Kadec-Pełczyński decomposition for Haagerup \(L^p\)-spaces (Q2777982)

The author generalizes the Kadec-Pełczyński subsequence decomposition property of classical \(L^p\) spaces to the case of Haagerup spaces. Namely, if \(M\) is an arbitrary von Neumann algebra, \(1\leq p<\infty\) and \((\phi_n)\) is a bounded sequence in \(L^p(M)\), then there exists a subsequence \((\phi_{n_k})\) of \((\phi_n)\) which decomposes into the sum of two bounded sequences \((y_k)\) and \((z_k)\) in \(L^p(M)\) such that \(\{(y_k), k\geq 1\}\) is uniformly integrable in \(L^p(M)\) and, for some sequence of projections \((e_k)\) with \(e_k\downarrow0\), \(z_k=e_kz_ke_k\) for each \(k\geq1\). One can also choose the sequence \((e_k)\) so that the projections are mutually orthogonal and \(e_ky_ke_k=0\) for each \(k\geq 1\). Uniform integrability of a bounded subset \(K\) of \(L^p(M)\) means here that \(\lim_{n\to\infty}\sup_{\phi\in K}\| e_n\phi e_n\| _p=0\) for every sequence of projections \((e_n)\) in \(M\) with \(e_n\downarrow0\).NEWLINENEWLINEThe theorem is especially attractive for \(p=1\), that is for preduals of von Neumann algebras, where uniform integrability of a bounded subset is equivalent to its relative weak compactness. Using this case, the author proves an interesting corollary of the decomposition property -- every non-reflexive subspace of the dual \(A^*\) of a \(C^*\)-algebra \(A\) contains asymptotically isometric copies of \(l_1\).