  For a C*-algebra A, G. Pedersen defined the concept of universal
measurability for self-adjoint elements of A**, the universal enveloping
algebra of A. Although he was unable to show that U, the set of universally
measurable elements, is a Jordan algebra, he showed that U contains a large
Jordan algebra, U_0. We show by means of a 2 x 2 matrix trick that U_0 is in
fact the real part of a C*-algebra. If it is ever shown that U is always a
C*-algebra, then the same trick will show that U is the real part of a
C*-algebra.
