REPRESENTATION OF A REAL B-ASTERISK-ALGEBRA ON A QUATERNIONIC HILBERT-SPACE

Let A be a real B*-algebra containing a *-subalgebra that is *-isomorphic to the real quaternion algebra H. Suppose the spectrum of every self-adjoint element in A is contained in the real line. Then it is proved that there exists a quaternionic Hilbert space X and an isometric *-isomorphism pi of A onto a closed *-subalgebra of BL(X) , the algebra of all bounded linear operators on X. If, in addition to the above hypotheses, every element in A is normal, then A is also proved to be isometrically *-isomorphic to C(Y, H), the algebra of all continuous H-valued functions on a compact Hausdorff space Y.