On quaternionic functional analysis

Details On quaternionic functional analysis On quaternionic functional analysis

2006-09-06

arxiv.org/abs/math/0609160v1

Mathematics

Operator Algebras

to appear in the Mathematical Proceedings of the Cambridge Philosophical Society

Scientific paper

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.

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