2000/05/15 by U. Haagerup, Uffe Haagerup, Haskell P. Rosenthal +6 · 1 citation
Mathematics · #46B20 #46L10 #46L52 #47L25 (Primary) #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.FA #msc:46B20 #msc:46L10 #msc:46L52 #msc:47L25
paper · pdf · doi:10.48550/arxiv.math/0005150
54 pp., LaTeX
arxiv created 2000/05/15 · openalex publication_date 2000/05/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let N and M be von Neumann algebras. It is proved that Lp(N) does not Banach embed in Lp(M) for N infinite, M finite, 1 < or = p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class Cp embeds in Lp(N) for N infinite). Theorem: Let 1 < or = p < 2 and let X be a Banach space with a spanning set (xij) so that for some C < or = 1: (i) any row or column is C-equivalent to the usual ell2-basis; (ii) (xik,jk) is C-equivalent to the usual ellp-basis, for any i1 < i2 < ... and j1 < j2 < ... . Then X is not isomorphic to a subspace of Lp(M), for M finite. Complements on the Banach space structure of non-commutative Lp-spaces are obtained, such as the p-Banach-Saks property and characterizations of subspaces of Lp(M) containing ellp isomorphically. The spaces Lp(N) are classified up to Banach isomorphism, for N infinite-dimensional, hyperfinite and semifinite, 1 < or = p< infty, p not= 2. It is proved that there are exactly thirteen isomorphism types; the corresponding embedding properties are determined for p < 2 via an eight level Hasse diagram. It is also proved for all 1 < or = p < infty that Lp(N) is completely isomorphic to Lp(M) if N and M are the algebras associated to free groups, or if N and M are injective factors of type IIIlambda and IIIlambda' for 0 < lambda, lambda' < or = 1.