Roundness and Metric Type

✍ Scribed by C. Lennard; A. Tonge; A. Weston

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 88 KB
• Volume: 252
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2000.6917

✦ Synopsis

We prove that if X is a Banach space containing l n uniformly in n, and if Y is a p metric space with metric type q ) p, then the inverse of any uniform homeomorphism T from X onto Y cannot satisfy a Lipschitz condition for large distances of order ␣qrp. It follows that if Y is a midpoint-convex subset of a Banach space Z with type q larger than the type supremum of a Banach space X, then X and Y cannot be uniformly homeomorphic. In particular, we prove the non-existence of uniform homeomorphisms between certain non-commutative L -spaces and midp point-convex subsets of another such space. We also prove that if a Banach space X has cotype infimum q larger than two, then it has maximal generalized roundness zero and maximal roundness at most q X . As a consequence, infinite-dimensional C U -algebras are seen to have maximal generalized roundness zero and maximal roundness one.