Hahn-Banach type extension theorems on p-operator spaces

Let $V\subseteq W$ be two operator spaces. Arveson-Wittstock-Hahn-Banach theorem asserts that every completely contractive map $\varphi:V\to $\mathcal{B}$(H)$hasacompletelycontractiveextension$\tilde{\varphi}:W\to $\mathcal{B}$(H)$,where$$\mathcal{B}$(H)$ 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 $V\subseteq W$, an $S{Q}_p$ space $E$, and a $p$-completely contractive map $\phi :V\to \mathcal{B}(E)$ such that $\phi$ 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 $\phi :V\to \mathcal{B}(L_p(\mu ))$ has a completely contractive extension $\tilde{\phi }:W\to \mathcal{B}(L_p(\mu ))$.