On quaternionic functional analysis

Abstract

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 $B^{\ast }$-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real $C^{\ast }$-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 $B^{\ast }$-algebras). On our way to these results, we compare, clarify and unify the term "quaternion Hilbert spaces" in the literatures.

to appear in the Mathematical Proceedings of the Cambridge Philosophical Society

Cited by in corpus (5)

• Continuous slice functional calculus in quaternionic Hilbert spaces
• Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley-Dickson algebras
• Quaternionic quantum theory admits universal dynamics only for two-level systems
• Non-associative Categories of Octonionic Bimodules
• Octonionic isometric isomorphisms and partial isometry