The radical of a Banach algebra with a closed cone (Q912360)

In this very short paper the author gives a very elementary proof of the following result: Theorem. Let A be a real Banach algebra with a closed cone C containing all the squares. Then the radical of A consists of all elements whose cubes are zero. In particular A is semi-simple if and only if the square of every non-zero element is different from zero. If A has a two-sided approximate identity then A is semi-simple. This implies in particular a result of \textit{E. Scheffold} [FF- Banachverbandsalgebren, Math. Z. 177, 193-205 (1981; Zbl 0439.46037)].