Representations of a class of real $B^ *$-algebras as algebras of quaternion-valued functions

Abstract:

For a compact Hausdorff space $X$, let $C(X,{\mathbf {H}})$ denote the set of all quaternion-valued functions on $X$. It is proved that if a real ${B^*}$-algebra $A$ satisfies the following conditions: (i) the spectrum of every selfadjoint element is contained in the real line and (ii) every element in $A$ is normal, then $A$ is isometrically $*$-isomorphic to a closed $*$-subalgebra of $C(X,{\mathbf {H}})$ for some compact Hausdorff $X$. In particular, a real ${C^*}$-algebra in which every element is normal is isometrically $*$-isomorphic to a closed $*$-subalgebra of $C(X,{\mathbf {H}})$.