Hahn-Banach type extension theorems on p-operator spaces

TL;DR

This paper investigates the failure of the Hahn-Banach extension property for p-operator spaces, showing that such extensions are not always possible, unlike in classical operator space theory, with specific conditions identified for Lp spaces.

Contribution

The paper demonstrates that the classical Hahn-Banach extension theorem does not generally hold for p-operator spaces and identifies conditions under which extensions are possible for Lp spaces.

Findings

Extension fails for certain p-operator spaces and maps.

Conditions for extension exist when restricting to Lp spaces.

Counterexamples highlight differences from classical operator space theory.

Abstract

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