Banach embedding properties of non-commutative L^p-spaces

Details Banach embedding properties of non-commutative L^p-spaces Banach embedding properties of non-commutative L^p-spaces

2000-05-15

arxiv.org/abs/math/0005150v1

Mathematics

Functional Analysis

54 pp., LaTeX

Scientific paper

Let N and M be von Neumann algebras. It is proved that L^p(N) does not Banach embed in L^p(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 C_p embeds in L^p(N) for N infinite). Theorem: Let 1 < or = p < 2 and let X be a Banach space with a spanning set (x_{ij}) so that for some C < or = 1: (i) any row or column is C-equivalent to the usual ell^2-basis; (ii) (x_{i_k,j_k}) is C-equivalent to the usual ell^p-basis, for any i_1 < i_2 < ... and j_1 < j_2 < ... . Then X is not isomorphic to a subspace of L^p(M), for M finite. Complements on the Banach space structure of non-commutative L^p-spaces are obtained, such as the p-Banach-Saks property and characterizations of subspaces of L^p(M) containing ell^p isomorphically. The spaces L^p(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 L^p(N) is completely isomorphic to L^p(M) if N and M are the algebras associated to free groups, or if N and M are injective factors of type III_lambda and III_{lambda'} for 0 < lambda, lambda' < or = 1.

Affiliated with

Also associated with

No associations

Rating

Banach embedding properties of non-commutative L^p-spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Banach embedding properties of non-commutative L^p-spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Banach embedding properties of non-commutative L^p-spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-103304