Tight embeddability of proper and stable metric spaces

We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for $p\in [1,\infty]$, every proper subset of $L_p$ is almost Lipschitzly embeddable into a Banach space $X$ if and only if $X$ contains uniformly the $\ell_p^n$'s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.Comment: 19 page

Embeddings of locally finite metric spaces into Banach spaces

We show that if X is a Banach space without cotype, then every locally finite metric space embeds metrically into X.Comment: 6 pages, to appear in Proceedings of the AM

Approximating quantum channels by completely positive maps with small Kraus rank

We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space $\mathrm{A}$ to states on some output Hilbert space $\mathrm{B}$ can be compressed into one with order $d\log d$ Kraus operators, where $d=\max(|\mathrm{A}|,|\mathrm{B}|)$, hence much less than $|\mathrm{A}||\mathrm{B}|$. In the case where the channel's outputs are all very mixed, this can be improved to order $d$. We discuss the optimality of this result as well as some consequences.Comment: 13 page

Regulation of Arabidopsis 14-3-3 gene expression by GABA.

The function in plants of the non-protein amino acid, gamma-aminobutyric acid (GABA) is poorly understood. In this study, we show that GABA down-regulates the expression of a large sub-set of 14-3-3 gene family members in Arabidopsis thaliana seedlings in a calcium, ethylene and abscisic acid-dependent manner. Gene expression is not affected when seedlings are supplied with glutamate, a precursor of GABA. The repression of 14-3-3 gene expression by GABA is dependent on functional ethylene and abscisic acid signalling pathways, since the response is lost in the etr1-1, abi1-1 and abi2-1 mutants. Calcium measurements show that in contrast to glutamate, GABA does not elicit a cytoplasmic calcium elevation, suggesting that the GABA response is unlikely to be mediated by glutamate receptors, as has been suggested previously. We suggest that in addition to its role as a stress-related metabolite, GABA may regulate gene expression in Arabidopsis, including members of the 14-3-3 gene family

Approximation properties and Schauder decompositions in Lipschitz-free spaces

We prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. We also show that the Lipschitz-free spaces over $\ell_1^N$ or $\ell_1$ have monotone finite-dimensional Schauder decompositions