A note on the column-row property

In this article, we study the following question asked by Michael Hartz in a recent paper \cite{Hartz}: $\mathit{\text{which}}\mathit{\text{operator}}\mathit{\text{spaces}}\mathit{\text{satisfy}}\mathit{\text{the}}\mathit{\text{column-row}}\mathit{\text{property?}}$ We provide a complete classification of the column-row property for non-commutative $L_p$-spaces over semifinite von Neumann algebras. We study other relevant properties of operator spaces that are related to the column-row property and discuss their existence and non-existence for various natural examples of operator spaces.