THE GROUP OF INVERTIBLE ELEMENTS OF A REAL BANACH ALGEBRA

The following result is proved: Let A be a commutative real Banach algebra with unit 1. Let G denote the group of invertible elements of A and let G(1) be the connected component of G containing 1. If the quotient group G/G(1) contains an element of finite order other than G(1), then the order of such an element must be 2. If the group G/G(1) is of finite order, then its order must be 2(n) for some nonnegative integer n.