Hahn-Banach type extension theorems on p-operator spaces

DOI10.7153/OAM-09-40zbMATH Open1343.46061arXiv1303.3513OpenAlexW2962924945MaRDI QIDQ3458066

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Publication date: 8 December 2015

Published in: (Search for Journal in Brave)

Abstract: Let $VsubseteqW$ be two operator spaces. Arveson-Wittstock-Hahn-Banach theorem asserts that every completely contractive map $varphi:VomathcalB(H)$ has a completely contractive extension $ildevarphi:WomathcalB(H)$, where $mathcalB(H)$ denotes the space of all bounded operators from a Hilbert space $H$ to itself. In this paper, we show that this is not in general true for $p$-operator spaces, that is, we show that there are $p$-operator spaces $VsubseteqW$, an $S{Q}_p$ space $E$, and a $p$-completely contractive map $varphi:VomathcalB(E)$ such that $varphi$ does not extend to a $p$-completely contractive map on $W$. Restricting $E$ to $L_p$ spaces, we also consider a condition on $W$ under which every completely contractive map $varphi:VomathcalB(L_p(mu))$ has a completely contractive extension $ildevarphi:WomathcalB(L_p(mu))$.

Full work available at URL: https://arxiv.org/abs/1303.3513

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