A geometric characterisation of real C*-algebras

We characterise the positive cone of a real C*-algebra geometrically. Given an open cone O in a real Banach space V, with the closure (O) over bar, we show that O 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 the positive cone (O) over bar.