Representation of a real $B^ *$-algebra on a quaternionic Hilbert space

Abstract:

Let A be a real ${B^ \ast }$-algebra containing a $\ast$-subalgebra that is $\ast$-isomorphic to the real quaternion algebra $\mathbb {H}$. 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 $\ast$-isomorphism $\pi$ of A onto a closed $\ast$-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 $\ast$-isomorphic to $C(Y,\mathbb {H})$, the algebra of all continuous $\mathbb {H}$-valued functions on a compact Hausdorff space Y.