leq \|u(x)\|_q\leq (1+\varepsilon) 2^{(1/p+ 1/q)}\|x\|_p \] for all \(x\in S^n_p\). Here \(m\) is a natural number less or equal \(C(q,p,\varepsilon) n^{n+ 1}\) where the constant \(C(q,p,\varepsilon)\) does not depend on \(n\).
Moreover, it is further established that a non-commutative \(L_p\)-space, \(1< p<2\), associated with the finite, hyperfinite factor \(R\) is isomorphic to a subspace of the predual \(R_*\).