Complexifications of real operator spaces

Abstract

We study the complexifications of real operator spaces. We show that for every real operator space $V$ there exists a unique complex operator space matrix norm $\{\|\cdot\|_n\}$ on its complexification $V_c = V \+{\rm i} V$ which extends the original matrix norm on $V$ and satisfies the condition $\|x +{\rm i}y\|_n = \|x -{\rm i}y\|_n$ for all $x + {\rm i} y \in M_n(V_c) = M_n(V) \+ {\rm i} M_n(V)$. As a consequence of this result, we characterize complex operator spaces which can be expressed as the complexification of some real operator space. Finally, we show that some properties of real operator spaces are closely related to the corresponding properties of their complexifications.