  In this article, we will show that the category of quaternion vector spaces,
the category of (both one-sided and two sided) quaternion Hilbert spaces and
the category of quaternion $B^*$-algebras are equivalent to the category of
real vector spaces, the category of real Hilbert spaces and the category of
real $C^*$-algebras respectively. We will also give a Riesz representation
theorem for quaternion Hilbert spaces and will extend two results of Kulkarni
(namely, we will give the full versions of the Gelfand-Naimark theorem and the
Gelfand theorem for quaternion $B^*$-algebras). On our way to these results, we
compare, clarify and unify the term "quaternion Hilbert spaces" in the
literatures.
