A geometric characterisation of real C*-algebras

We characterise the positive cone of a real C*-algebra geometrically. Given an open cone $\mathrm{\Omega }$ in a real Banach space $V$, with closure $\bar{\mathrm{\Omega }}$, we show that $\mathrm{\Omega }$ is the interior of the positive cone of a unital real C*-algebra if and only if it is a Finsler symmetric cone with an orientable extension, which is equivalent to the condition that $V$ is, in an equivalent norm, the hermitian part of a unital real C*-algebra with positive cone $\bar{\mathrm{\Omega }}$.