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April 2012

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Abstract of a paper by Casey Kelleher, Daniel Miller, Trenton Osborn and Anthony Weston
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "Polygonal equalities and virtual degeneracy in $L$-spaces" by Casey Kelleher, Daniel Miller, Trenton Osborn and Anthony Weston. Abstract: Cases of equality in the classical $p$-negative type inequalities for $L_{p}(\mu)$-spaces are characterized for each $p \in (0,2)$ according to a new property called virtual degeneracy. For each $p \in (0,2)$, this leads to a complete classification of the subsets of $L_{p}$-spaces that have strict $p$-negative type. It follows that if $0 < p < q \leq 2$ and if $(\Omega_{1}, \mu_{1})$ and $(\Omega_{2}, \mu_{2})$ are measure spaces, then no subset of $L_{q}(\Omega_{2}, \mu_{2})$ is isometric to any linear subspace $W$ of $L_{p}(\Omega_{1}, \mu_{1})$ that contains a pair of disjointly supported unit vectors. Under these circumstances it is also the case that no subset of $L_{q}(\Omega_{2}, \mu_{2})$ is isometric to any subset of $L_{p}(\Omega_{1}, \mu_{1})$ that has non-empty interior. We conclude the paper by examining virtually degenerate subspaces of $L_{p}(\mu)$-spaces. Archive classification: math.FA Mathematics Subject Classification: 46B04 Remarks: 9 pages Submitted from: westona(a)canisius.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.5837 or http://arXiv.org/abs/1203.5837

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